# Homo Deus Lab — Interactive Scientific Simulations

> Open-source library of 1,082 interactive scientific simulations across 217 topics.
> Physics, chemistry, biology, engineering, mathematics, computer science, earth sciences, and more.
> Each simulation runs a live Canvas 2D engine with adjustable parameters, computed outputs, and educational content.
> Website: https://homo-deus.com
> GitHub: github.com/homo-techno/simulations

## applied-science (2 topics, 10 simulations)

### Agriculture & Crop Science (5 simulations)
- [Crop Rotation & Soil Nutrient Cycles](/lab/agriculture/crop-rotation): Simulate multi-year crop rotation strategies to see how nitrogen fixation, nutrient depletion, and fallow periods affect long-term soil health and yield Formula: N_soil(t+1) = N_soil(t) - crop_uptake + fixation + mineralization - leaching. 4 parameters.
- [Crop Yield Prediction & Weather Simulator](/lab/agriculture/yield-prediction): Predict crop yields based on temperature, rainfall, and growing degree days — see how weather variability impacts harvest outcomes Formula: GDD = Σ max(T_avg(day) - T_base, 0) for each day in growing season. 4 parameters.
- [Integrated Pest Management Model](/lab/agriculture/pest-control): Simulate pest population dynamics under integrated pest management — balancing biological control, pesticide application, and economic thresholds Formula: Pest(t+1) = Pest(t) × growth_rate - Predators(t) × kill_rate - spray_kill. 4 parameters.
- [Irrigation Water Efficiency Simulator](/lab/agriculture/irrigation-optimization): Optimize irrigation schedules by balancing water application, evapotranspiration, and root zone soil moisture to maximize crop water use efficiency Formula: Soil_moisture(t+1) = min(Soil_moisture(t) + irrigation - ET₀ × Kc, field_capacity). 4 parameters.
- [Soil NPK Nutrient Balance Simulator](/lab/agriculture/soil-nutrients): Model the balance of nitrogen, phosphorus, and potassium in soil to understand Liebig's law of the minimum and optimize fertilizer application Formula: Yield_potential = min(N_supply / N_demand, P_supply / P_demand, K_supply / K_demand) × 100. 4 parameters.

### Forensic Science & Criminal Investigation (5 simulations)
- [Blood Spatter Angle Analysis](/lab/forensic-science/blood-spatter): Reconstruct the impact angle and origin of bloodstains from droplet geometry — the trigonometry behind bloodstain pattern analysis Formula: Impact_angle = arcsin(stain_width / stain_length). 4 parameters.
- [Bullet Trajectory & Forensic Ballistics](/lab/forensic-science/ballistics): Simulate bullet trajectories accounting for gravity, drag, and wind to reconstruct shooting positions and distances in forensic investigations Formula: F_drag = 0.5 × ρ_air × Cd × A × v². 4 parameters.
- [DNA Fingerprinting & Match Probability](/lab/forensic-science/dna-fingerprinting): Calculate the probability of a random DNA profile match using STR loci allele frequencies — the statistical backbone of forensic genetics Formula: RMP = ∏(2 × p_i × q_i) for heterozygous loci, ∏(p_i²) for homozygous loci. 4 parameters.
- [Fingerprint Minutiae Matching Simulator](/lab/forensic-science/fingerprint-matching): Score fingerprint similarity by comparing minutiae points — ridge endings, bifurcations, and their spatial relationships used in automated fingerprint identification Formula: Match_score = (matched_minutiae / total_minutiae) × spatial_agreement × angular_agreement × 100. 4 parameters.
- [Time of Death Estimation (Henssge Nomogram)](/lab/forensic-science/time-of-death): Estimate postmortem interval from body temperature using the Henssge cooling model — the standard forensic method for time-of-death determination Formula: T(t) = T_env + (T₀ - T_env) × (1.25 × exp(-k₁t) - 0.25 × exp(-5k₁t)). 4 parameters.

## arts (1 topics, 5 simulations)

### Music Theory & Sound (5 simulations)
- [Chord Progression & Harmony Simulator](/lab/music-theory/chord-progressions): Build and analyze chord progressions using Roman numeral analysis — explore voice leading, harmonic tension, and resolution in real time Formula: Dissonance score = Σ |interval_class - nearest_consonance| for all note pairs. 4 parameters.
- [Circle of Fifths Explorer](/lab/music-theory/circle-of-fifths): Navigate the circle of fifths to visualize key signatures, relative majors and minors, and the 3:2 frequency ratio that structures all of Western harmony Formula: Fifth frequency = root × 3/2 (just intonation). 4 parameters.
- [Musical Scales & Modes Explorer](/lab/music-theory/scale-modes): Explore the seven diatonic modes from bright Ionian to dark Locrian — visualize interval patterns, semitone positions, and modal color on a keyboard and circle Formula: Mode intervals (in semitones from root): Ionian [0,2,4,5,7,9,11], Dorian [0,2,3,5,7,9,10], etc.. 4 parameters.
- [Polyrhythm & Time Signature Simulator](/lab/music-theory/rhythm-patterns): Layer two independent rhythmic cycles to create polyrhythms — watch how 3:2, 4:3, and complex ratios generate emergent rhythmic patterns Formula: LCM(a, b) = (a × b) / GCD(a, b). 4 parameters.
- [Tuning Systems Comparison](/lab/music-theory/tuning-systems): Compare equal temperament and just intonation side by side — hear and see the cent differences between pure frequency ratios and the compromises of modern tuning Formula: Equal-tempered frequency = base × Math.pow(2, interval/12). 4 parameters.

## behavioral-science (1 topics, 5 simulations)

### Psychology & Cognition (5 simulations)
- [Cognitive Load & Working Memory Capacity](/lab/psychology/cognitive-load): Explore the limits of working memory — visualize how cognitive load affects performance and discover Miller's magical number 7±2 Formula: Effective Capacity = memory_span × (1 - complexity_penalty) × (1 - distraction_penalty). 4 parameters.
- [Dunning-Kruger Effect Simulator](/lab/psychology/dunning-kruger): Visualize the Dunning-Kruger curve — see how low performers drastically overestimate their ability while top performers slightly underestimate theirs Formula: Overestimation = self_rated_percentile - actual_percentile. 4 parameters.
- [Milgram Obedience Experiment Simulation](/lab/psychology/milgram-obedience): Simulate Milgram's shocking obedience experiment — watch how authority pressure drives compliance rates to 65% even when participants believe they're harming others Formula: Obedience Rate = subjects_reaching_450V / total_subjects × 100%. 4 parameters.
- [Spacing Effect & Optimal Study Intervals](/lab/psychology/spacing-effect): Discover why spaced repetition crushes cramming — simulate memory retention curves and find the optimal study schedule for long-term learning Formula: Retention(t) = a × Math.pow(1 + t, -b) where a=initial_strength, b=decay_rate, t=time. 4 parameters.
- [Stroop Effect Reaction Time Test](/lab/psychology/stroop-effect): Measure how conflicting word-color information slows your brain — the classic Stroop effect demonstrates automatic reading interference with color naming Formula: Stroop Interference = RT_incongruent - RT_congruent. 4 parameters.

## biology (16 topics, 79 simulations)

### Behavioral Ecology & Animal Strategy (5 simulations)
- [Hawk-Dove Evolutionary Game](/lab/behavioral-ecology/hawk-dove-game): Play the hawk-dove game — adjust resource value, injury cost, display cost, and initial hawk fraction to watch evolutionary dynamics converge to the ESS mixed strategy Formula: p* = V/C (ESS hawk frequency when V < C). 4 parameters.
- [Kin Selection & Hamilton's Rule](/lab/behavioral-ecology/kin-selection): Model kin selection and altruism — adjust relatedness, benefit to recipient, cost to altruist, and group size to explore when self-sacrifice evolves through inclusive fitness Formula: rB > C (Hamilton's rule for altruism). 4 parameters.
- [Optimal Foraging Theory](/lab/behavioral-ecology/optimal-foraging): Simulate optimal foraging decisions — adjust prey energy value, handling time, encounter rate, and travel cost to find when a forager should be selective vs. generalist Formula: R = λE / (1 + λh) (Holling's disc equation, energy intake rate). 4 parameters.
- [Predator-Prey Behavioral Strategy](/lab/behavioral-ecology/predator-prey-strategy): Simulate predator-prey behavioral arms races — adjust predator speed, prey vigilance, group size, and escape probability to model the landscape of fear and optimal anti-predator behavior Formula: P_detect = 1 - (1-v)^G (many-eyes group detection). 4 parameters.
- [Territory Defense Economics](/lab/behavioral-ecology/territory-defense): Model territory defense trade-offs — adjust territory size, resource density, defense cost, and intruder pressure to find the economically optimal territory size Formula: Net benefit B = ρA - c×2√(πA) - I×c_chase. 4 parameters.

### Biochemistry & Molecular Dynamics (5 simulations)
- [ATP Synthesis Energy Cycle](/lab/biochemistry/atp-cycle): Visualize the ATP-ADP energy cycle — the universal currency of cellular energy that powers every living process from muscle contraction to DNA replication Formula: ΔG = ΔG°' + RT × ln([ADP][Pi] / [ATP]). 4 parameters.
- [Chemical Reaction Rate & Arrhenius Equation](/lab/biochemistry/reaction-rate): Explore how temperature governs chemical reaction rates through the Arrhenius equation — visualize the exponential relationship between temperature and rate constant Formula: k = A × exp(−Ea / RT) (Arrhenius equation). 4 parameters.
- [Michaelis-Menten Enzyme Kinetics](/lab/biochemistry/enzyme-kinetics): Simulate enzyme kinetics using the Michaelis-Menten model — explore how substrate concentration, Km, and Vmax determine reaction velocity Formula: v = Vmax × [S] / (Km + [S]). 4 parameters.
- [pH Buffer Equilibrium](/lab/biochemistry/ph-buffer): Explore how buffer solutions resist pH changes — simulate acid/base titration and the Henderson-Hasselbalch equation in real time Formula: pH = pKa + log₁₀([A⁻] / [HA]) (Henderson-Hasselbalch). 4 parameters.
- [Protein Folding Energy Landscape](/lab/biochemistry/protein-folding): Visualize the protein folding energy landscape — a rugged funnel where polypeptide chains navigate toward their minimum-energy native state Formula: ΔG = ΔH − TΔS (free energy of folding). 4 parameters.

### Biomechanics & Human Movement (5 simulations)
- [Gait Cycle Analysis](/lab/biomechanics/gait-analysis): Simulate the human gait cycle — explore how walking speed, stride length, leg length, and cadence affect stance phase, swing phase, and ground reaction forces Formula: Cadence = (v / stride_length) × 120 steps/min. 4 parameters.
- [Joint Reaction Forces](/lab/biomechanics/joint-loading): Simulate joint reaction forces — explore how body weight, muscle force, lever arms, and joint angle determine compressive and shear loads on the hip and knee Formula: ΣM = 0: Fm × dm = W × dw × cos(θ). 4 parameters.
- [Muscle Force-Velocity Relationship](/lab/biomechanics/muscle-force): Simulate the Hill muscle model — explore how contraction velocity, activation level, fiber length, and maximum isometric force determine muscle output Formula: F = a × F₀ × fl(L/L₀) × fv(v). 4 parameters.
- [Postural Balance & Stability](/lab/biomechanics/balance-stability): Simulate human balance — explore how center-of-mass position, base-of-support width, body height, and perturbation magnitude determine stability margin and fall risk Formula: SM = BOS/2 − x_COM. 4 parameters.
- [Vertical Jump Projectile Analysis](/lab/biomechanics/projectile-jump): Simulate the biomechanics of vertical jumping — explore how takeoff velocity, body mass, push-off depth, and air time determine jump height and ground reaction forces Formula: h = v₀² / (2g). 4 parameters.

### Biophysics & Molecular Mechanics (5 simulations)
- [DNA Mechanics & Elasticity](/lab/biophysics/dna-mechanics): Simulate DNA stretching mechanics — explore the worm-like chain model, persistence length, and force-extension behavior of double-stranded DNA Formula: F = (kBT / Lp) × [1/(4(1-x/Lc)²) - 1/4 + x/Lc] (Marko-Siggia WLC interpolation). 4 parameters.
- [Ion Channel Gating Dynamics](/lab/biophysics/ion-channels): Simulate voltage-gated ion channel kinetics — explore how membrane voltage, gating charge, and temperature control channel opening probability and conductance Formula: p_open = 1 / (1 + exp(-(Vm - V½) / k)). 4 parameters.
- [Membrane Transport & Nernst Potential](/lab/biophysics/membrane-transport): Simulate ion transport across biological membranes — explore how concentration gradients, membrane permeability, and voltage drive the Nernst and Goldman equations Formula: E_ion = (RT / zF) × ln([ion]_out / [ion]_in). 4 parameters.
- [Molecular Motors & Kinesin Stepping](/lab/biophysics/molecular-motors): Simulate molecular motor stepping along microtubules — explore how ATP concentration, load force, and temperature govern kinesin velocity and processivity Formula: v = d × kcat × [ATP] / (Km + [ATP]) × exp(-F × d_load / (kBT)). 4 parameters.
- [Protein Folding Energy Landscape](/lab/biophysics/protein-folding): Simulate the protein folding energy landscape — explore how temperature, hydrophobic forces, and chain length determine folding kinetics and native state stability Formula: deltaG = deltaH - T × deltaS. 4 parameters.

### Biotechnology & Genetic Engineering (5 simulations)
- [Bioreactor Growth Dynamics & Scale-Up](/lab/biotechnology/bioreactor-design): Simulate microbial and cell culture growth in bioreactors — model Monod kinetics, substrate consumption, oxygen transfer, and scale-up from bench to production Formula: μ = μmax × S/(Ks + S) (Monod equation). 4 parameters.
- [CRISPR-Cas9 Gene Editing Efficiency](/lab/biotechnology/crispr-editing): Simulate CRISPR-Cas9 gene editing — explore guide RNA design, on-target efficiency, off-target probability, and indel outcomes for different genomic targets Formula: P(off-target) ≈ P₀ × e^(-k × mismatches) (exponential mismatch penalty). 4 parameters.
- [Gel Electrophoresis DNA Separation](/lab/biotechnology/gel-electrophoresis): Simulate gel electrophoresis — visualize how DNA fragments separate by size through an agarose matrix under an electric field, with band migration and ladder comparison Formula: μ = v/E = d/(t·E) (electrophoretic mobility). 4 parameters.
- [Gene Expression & Transcription Regulation](/lab/biotechnology/gene-expression): Simulate gene expression dynamics — explore promoter strength, mRNA degradation, protein synthesis rates, and feedback regulation in a single-gene expression model Formula: dm/dt = kTx - δm·m (mRNA dynamics). 4 parameters.
- [PCR Amplification & Thermal Cycling](/lab/biotechnology/pcr-amplification): Simulate polymerase chain reaction — visualize exponential DNA amplification through thermal cycling, explore primer annealing, and calculate copy numbers across cycles Formula: N = N₀ × (1 + E)ⁿ (exponential amplification). 4 parameters.

### Chronobiology & Biological Rhythms (5 simulations)
- [Circadian Rhythm Oscillator](/lab/chronobiology/circadian-rhythm): Simulate the 24-hour circadian clock — adjust light exposure, clock period, and coupling strength to see how the body's master oscillator entrains to the day-night cycle Formula: dθ/dt = (2π/τ) + K × sin(θ_light − θ) (phase oscillator model). 4 parameters.
- [Jet Lag Recovery Model](/lab/chronobiology/jet-lag-model): Model jet-lag recovery after crossing time zones — adjust the number of zones crossed, direction of travel, and light strategy to predict how many days until re-entrainment Formula: Recovery days ≈ zones / (rate × light_factor × age_factor). 4 parameters.
- [Seasonal Affective Disorder Model](/lab/chronobiology/seasonal-affective): Model how photoperiod changes drive seasonal mood variation — adjust latitude, light therapy dose, and melatonin sensitivity to explore SAD mechanisms Formula: Photoperiod(lat, day) = 24 − (24/π) × arccos([−tan(lat)×tan(δ)]). 4 parameters.
- [Sleep Cycle Architecture](/lab/chronobiology/sleep-cycle): Visualize sleep architecture through the night — adjust sleep duration, REM pressure, and slow-wave drive to see how NREM and REM stages cycle in 90-minute ultradian rhythms Formula: Cycles ≈ sleep_duration / cycle_period. 4 parameters.
- [Ultradian Rhythm Oscillator](/lab/chronobiology/ultradian-rhythm): Explore ultradian rhythms — the 90–120 minute cycles of alertness, hormone release, and cognitive performance that pattern your waking and sleeping hours Formula: Alertness(t) = A × sin(2πt / period) × [1 + C × sin(2πt / 24h)] + caffeine_offset. 4 parameters.

### Ecology (5 simulations)
- [Ecological Succession](/lab/ecology/ecological-succession): Watch a landscape recover from disturbance — from bare ground through pioneer species to mature forest, with biodiversity and biomass tracking over time. Formula: Shannon diversity: H = -sum(p_i * ln(p_i)). 4 parameters.
- [Food Web & Trophic Cascades](/lab/ecology/food-web): Interactive food web model showing how species removal triggers trophic cascades — watch energy flow through the network and see which species are truly keystone. Formula: Connectance C = L / (S(S-1)/2). 4 parameters.
- [Island Biogeography Theory](/lab/ecology/island-biogeography): Explore MacArthur-Wilson theory of island biogeography — watch species immigrate and go extinct until equilibrium is reached. Formula: S_eq = P * i / (i + e * d / A^z). 4 parameters.
- [Population Growth & Carrying Capacity](/lab/ecology/carrying-capacity): Watch a population grow through exponential and logistic phases — explore how carrying capacity limits growth and how harvesting affects sustainability. Formula: dN/dt = rN(1 - N/K). 4 parameters.
- [Species-Area Relationship](/lab/ecology/species-area): Explore the power-law relationship between habitat area and species richness — see why halving habitat doesn't halve biodiversity, and calculate extinction debt. Formula: S = c * A^z (species-area power law). 4 parameters.

### Epidemiology & Disease Dynamics (5 simulations)
- [Basic Reproduction Number R₀ Dynamics](/lab/epidemiology/r-number): Manipulate the basic reproduction number R₀ and observe how interventions like masking, distancing, and vaccination drive the effective reproduction number below 1 Formula: R_eff = R₀ × S/N × (1 - Σ intervention_effects) (effective reproduction number). 4 parameters.
- [Contact Tracing Network Simulation](/lab/epidemiology/contact-tracing): Watch a disease spread through a social network and see how contact tracing identifies and isolates infectious individuals to break transmission chains Formula: R_effective = R₀ × (1 - trace_efficiency × isolation_compliance). 4 parameters.
- [Epidemic Curve Flattening Strategies](/lab/epidemiology/epidemic-curve): Model SIR epidemic dynamics and observe how timing and intensity of interventions flatten the curve to keep cases below healthcare capacity Formula: dS/dt = -β × S × I / N (susceptible depletion rate). 4 parameters.
- [Herd Immunity Threshold Calculator](/lab/epidemiology/herd-immunity): Calculate the herd immunity threshold for any disease — visualize how vaccination coverage protects unvaccinated individuals through population-level immunity Formula: HIT = 1 - 1/R₀ (herd immunity threshold). 4 parameters.
- [Vaccine Efficacy & Trial Design](/lab/epidemiology/vaccine-efficacy): Design a vaccine clinical trial — set sample sizes, observe attack rates in treatment and control groups, and calculate efficacy with confidence intervals Formula: VE = 1 - (ARV / ARP) = 1 - RR (relative risk reduction). 4 parameters.

### Evolution & Population Genetics (4 simulations)
- [Genetic Drift Simulator](/lab/evolution/genetic-drift): Wright-Fisher model demonstrating how random sampling in finite populations causes allele frequencies to fluctuate, leading to fixation or loss Formula: X(t+1) ~ Binomial(2N, p(t)). 4 parameters.
- [Mutation Accumulation & Cancer Risk Simulator](/lab/evolution/mutation-accumulation): Armitage-Doll multi-hit model showing how somatic mutations accumulate with age and drive cancer incidence, with animated DNA visualization Formula: P(cancer by age t) ≈ 1 − exp(−μ^k · N · t^k / k!). 5 parameters.
- [Natural Selection Simulator](/lab/evolution/natural-selection): Interactive Hardy-Weinberg selection model showing how fitness differences drive allele frequency changes across generations Formula: p' = (p²·w_AA + p·q·w_Aa) / w̄. 5 parameters.
- [Predator-Prey Dynamics Simulator](/lab/evolution/predator-prey): Lotka-Volterra model showing oscillating population dynamics between predator and prey species with phase portrait visualization Formula: dx/dt = αx − βxy. 6 parameters.

### Genetics (5 simulations)
- [CRISPR-Cas9 Gene Editing](/lab/genetics/crispr-mechanism): Visualize CRISPR-Cas9 in action — guide RNA targets a DNA sequence, Cas9 cuts the double strand, and the cell repairs via NHEJ or HDR pathways Formula: On-target efficiency ≈ activity × (1 − mismatches/guide_length)². 4 parameters.
- [DNA Replication Fork](/lab/genetics/dna-replication): Watch DNA unwind and replicate — visualize the replication fork with helicase, primase, and DNA polymerase synthesizing leading and lagging strands in real time Formula: Errors per genome = genome_size × 10^(error_rate_exponent). 4 parameters.
- [Genetic Code & Protein Translation](/lab/genetics/genetic-code): Decode mRNA into proteins — watch ribosomes translate codons into amino acids, introduce mutations, and discover how frameshift errors devastate protein structure Formula: 64 codons = 4³ (three positions, four possible nucleotides each). 4 parameters.
- [Hardy-Weinberg Equilibrium](/lab/genetics/hardy-weinberg): Explore population genetics — watch allele frequencies evolve across generations under genetic drift, natural selection, and the Hardy-Weinberg equilibrium principle Formula: p² + 2pq + q² = 1 (Hardy-Weinberg genotype frequencies). 4 parameters.
- [Punnett Square & Mendelian Inheritance](/lab/genetics/mendelian-inheritance): Simulate Mendelian inheritance with interactive Punnett squares — predict offspring genotype ratios, test complete vs incomplete dominance, and validate results with chi-squared analysis Formula: P(AA) = p² where p = frequency of dominant allele. 4 parameters.

### Immunology & Immune Response (5 simulations)
- [Antibody-Antigen Binding Kinetics](/lab/immunology/antibody-binding): Simulate antibody-antigen binding — explore how affinity, concentration, and valence affect immune complex formation and pathogen neutralization Formula: θ = [Ab] / (Kd + [Ab]) (Langmuir binding isotherm). 4 parameters.
- [Autoimmune Threshold & Tolerance Breakdown](/lab/immunology/autoimmune-threshold): Simulate autoimmune dynamics — explore how self-reactivity levels, regulatory T-cell suppression, and molecular mimicry affect the threshold between tolerance and autoimmune disease Formula: A_net = R × (1 + Mm) - Treg × k_supp (net autoreactivity). 4 parameters.
- [Cytokine Cascade & Signaling Networks](/lab/immunology/cytokine-cascade): Simulate cytokine signaling cascades — explore how initial stimulus, amplification gain, and negative feedback shape immune signaling dynamics Formula: dC/dt = k_prod × S - k_deg × C (production-degradation balance). 4 parameters.
- [T-Cell Activation & Clonal Expansion](/lab/immunology/t-cell-response): Simulate T-cell immune response — explore how antigen dose, costimulation strength, and cytokine environment shape clonal expansion and memory formation Formula: N(t) = N₀ × 2^(t/τ_div) (exponential expansion). 4 parameters.
- [Vaccine-Induced Immunity Timeline](/lab/immunology/vaccine-immunity): Simulate vaccine immune response — explore how antigen dose, adjuvant strength, and booster timing affect antibody titers and protection duration Formula: T(t) = T_peak × e^(-ln(2) × t / t½) (antibody decay). 4 parameters.

### Marine Biology & Ocean Life (5 simulations)
- [Bioluminescence & Light Production](/lab/marine-biology/bioluminescence): Simulate the biochemistry and ecology of bioluminescence — watch luciferin-luciferase reactions produce light and see how organisms use it for communication, camouflage, and predation. Formula: Luciferin + O₂ →(luciferase) Oxyluciferin + Light (hν). 4 parameters.
- [Coral Reef Growth & Bleaching](/lab/marine-biology/coral-reef): Watch a coral reef grow, compete for space, and bleach under thermal stress — modeling the symbiotic relationship between coral polyps and zooxanthellae algae that sustains reef ecosystems. Formula: Calcification rate G = G_max × Ω_arag × f(T) × f(light). 4 parameters.
- [Deep Sea Pressure & Adaptation](/lab/marine-biology/deep-sea-pressure): Visualize how hydrostatic pressure increases with ocean depth and how deep-sea organisms adapt their biochemistry — from flexible membranes to piezolyte molecules that resist crushing forces. Formula: Hydrostatic pressure: P = ρgh (≈ 1 atm per 10.3m depth). 4 parameters.
- [Fish Schooling & Collective Behavior](/lab/marine-biology/fish-schooling): Watch fish schools emerge from simple individual rules — alignment, cohesion, and separation create mesmerizing collective motion that confuses predators and improves foraging. Formula: Alignment: v_i += w_a × mean(v_neighbors). 4 parameters.
- [Ocean Currents & Thermohaline Circulation](/lab/marine-biology/ocean-currents): Visualize global ocean circulation driven by temperature and salinity differences — the thermohaline conveyor belt that distributes heat, nutrients, and marine life across the planet. Formula: Density: ρ = ρ₀(1 - αT + βS) where α,β are expansion coefficients. 4 parameters.

### Medical Science (5 simulations)
- [Cardiovascular Blood Flow](/lab/medical-science/blood-circulation): Model the cardiovascular system — visualize cardiac output, blood pressure waveforms, and the hemodynamics of circulation Formula: Cardiac output: CO = HR × SV. 4 parameters.
- [Drug Pharmacokinetics (ADME)](/lab/medical-science/pharmacokinetics): Model drug absorption, distribution, metabolism, and elimination — visualize plasma concentration curves, therapeutic windows, and multi-dose accumulation Formula: Single-dose concentration: C(t) = F·D·ka/(Vd·(ka-ke)) · (e^(-ke·t) - e^(-ka·t)). 5 parameters.
- [Immune Response Dynamics](/lab/medical-science/immune-response): Model the innate and adaptive immune response to infection — compare naive and vaccinated immune dynamics with pathogen clearance curves Formula: Pathogen growth: dP/dt = r·P·(1 - P/K) - α_innate·I·P - α_adaptive·A·P. 4 parameters.
- [Tumor Growth Dynamics](/lab/medical-science/tumor-growth): Model tumor progression with Gompertzian growth — explore immune surveillance, treatment timing, and the race between growth and therapy Formula: Gompertz growth: N(t) = K · exp(-ln(K/N0) · exp(-r·t)). 4 parameters.
- [Viral Replication Cycle](/lab/medical-science/virus-replication): Simulate the viral lifecycle — binding, entry, replication, assembly, and budding — and watch the immune system fight back Formula: Viral load growth: dV/dt = p·I - c·V - k·E·V (production - clearance - immune killing). 4 parameters.

### Neuroscience (5 simulations)
- [Hebbian Learning Rule](/lab/neuroscience/hebbian-learning): Neurons that fire together wire together — watch synaptic connections strengthen in real time as correlated neurons form memory assemblies. Formula: Δwᵢⱼ = η × xᵢ × xⱼ (basic Hebbian rule). 4 parameters.
- [Neural Action Potential](/lab/neuroscience/action-potential): Watch a neuron fire — see how ion channels open and close to produce the all-or-nothing electrical spike that carries information in the brain. Formula: C_m × dV/dt = -g_Na × m³h × (V - E_Na) - g_K × n⁴ × (V - E_K) - g_L × (V - E_L) + I_ext. 4 parameters.
- [Neural Firing Patterns](/lab/neuroscience/neuron-firing-patterns): Explore the five distinct firing patterns neurons use to encode information — from steady tonic firing to complex bursting rhythms. Formula: CV(ISI) = σ(ISI) / μ(ISI). 4 parameters.
- [Simple Perceptron Training](/lab/neuroscience/neural-network): Train a perceptron to classify points — watch the decision boundary rotate and shift as gradient descent finds the optimal separation. Formula: y = σ(w₁x₁ + w₂x₂ + b) where σ is step function. 4 parameters.
- [Synaptic Plasticity & LTP/LTD](/lab/neuroscience/synaptic-plasticity): Explore spike-timing-dependent plasticity — see how millisecond differences in spike timing determine whether synapses strengthen or weaken. Formula: Δw = A₊ × exp(-Δt/τ₊) if Δt > 0 (LTP). 4 parameters.

### Nutrition Science (5 simulations)
- [Calorie Balance & Metabolic Rate Calculator](/lab/nutrition-science/calorie-balance): Calculate your Basal Metabolic Rate and simulate how calorie surplus or deficit drives weight change over weeks and months Formula: BMR (Mifflin-St Jeor) = 10 × weight(kg) + 6.25 × height(cm) - 5 × age + s (s = +5 male, -161 female). 6 parameters.
- [Glycemic Index & Blood Sugar Response Simulator](/lab/nutrition-science/glycemic-index): Simulate how different glycemic index values affect blood sugar curves over time, comparing high-GI and low-GI meals Formula: Glycemic Load (GL) = GI × carb_grams / 100. 5 parameters.
- [Hydration & Electrolyte Balance Model](/lab/nutrition-science/hydration-model): Model water intake, sweat loss, and electrolyte balance to understand how hydration affects performance and health Formula: Base water need = body_weight × 33 (mL/day). 5 parameters.
- [Macronutrient Ratio Optimizer](/lab/nutrition-science/macronutrient-ratio): Visualize how protein, carbohydrate, and fat ratios affect total calories, thermic effect, and body composition outcomes Formula: Protein (g) = total_calories × protein_pct / 100 / 4. 4 parameters.
- [Vitamin Absorption & Bioavailability Simulator](/lab/nutrition-science/vitamin-absorption): Explore how fat-soluble and water-soluble vitamins are absorbed differently, and how meal composition affects bioavailability Formula: Absorption rate = base_rate × fat_modifier × gut_health × (1 - 0.4 × competing_nutrients). 5 parameters.

### Population Dynamics (5 simulations)
- [Age-Structured Population Model](/lab/population-dynamics/age-structure): Model how age-specific birth and death rates shape a population's age distribution over generations using Leslie matrix dynamics Formula: n(t+1) = L · n(t) where L is the Leslie matrix. 6 parameters.
- [Demographic Transition Stages](/lab/population-dynamics/demographic-transition): Simulate the four stages of demographic transition — from high birth/death rates to low rates — and watch population explode then stabilize Formula: Natural increase rate = birth rate − death rate. 5 parameters.
- [Malthusian vs Logistic Growth](/lab/population-dynamics/malthusian-growth): Compare unlimited exponential growth against resource-limited logistic growth — see why Malthus was right about the math but wrong about the outcome Formula: Malthusian: N(t) = N₀ · e^(r·t). 4 parameters.
- [Population Pyramid Dynamics](/lab/population-dynamics/population-pyramid): Build animated population pyramids — watch how birth rates, mortality, and migration reshape the age-sex distribution over decades Formula: Dependency ratio = (P₀₋₁₄ + P₆₅₊) / P₁₅₋₆₄ × 100. 5 parameters.
- [SIR Epidemic Model Simulator](/lab/population-dynamics/sir-epidemic): Simulate how infectious diseases spread through populations using the SIR model — adjust transmission and recovery rates to flatten the curve Formula: dS/dt = −β·S·I/N. 5 parameters.

## chemistry (9 topics, 45 simulations)

### Astrochemistry & Molecular Space (5 simulations)
- [Deuterium Enrichment in Cold Clouds](/lab/astrochemistry/isotope-fractionation): Simulate isotope fractionation reactions that enrich cold molecular clouds in deuterium, tracing the chemical heritage of our solar system Formula: H₃⁺ + HD → H₂D⁺ + H₂ + 232 K (exothermic fractionation). 4 parameters.
- [Grain Surface Ice Chemistry](/lab/astrochemistry/ice-mantle): Model the growth of molecular ice mantles on interstellar dust grains through accretion, surface reactions, and thermal desorption Formula: R_accretion = n_i · v_thermal · σ_grain · S (molecules/s per grain). 4 parameters.
- [Interstellar Molecular Cloud Chemistry](/lab/astrochemistry/molecular-cloud): Simulate gas-phase ion-molecule reaction networks inside a dense molecular cloud and watch molecular abundances evolve over millions of years Formula: dn(CO)/dt = k₁·n(C⁺)·n(OH) - k₂·n(He⁺)·n(CO) - k_freeze·n(CO). 4 parameters.
- [Photodissociation Region (PDR) Chemistry](/lab/astrochemistry/photodissociation-region): Explore UV-driven chemistry at the boundary between ionized gas and molecular clouds where photodissociation and photoionization shape the chemical structure Formula: G₀ · exp(-γ·A_V) = photodissociation rate attenuation into cloud. 4 parameters.
- [Rotational Spectral Line Identification](/lab/astrochemistry/spectral-line-id): Match observed radio/submillimeter spectral lines to molecular species using rotational transition frequencies and quantum numbers Formula: ν(J→J-1) = 2BJ for linear rotor, B = ℏ/(4πI). 4 parameters.

### Brewing Science & Beer Chemistry (5 simulations)
- [Carbonation & Henry's Law](/lab/brewing-science/carbonation): Simulate beer carbonation — calculate CO2 volumes, equilibrium pressure, and priming sugar dose using Henry's law and temperature dependence Formula: C = k_H(T) × P_CO2 (Henry's law). 4 parameters.
- [Hop Utilization & IBU Calculation](/lab/brewing-science/hop-utilization): Simulate hop bitterness extraction — calculate IBU from alpha acids, boil time, wort gravity, and hop form using the Tinseth model Formula: U = 1.65 × 0.000125^(OG-1) × (1 - e^(-0.04t)) / 4.15. 4 parameters.
- [Mash Conversion & Enzyme Kinetics](/lab/brewing-science/mash-conversion): Simulate starch-to-sugar conversion during mashing — explore how temperature, pH, and time affect alpha- and beta-amylase activity Formula: Rate = V_max × [S] / (K_m + [S]) × exp(-k_d × t). 4 parameters.
- [Water Chemistry & Mash pH](/lab/brewing-science/water-chemistry): Simulate brewing water adjustments — calculate mash pH from grain bill, water mineral content, and acid or salt additions Formula: RA = Alkalinity - Ca/1.4 - Mg/1.7 (Kolbach). 4 parameters.
- [Yeast Attenuation & Fermentation](/lab/brewing-science/yeast-attenuation): Simulate yeast fermentation — predict apparent and real attenuation, final gravity, and alcohol content from OG and yeast characteristics Formula: FG = OG - (OG - 1) × AA / 100. 4 parameters.

### Chemical Kinetics & Reaction Rates (5 simulations)
- [Arrhenius Equation & Activation Energy](/lab/chemical-kinetics/arrhenius-equation): Explore how temperature and activation energy control reaction rate constants through the Arrhenius equation with interactive energy diagrams Formula: k = A × exp(−Ea / RT). 4 parameters.
- [Catalysis Energy Diagram & Turnover](/lab/chemical-kinetics/catalysis-mechanism): Visualize how catalysts lower activation energy, compare catalyzed vs uncatalyzed reaction profiles, and calculate turnover frequency Formula: k_cat/k_uncat = exp[(Ea − Ea*) / RT]. 4 parameters.
- [Dynamic Equilibrium & Le Chatelier's Principle](/lab/chemical-kinetics/equilibrium-kinetics): Visualize forward and reverse reaction rates reaching dynamic equilibrium, and observe Le Chatelier shifts when conditions change Formula: Keq = kf / kr = [B]eq / [A]eq. 4 parameters.
- [Radical Chain Reactions & Explosion Limits](/lab/chemical-kinetics/chain-reaction): Model radical chain propagation, branching factors, and the transition from steady combustion to thermal explosion Formula: φ = f_branching − k_termination (net branching). 4 parameters.
- [Reaction Order & Concentration Decay](/lab/chemical-kinetics/reaction-order): Compare zero, first, and second-order kinetics side by side — concentration decay, half-life dependence, and integrated rate law plots Formula: [A] = [A]₀ − kt (zero order). 4 parameters.

### Electrochemistry & Cell Design (5 simulations)
- [Battery Discharge Curve Simulator](/lab/electrochemistry/battery-discharge): Visualize battery discharge curves — explore how load current, internal resistance, and capacity affect voltage over time for different cell chemistries Formula: V_terminal = V_ocv(SOC) − I × R_internal. 4 parameters.
- [Cyclic Voltammetry Simulator](/lab/electrochemistry/cyclic-voltammetry): Simulate cyclic voltammetry scans — sweep electrode potential and observe the characteristic duck-shaped voltammogram revealing redox peak positions and reversibility Formula: ip = 0.4463 × n^(3/2) × F^(3/2) × A × C × (Dν / RT)^(1/2). 4 parameters.
- [Electrochemical Impedance Spectroscopy](/lab/electrochemistry/impedance-spectroscopy): Explore Nyquist plots and impedance spectra — adjust solution resistance, charge transfer resistance, and double-layer capacitance to model electrode behavior Formula: Z(ω) = Rs + Rct / (1 + jω·Rct·Cdl) + σω^(−1/2)(1 − j). 4 parameters.
- [Electrolysis & Faraday's Laws](/lab/electrochemistry/electrolysis): Simulate electrolysis — see how current, time, and molar mass determine the mass deposited at an electrode according to Faraday's laws Formula: m = (M × I × t) / (z × F). 4 parameters.
- [Nernst Equation & Cell Potential](/lab/electrochemistry/nernst-equation): Calculate electrochemical cell potential under non-standard conditions using the Nernst equation — adjust temperature, ion concentrations, and electron transfer count Formula: E = E° − (RT / nF) × ln(Q). 4 parameters.

### Photochemistry & Light-Driven Reactions (5 simulations)
- [Beer-Lambert Law & Absorption Spectra](/lab/photochemistry/photon-absorption): Simulate photon absorption through solutions using the Beer-Lambert law — explore how concentration, path length, and molar absorptivity affect transmittance Formula: A = ε × c × l (Beer-Lambert law). 4 parameters.
- [Fluorescence Quantum Yield Simulator](/lab/photochemistry/quantum-yield): Simulate fluorescence quantum yield — explore radiative and non-radiative decay rates, Stokes shift, and emission spectra Formula: Φ = k_r / (k_r + k_nr) — quantum yield. 4 parameters.
- [Photosynthesis Light Reactions & Electron Transport](/lab/photochemistry/photosynthesis-light): Simulate the light reactions of photosynthesis — model electron flow through PSII, cytochrome b6f, and PSI to produce ATP and NADPH Formula: ETR = PAR × f_PSII × α_leaf × Φ_PSII. 4 parameters.
- [TiO₂ Photocatalytic Degradation](/lab/photochemistry/photocatalysis): Model TiO2 photocatalytic degradation of organic pollutants — explore UV intensity, catalyst loading, and Langmuir-Hinshelwood kinetics Formula: r = k_r × K × C / (1 + K × C) — Langmuir-Hinshelwood rate law. 4 parameters.
- [UV Weathering & Polymer Degradation](/lab/photochemistry/uv-degradation): Model UV-induced polymer degradation — predict material lifetime from UV dose, stabilizer concentration, and temperature using Arrhenius kinetics Formula: k = k₀ × exp(-Ea/RT) × I_UV^n — Arrhenius-UV rate constant. 4 parameters.

### Polymer Science & Plastics (5 simulations)
- [Glass Transition Temperature (Tg)](/lab/polymer-science/glass-transition): Simulate the glass transition — explore how temperature, cooling rate, plasticizer content, and molecular weight control the shift from rubbery to glassy polymer behavior Formula: Tg(Mₙ) = Tg∞ - K / Mₙ  (Fox–Flory). 4 parameters.
- [Polymer Crystallinity & Spherulite Growth](/lab/polymer-science/polymer-crystallinity): Simulate crystallization in semicrystalline polymers — explore how undercooling, nucleation density, and chain regularity control spherulite growth and crystallinity fraction Formula: Xc(t) = 1 - exp(-K · t^n)  (Avrami). 4 parameters.
- [Polymerization Kinetics & Chain Growth](/lab/polymer-science/polymerization-kinetics): Simulate free-radical polymerization — explore how initiator concentration, rate constants, and temperature control molecular weight and conversion over time Formula: Rₚ = kₚ · [M] · √(f · kd · [I] / kₜ). 4 parameters.
- [Rubber Elasticity & Entropic Springs](/lab/polymer-science/rubber-elasticity): Simulate rubber elasticity — explore how crosslink density, temperature, chain length, and strain produce the unique stress-strain behavior of elastomeric networks Formula: σ_true = νkT(λ² - 1/λ)  (neo-Hookean). 4 parameters.
- [Viscoelasticity & Stress Relaxation](/lab/polymer-science/viscoelasticity): Simulate viscoelastic behavior — explore how elastic modulus, viscosity, strain rate, and temperature produce creep, relaxation, and hysteresis in polymers Formula: σ(t) = σ₀ · exp(-t·E/η)  (Maxwell relaxation). 4 parameters.

### Stereochemistry & Molecular Chirality (5 simulations)
- [Chirality Center: R/S Configuration & CIP Rules](/lab/stereochemistry/chirality-center): Assign R/S configuration at a chiral center using Cahn-Ingold-Prelog priority rules with interactive 3D-style visualization Formula: CIP Priority: rank by atomic number at point of attachment. 4 parameters.
- [Conformational Analysis & Newman Projections](/lab/stereochemistry/conformational-analysis): Visualize Newman projections and rotational energy barriers for ethane and substituted alkanes with animated dihedral rotation Formula: E(φ) = (V₀/2)(1 − cos(3φ)) for threefold barrier. 4 parameters.
- [Enantiomer Separation & Chiral Resolution](/lab/stereochemistry/enantiomer-separation): Simulate chiral resolution methods and calculate enantiomeric excess from chromatographic peak areas with real-time visualization Formula: ee = |A_R − A_S| / (A_R + A_S) × 100%. 4 parameters.
- [Fischer Projection & Sugar Stereochemistry](/lab/stereochemistry/fischer-projection): Build Fischer projections of aldoses and explore D/L notation with interactive carbon chain configuration and sugar classification Formula: Stereoisomers = 2^(n−2) for n-carbon aldose. 4 parameters.
- [Optical Rotation & Polarimetry](/lab/stereochemistry/optical-rotation): Measure specific rotation and optical activity of chiral compounds using a virtual polarimeter with animated light polarization Formula: α_obs = [α]_D × c × l × (ee / 100). 4 parameters.

### Surface Chemistry & Adsorption (5 simulations)
- [BET Isotherm & Multilayer Adsorption](/lab/surface-chemistry/bet-isotherm): Simulate the BET multilayer adsorption model — explore how relative pressure and BET constant determine surface area and layer formation Formula: V = V_m × C × x / ((1 - x)(1 - x + Cx)), where x = P/P₀. 4 parameters.
- [Contact Angle & Wettability](/lab/surface-chemistry/contact-angle): Simulate wettability using Young's equation — explore how surface energies determine contact angle and droplet shape Formula: cos θ = (γ_SV - γ_SL) / γ_LV (Young's equation). 4 parameters.
- [Heterogeneous Catalysis & Sabatier Principle](/lab/surface-chemistry/heterogeneous-catalysis): Simulate heterogeneous catalysis — explore the volcano plot, Sabatier principle, and how binding energy controls catalytic activity Formula: TOF = A × exp(-E_a/RT) × θ × (1 - θ). 4 parameters.
- [Langmuir Isotherm & Monolayer Adsorption](/lab/surface-chemistry/langmuir-isotherm): Simulate the Langmuir adsorption isotherm — explore how pressure, binding energy, and temperature control monolayer surface coverage Formula: θ = KP / (1 + KP). 4 parameters.
- [Surface Tension, Laplace Pressure & Capillary Rise](/lab/surface-chemistry/surface-tension): Simulate surface tension effects — explore Laplace pressure in bubbles and capillary rise in narrow tubes Formula: h = 2γ cos θ / (ρgr) (Jurin's law). 4 parameters.

### Tribochemistry & Surface Reactions (5 simulations)
- [Flash Temperature at Sliding Contacts](/lab/tribochemistry/flash-temperature): Simulate transient flash temperatures at asperity contacts — explore how sliding velocity, friction coefficient, load, and thermal conductivity determine the extreme temperature spikes that drive trib Formula: ΔT = μ × Fn × v / (4 × k × a). 5 parameters.
- [Friction Energy Dissipation](/lab/tribochemistry/friction-energy): Simulate frictional energy budgets — explore how normal load, sliding velocity, and friction coefficient partition energy into heat, plastic deformation, and chemical activation at a tribological cont Formula: Ff = μ × Fn. 4 parameters.
- [Mechanochemical Reaction Kinetics](/lab/tribochemistry/mechanochemical-reaction): Simulate mechanochemical bond activation — explore how shear stress, temperature, and contact pressure lower reaction barriers and drive tribochemical transformations at sliding interfaces Formula: Ea_eff = Ea - τ × V*. 4 parameters.
- [Tribofilm Growth Dynamics](/lab/tribochemistry/tribofilm-growth): Simulate protective tribofilm formation — explore how contact pressure, temperature, additive concentration, and sliding cycles govern the nucleation and growth of tribochemical surface films Formula: h = h_max × (1 - exp(-k × N)). 4 parameters.
- [Wear Debris Generation & Morphology](/lab/tribochemistry/wear-debris): Simulate wear particle formation — explore how load, hardness ratio, sliding distance, and wear mechanism govern the size, shape, and volume of debris ejected from tribological contacts Formula: V = K × Fn × L / H. 4 parameters.

## computer-science (13 topics, 64 simulations)

### AI Risk & Alignment (4 simulations)
- [AI Governance Race Dynamics Simulator](/lab/ai-risk/ai-governance-race): Model the multi-actor AI development race where competitive pressure, safety investment, and international coordination determine catastrophic risk. Explore how race dynamics create collective action  Formula: P(\text{catastrophe}) = 1 - \prod_{i=1}^{n} (1 - r_i). 5 parameters.
- [Alignment Tax Calculator](/lab/ai-risk/alignment-tax): Quantify the tradeoff between AI capability and safety. The 'alignment tax' is the capability cost of building safe AI systems. This simulator models how alignment overhead affects capability growth,  Formula: \text{Capability}_{\text{aligned}}(t) = R \cdot (1 - o)^t. 5 parameters.
- [Intelligence Explosion Simulator](/lab/ai-risk/intelligence-explosion): Explore recursive self-improvement dynamics and the conditions under which artificial intelligence undergoes slow, exponential, or hyperbolic (FOOM) takeoff. Based on I.J. Good's intelligence explosio Formula: I(t+1) = I(t) + \eta \cdot I(t)^{\alpha}. 4 parameters.
- [Mesa-Optimization & Goodhart's Law Simulator](/lab/ai-risk/mesa-optimization): Visualize how optimizing a proxy objective diverges from the true objective under distribution shift, and how mesa-optimizers can become deceptively aligned when capability exceeds oversight. Combines Formula: \text{True Perf} = P_{\text{proxy}} \cdot \rho - \delta \cdot P_{\text{opt}}^2. 5 parameters.

### Animation Physics & Simulation (5 simulations)
- [Bezier Curve Animation Paths Simulator](/lab/animation-physics/bezier-curves): Design and visualize cubic Bezier curves used for animation easing, motion paths, and vector graphics — drag control points and see De Casteljau's algorithm in action Formula: B(t) = (1-t)³P₀ + 3(1-t)²tP₁ + 3(1-t)t²P₂ + t³P₃, t ∈ [0,1]. 4 parameters.
- [Cloth Simulation With Constraints Simulator](/lab/animation-physics/cloth-simulation): Drape a particle-based cloth mesh over obstacles using distance constraints and Verlet integration — the technique behind capes, flags, and fabric in real-time graphics Formula: Verlet: x_new = 2x - x_old + a × dt². 4 parameters.
- [Particle System Effects Simulator](/lab/animation-physics/particle-effects): Spawn, animate, and blend thousands of particles to create fire, smoke, sparks, and explosions — the visual effects system powering every game engine Formula: Steady-state count = emit_rate × average_lifetime. 4 parameters.
- [Ragdoll Physics Joint Constraints Simulator](/lab/animation-physics/ragdoll-physics): Build and simulate a ragdoll skeleton with angular joint constraints — the physics system that makes characters tumble, collapse, and react to impacts in games Formula: Angular constraint: θ_min ≤ angle(bone_a, bone_b) ≤ θ_max. 4 parameters.
- [Spring Dynamics & Verlet Integration Simulator](/lab/animation-physics/spring-dynamics): Simulate spring-mass systems using Verlet integration — the stable numerical method behind hair, jelly, and soft-body physics in games and film Formula: F = -kx (Hooke's Law). 4 parameters.

### Automata Theory & Formal Languages (5 simulations)
- [Cellular Automata & Emergent Complexity](/lab/automata-theory/cellular-automata): Explore 1D and 2D cellular automata — watch simple local rules generate fractals, gliders, and Turing-complete computation from random initial conditions Formula: s(i, t+1) = f(s(i-1, t), s(i, t), s(i+1, t)). 4 parameters.
- [Deterministic Finite Automaton (DFA)](/lab/automata-theory/finite-automaton): Build and animate a deterministic finite automaton — watch states transition on input symbols, visualize acceptance and rejection of strings in real time Formula: DFA = (Q, Σ, δ, q₀, F). 4 parameters.
- [Pushdown Automaton & Context-Free Parsing](/lab/automata-theory/pushdown-automaton): Animate a pushdown automaton with a visible stack — trace how it recognizes nested parentheses, palindromes, and context-free grammars symbol by symbol Formula: PDA = (Q, Σ, Γ, δ, q₀, Z₀, F). 4 parameters.
- [Regular Expressions & Pattern Matching](/lab/automata-theory/regular-expressions): Visualize how a regular expression compiles to an NFA and matches strings — watch states activate as the engine processes each character Formula: L(r₁|r₂) = L(r₁) ∪ L(r₂). 4 parameters.
- [Turing Machine & Universal Computation](/lab/automata-theory/turing-machine): Program a single-tape Turing machine — watch the head read, write, and move across an infinite tape as it computes step by step Formula: TM = (Q, Σ, Γ, δ, q₀, q_accept, q_reject). 4 parameters.

### Computer Science (5 simulations)
- [A* Pathfinding Visualizer](/lab/computer-science/pathfinding): Watch A*, Dijkstra, and BFS explore a grid to find the shortest path — see how heuristics dramatically reduce search space Formula: f(n) = g(n) + h(n) — A* evaluation function. 4 parameters.
- [Binary Search vs Linear Search](/lab/computer-science/binary-search): Race binary search against linear search to see why halving the search space is exponentially faster than scanning one by one Formula: Binary search: T(n) = O(log₂ n) — at most ⌈log₂(n)⌉ comparisons. 4 parameters.
- [Conway's Game of Life](/lab/computer-science/game-of-life): Watch complexity emerge from four simple rules — gliders, oscillators, and spaceships in the most famous cellular automaton Formula: Next state: alive if (alive AND neighbors ∈ {2,3}) OR (dead AND neighbors = 3). 4 parameters.
- [Hash Table Collision Visualization](/lab/computer-science/hash-table): See how hash functions map keys to buckets and how collisions are resolved through chaining or open addressing Formula: Expected lookup (chaining): O(1 + α) where α = n/m is the load factor. 4 parameters.
- [Sorting Algorithm Race](/lab/computer-science/sorting-algorithms): Watch sorting algorithms race in real time — see why O(n log n) beats O(n²) as arrays grow larger Formula: Bubble/Selection/Insertion worst case: T(n) = O(n²). 4 parameters.

### Cryptocurrency & Blockchain (5 simulations)
- [Blockchain Mining Simulator](/lab/cryptocurrency/blockchain-mining): Mine blocks interactively — adjust hash rate, difficulty, and block size to see how miners compete to extend the blockchain Formula: E[hashes] = 2^difficulty. 4 parameters.
- [Consensus Algorithm Visualizer](/lab/cryptocurrency/consensus-algorithm): Watch distributed nodes reach consensus — simulate message passing, Byzantine faults, and finality in real-time network animations Formula: N ≥ 3f + 1 (BFT threshold). 4 parameters.
- [Cryptographic Hash Function Explorer](/lab/cryptocurrency/hash-function): Explore cryptographic hash functions — visualize the avalanche effect, collision resistance, and bit distribution as you modify inputs character by character Formula: Avalanche ratio = Hamming(H(x), H(x⊕1bit)) / bits. 4 parameters.
- [Merkle Tree Builder](/lab/cryptocurrency/merkle-tree): Build a Merkle tree from transactions — visualize hash propagation, verify inclusion proofs, and understand how blockchains compress transaction integrity Formula: root = H(H(H(tx₀||tx₁) || H(tx₂||tx₃)) || ...). 4 parameters.
- [Proof-of-Work Puzzle](/lab/cryptocurrency/proof-of-work): Solve proof-of-work puzzles interactively — watch nonce iteration, hash computation, and difficulty target comparison in real time Formula: E[attempts] = 2^difficulty. 4 parameters.

### Cryptography (5 simulations)
- [Caesar Cipher Visualizer](/lab/cryptography/caesar-cipher): Explore the oldest known encryption — rotate alphabet rings and watch plaintext transform into ciphertext with this interactive Caesar cipher. Formula: E(x) = (x + k) mod 26. 4 parameters.
- [Cryptographic Hash Visualizer](/lab/cryptography/hash-function): See why cryptographic hashes are one-way streets — watch the avalanche effect flip half the bits from a single input change. Formula: P(collision) ≈ 1 - e^(-n²/(2 × 2^b)) (birthday paradox). 4 parameters.
- [Diffie-Hellman Key Exchange](/lab/cryptography/diffie-hellman): Watch Alice and Bob agree on a shared secret over a public channel — the magic of Diffie-Hellman key exchange visualized step by step. Formula: A = g^a mod p (Alice's public value). 4 parameters.
- [One-Time Pad / Perfect Secrecy](/lab/cryptography/one-time-pad): The only provably unbreakable cipher — see how XOR with a random key creates perfect secrecy, and why key reuse destroys it. Formula: C = M ⊕ K (encryption: XOR message with key). 4 parameters.
- [RSA Public Key Encryption](/lab/cryptography/rsa-encryption): Watch how two prime numbers create a public-private key pair and encrypt a message step by step using RSA. Formula: C = M^e mod n (encryption). 4 parameters.

### Data Science & Machine Learning (5 simulations)
- [Decision Tree Classifier](/lab/data-science/decision-tree): Watch a decision tree carve feature space into regions — see how splits are chosen to maximize information gain and classify data Formula: Gini(node) = 1 - Σᵢ pᵢ². 4 parameters.
- [Gradient Descent Optimization](/lab/data-science/gradient-descent): Watch gradient descent navigate a loss landscape — see how learning rate and momentum affect convergence to the minimum of a function Formula: θₜ₊₁ = θₜ - α × ∇f(θₜ). 4 parameters.
- [K-Means Clustering Algorithm](/lab/data-science/k-means-clustering): Watch K-means discover clusters in real time — see centroids shift and data points reassign as the algorithm converges to a partition Formula: Inertia (WCSS) = Σᵢ Σⱼ∈Cᵢ ||xⱼ - μᵢ||². 4 parameters.
- [Linear Regression & Least Squares Fit](/lab/data-science/linear-regression): Watch a regression line snap to data points in real time — explore how least squares minimization finds the best-fit line through noisy data Formula: β₁ = Σ(xᵢ - x̄)(yᵢ - ȳ) / Σ(xᵢ - x̄)². 4 parameters.
- [PCA Dimensionality Reduction](/lab/data-science/pca-reduction): Watch PCA compress high-dimensional data into its essential components — see eigenvectors emerge and variance explained accumulate Formula: Covariance Matrix: C = (1/n) Xᵀ X (where X is mean-centered). 4 parameters.

### Game Design & Systems (5 simulations)
- [2D Physics Engine (Rigid Body Collisions)](/lab/game-design/physics-engine): Simulate rigid body physics with gravity, elasticity, and friction — see how real-time collision detection and resolution work under the hood Formula: Position: x(t+dt) = x(t) + v(t)×dt + 0.5×a×dt². 6 parameters.
- [Dynamic Difficulty Adjustment Curve](/lab/game-design/difficulty-curve): Visualize how difficulty curves shape player engagement — from linear ramps to adaptive systems that respond to player skill Formula: Linear: difficulty(level) = base + (max - base) × level / num_levels. 6 parameters.
- [Loot Table Probability & Drop Rates](/lab/game-design/loot-tables): Simulate thousands of loot drops to visualize how weighted probability tables create the reward psychology behind every RPG and looter game Formula: Drop probability = item_weight / total_weight. 7 parameters.
- [Navigation Mesh Pathfinding (A*)](/lab/game-design/pathfinding-navmesh): Visualize A* pathfinding on a navigation mesh grid — watch how obstacles, heuristics, and grid size affect NPC pathfinding in real time Formula: f(n) = g(n) + w × h(n), where w = heuristic_weight. 6 parameters.
- [Procedural Terrain Generation (Perlin Noise)](/lab/game-design/procedural-generation): Generate infinite terrain landscapes using layered Perlin noise — the algorithm behind Minecraft, No Man's Sky, and countless open-world games Formula: noise(x,y) = Σ(i=0 to octaves-1) persistence^i × perlin(x × frequency × lacunarity^i, y × frequency × lacunarity^i). 6 parameters.

### Logic Circuits & Digital Systems (5 simulations)
- [Boolean Logic Gates Simulator](/lab/logic-circuits/boolean-gates): Visualize AND, OR, NOT, XOR, and NAND gates in real time — toggle inputs and watch outputs change instantly to build intuition for digital logic Formula: AND: Q = A · B. 4 parameters.
- [Finite State Machine Simulator](/lab/logic-circuits/finite-automaton): Design and run finite automata — define states and transitions, feed input strings, and watch the machine accept or reject patterns in real time Formula: DFA: M = (Q, Σ, δ, q₀, F) where δ: Q × Σ → Q. 4 parameters.
- [Half Adder & Full Adder Circuit Simulator](/lab/logic-circuits/half-adder): Watch binary addition happen gate-by-gate — see carry bits propagate through half adders and full adders to build multi-bit arithmetic Formula: Half adder: Sum = A ⊕ B, Carry = A · B. 4 parameters.
- [SR & JK Flip-Flop Memory Circuit Simulator](/lab/logic-circuits/flip-flop): Observe how flip-flops store a single bit of memory — toggle Set, Reset, and Clock inputs to see state transitions in SR and JK circuits Formula: SR: Q(next) = S + R'·Q, constraint S·R = 0. 4 parameters.
- [Truth Table Generator & Evaluator](/lab/logic-circuits/truth-table): Generate and visualize truth tables for any combination of Boolean operations — see every possible input-output mapping at a glance Formula: Number of rows = 2^N where N = number of inputs. 4 parameters.

### Machine Learning & AI Algorithms (5 simulations)
- [Decision Tree Classifier Visualizer](/lab/machine-learning/decision-tree): Grow a decision tree on 2D data — adjust tree depth, split criteria, and data complexity to see how the tree partitions feature space Formula: Entropy: H(S) = −Σ p_i × log₂(p_i). 4 parameters.
- [Gradient Descent Optimizer Visualizer](/lab/machine-learning/gradient-descent): Watch gradient descent navigate a 2D loss landscape — adjust learning rate, momentum, and surface shape to see convergence, oscillation, and divergence Formula: Parameter update: θ_{t+1} = θ_t − η × ∇L(θ_t). 4 parameters.
- [K-Means Clustering Visualizer](/lab/machine-learning/clustering): Watch k-means clustering assign points to clusters — adjust k, data distribution, and initialization to see centroid convergence and Voronoi partitions Formula: Inertia: J = Σ_k Σ_{x∈C_k} ||x − μ_k||². 4 parameters.
- [Neural Network Forward Pass Visualizer](/lab/machine-learning/neural-network): Build a neural network and watch activations propagate — adjust layer sizes, activation functions, and input values to see how networks transform data Formula: Neuron output: y = σ(W·x + b). 4 parameters.
- [Overfitting & Bias-Variance Tradeoff](/lab/machine-learning/overfitting): Explore overfitting by fitting polynomials to noisy data — adjust model complexity, data size, and noise to see training vs test error diverge Formula: MSE = (1/n) × Σ (y_i − ŷ_i)². 4 parameters.

### Neuromorphic Computing (5 simulations)
- [Memristor Crossbar Array](/lab/neuromorphic-computing/memristor-crossbar): Simulate a memristive crossbar array — explore how conductance states, voltage pulses, and array size enable analog matrix-vector multiplication for neural networks Formula: I_j = Σ V_i × G_ij (Kirchhoff's current law at bit line j). 5 parameters.
- [Neuromorphic Vision (Event Camera)](/lab/neuromorphic-computing/neuromorphic-vision): Simulate a neuromorphic event camera — explore how contrast threshold, pixel bandwidth, and latency enable microsecond-resolution motion capture with minimal data Formula: Event triggered when |log(I(t)) - log(I(t_last))| > C. 4 parameters.
- [Reservoir Computing (Echo State Network)](/lab/neuromorphic-computing/reservoir-computing): Simulate a reservoir computer — explore how recurrent network size, spectral radius, and input scaling transform temporal signals into linearly separable representations Formula: x(t+1) = (1-α)x(t) + α × tanh(W_in × u(t) + W × x(t)). 4 parameters.
- [Spike-Timing-Dependent Plasticity (STDP)](/lab/neuromorphic-computing/stdp-learning): Simulate STDP learning — explore how the relative timing of pre- and post-synaptic spikes strengthens or weakens synaptic connections Formula: ΔW = A+ × exp(-Δt/τ+) for Δt > 0 (LTP). 5 parameters.
- [Spiking Neuron Model (LIF)](/lab/neuromorphic-computing/spiking-neuron): Simulate a leaky integrate-and-fire neuron — explore how input current, membrane time constant, and threshold voltage shape spike trains in neuromorphic hardware Formula: τ_m × dV/dt = -(V - V_rest) + R_m × I_in. 4 parameters.

### Quantum Computing & Qubits (5 simulations)
- [Quantum Entanglement & Bell States](/lab/quantum-computing/quantum-entanglement): Simulate quantum entanglement between two qubits — explore Bell states, CHSH inequality violations, and spooky action at a distance with real-time correlation analysis Formula: |Φ+⟩ = (|00⟩ + |11⟩)/√2. 4 parameters.
- [Quantum Error Correction Codes](/lab/quantum-computing/quantum-error-correction): Simulate quantum error correction — explore how surface codes, Shor codes, and Steane codes detect and fix bit-flip and phase-flip errors on encoded logical qubits Formula: p_L ≈ C × (p/p_th)^{(d+1)/2}. 4 parameters.
- [Quantum Gates & Circuit Operations](/lab/quantum-computing/quantum-gates): Simulate quantum gate operations on a qubit — apply Hadamard, Pauli, and phase gates and watch the state vector rotate on the Bloch sphere in real time Formula: H = (1/√2)[[1,1],[1,-1]]. 4 parameters.
- [Qubit Bloch Sphere Visualization](/lab/quantum-computing/qubit-bloch-sphere): Visualize a single qubit state on the Bloch sphere — explore how polar and azimuthal angles map to superposition amplitudes and measurement probabilities Formula: |ψ⟩ = cos(θ/2)|0⟩ + e^{iφ} sin(θ/2)|1⟩. 4 parameters.
- [Shor's Algorithm & Integer Factoring](/lab/quantum-computing/shor-algorithm): Simulate Shor's quantum factoring algorithm — explore how quantum Fourier transforms find the period of modular exponentiation to factor integers exponentially faster than classical methods Formula: a^r ≡ 1 (mod N) → period r via QFT. 4 parameters.

### Swarm Intelligence & Collective Behavior (5 simulations)
- [Ant Colony Optimization Pathfinding](/lab/swarm-intelligence/ant-colony): Watch virtual ants find the shortest path using pheromone trails — explore how evaporation rate, colony size, and pheromone influence shape emergent optimization Formula: P_ij = (τ_ij^α × η_ij^β) / Σ(τ^α × η^β) (transition probability). 4 parameters.
- [Bee Foraging & Waggle Dance Optimization](/lab/swarm-intelligence/bee-foraging): Model honey bee foraging strategy — explore how scout bees, waggle dance recruitment, and nectar quality drive optimal resource allocation across multiple food sources Formula: P_recruit(i) = fitness(i) / Σfitness(j) (probability of recruiting to source i). 4 parameters.
- [Boid Flocking: Emergent Collective Motion](/lab/swarm-intelligence/boid-flocking): Simulate Reynolds' boid flocking — explore how three simple rules (separation, alignment, cohesion) generate realistic bird-flock and fish-school behavior Formula: v_sep = -Σ(x_j − x_i) / |x_j − x_i|² for neighbors within r_sep. 4 parameters.
- [Firefly Algorithm: Light-Based Optimization](/lab/swarm-intelligence/firefly-algorithm): Visualize firefly agents synchronizing and converging on optima — explore how brightness attraction, light absorption, and randomness drive multi-modal optimization Formula: β(r) = β₀ × exp(−γ × r²) (attractiveness decays with distance). 4 parameters.
- [Particle Swarm Optimization on Function Landscapes](/lab/swarm-intelligence/particle-swarm): Visualize particles swarming across a 2D fitness landscape — explore how inertia, cognitive pull, and social pull guide the swarm toward global optima Formula: v_i ← w×v_i + c₁×r₁×(p_best_i − x_i) + c₂×r₂×(g_best − x_i). 4 parameters.

## cosmology (3 topics, 21 simulations)

### Astronomy (6 simulations)
- [Cosmic Microwave Background](/lab/astronomy/cosmic-microwave): Explore the cosmic microwave background radiation — the oldest light in the universe — and its blackbody spectrum that reveals conditions moments after the Big Bang. Formula: λ_max = 2.898 × 10⁻³ / T. 4 parameters.
- [Exoplanet Transit Detection](/lab/astronomy/exoplanet-transit): Simulate how astronomers detect exoplanets by measuring the tiny dip in starlight as a planet crosses in front of its host star. Formula: δ = (R_p / R_★)². 4 parameters.
- [Galaxy Rotation Curves & Dark Matter](/lab/astronomy/galaxy-rotation): Visualize how galaxy rotation curves deviate from Newtonian predictions, providing compelling evidence for dark matter halos surrounding galaxies. Formula: v(r) = √(G × M(r) / r). 4 parameters.
- [Hertzsprung-Russell Diagram](/lab/astronomy/hr-diagram): Explore the HR diagram — the astronomer's most powerful tool. Plot hundreds of stars by temperature and luminosity, and watch how cluster age determines the main sequence turnoff point. Formula: Turnoff mass ~ (10 / age_Gyr)^0.4 solar masses. 4 parameters.
- [Hubble's Law & Universe Expansion](/lab/astronomy/hubble-expansion): Explore how galaxies recede faster with distance, revealing the expansion of the universe through Hubble's Law and the cosmological redshift. Formula: v = H₀ × d. 4 parameters.
- [Star Life Cycle](/lab/astronomy/stellar-evolution): Watch a star evolve from birth to death. Adjust mass and metallicity to see how different stars live, age, and ultimately end as white dwarfs, neutron stars, or black holes. Formula: Lifetime ~ 10 * (M_sun / M)^2.5 billion years. 3 parameters.

### Black Holes (4 simulations)
- [Gravitational Lensing Simulator](/lab/black-holes/gravitational-lensing): Visualize how massive objects bend light, creating Einstein rings and gravitational arcs from distant sources Formula: θE = √(4GM·DLS/(c²·DL·DS)). 4 parameters.
- [Hawking Radiation Simulator](/lab/black-holes/hawking-radiation): Explore how black holes emit thermal radiation and slowly evaporate, with temperature, luminosity, and lifetime calculations Formula: T = ℏc³ / (8πGMkB). 1 parameters.
- [Schwarzschild Radius Calculator](/lab/black-holes/schwarzschild-radius): Interactive simulator for calculating the event horizon size of a black hole using the Schwarzschild metric Formula: Rs = 2GM/c². 2 parameters.
- [Tidal Forces & Spaghettification Simulator](/lab/black-holes/tidal-forces): Visualize how tidal forces near a black hole stretch and compress objects, leading to the phenomenon of spaghettification Formula: Δa = 2GMΔr/r³. 3 parameters.

### Fermi Paradox (11 simulations)
- [Civilization Self-Destruction Probability](/lab/fermi-paradox/self-destruction): Estimating the probability of a technological civilization's self-destruction across various scenarios Formula: P(survive) = (1 - P_nuke)(1 - P_ai)(1 - P_bio)(1 - P_climate)(1 - P_nano). 6 parameters.
- [Dark Forest Game Theory](/lab/fermi-paradox/dark-forest): Game theory: why civilizations may hide from each other in the cosmos Formula: P(survival) = (1 - P_aggression)^(N-1). 4 parameters.
- [Drake Equation Calculator](/lab/fermi-paradox/drake-equation): Interactive calculator for estimating the number of civilizations in the Galaxy using the Drake Equation Formula: N = R* × fp × ne × fl × fi × fc × L. 7 parameters.
- [Earth's Light Cone Visualization](/lab/fermi-paradox/light-cone): How far our radio signals have traveled into space in over 100 years Formula: R = c × t. 2 parameters.
- [Galaxy Colonization Simulation](/lab/fermi-paradox/galaxy-colonization): Visualization of a galactic colonization wave traveling at 0.1–2% the speed of light Formula: T = R_galaxy / (v × c). 4 parameters.
- [Grabby Aliens Expansion Model](/lab/fermi-paradox/grabby-aliens): Robin Hanson's model: how 'grabby' civilizations divide the universe among themselves Formula: Rate(t) ∝ t^n. 4 parameters.
- [Great Filter Probability Cascade](/lab/fermi-paradox/great-filter): Probability funnel: from dead matter to a starfaring civilization Formula: P(total) = P(life) × P(complex) × P(intelligence) × P(tech) × P(survive) × P(expand). 6 parameters.
- [Humanity's Radio Bubble Scale](/lab/fermi-paradox/radio-bubble): Scale visualization: the size of our radio bubble relative to the Galaxy and the universe Formula: R_bubble = c × t_radio ≈ 126 light-years. 2 parameters.
- [Kardashev Scale Slider](/lab/fermi-paradox/kardashev-scale): Interactive scale of civilizations classified by energy consumption level (Type I, II, III) Formula: K = (log₁₀P − 10) / 10. 2 parameters.
- [SETI Detection Probability (Monte Carlo)](/lab/fermi-paradox/seti-monte-carlo): Monte Carlo simulation: what is the probability of detecting an extraterrestrial signal with SETI? Formula: P(detect) = 1 - (1 - Ω/100)^N. 5 parameters.
- [Zoo Hypothesis Decision Tree](/lab/fermi-paradox/zoo-hypothesis): Decision tree: classifying hypotheses about why the aliens are silent Formula: P(zoo) = P(exist) × P(know) × P(care) × P(non-interference). 4 parameters.

## earth-science (3 topics, 11 simulations)

### Climate Science (4 simulations)
- [Carbon Cycle Simulator](/lab/climate-science/carbon-cycle): Model the global carbon cycle as a box model with atmosphere, ocean, land, and fossil fuel reservoirs. Explore how emission rates, ocean uptake, and deforestation determine atmospheric CO₂ concentrati Formula: \frac{dC_{\text{atm}}}{dt} = E - k_{\text{ocean}} \cdot (C_{\text{atm}} - C_{\text{eq}}) - k_{\text{land}} \cdot (C_{\text{atm}} - C_{\text{eq}}). 5 parameters.
- [Greenhouse Effect Simulator](/lab/climate-science/greenhouse-effect): Explore how CO₂ concentration, solar radiation, and albedo determine Earth's surface temperature through the greenhouse effect. Based on the Stefan-Boltzmann radiation law with optical depth parameter Formula: T_{\text{eff}} = \left(\frac{S(1-\alpha)}{4\sigma}\right)^{1/4}. 4 parameters.
- [Ice-Albedo Feedback Simulator](/lab/climate-science/ice-albedo-feedback): Visualize the ice-albedo positive feedback loop: warming melts ice, exposing dark ocean and land, which absorbs more radiation, causing further warming. Compare temperature trajectories with and witho Formula: \alpha(T) = \alpha_{\text{ice}} + (\alpha_{\text{land}} - \alpha_{\text{ice}}) \cdot \frac{T - T_{\text{ice}}}{T_{\text{melt}} - T_{\text{ice}}}. 4 parameters.
- [Sea Level Rise Simulator](/lab/climate-science/sea-level-rise): Model multi-component sea level rise from thermal expansion, glacier melt, and ice sheet disintegration. Visualize how warming scenarios translate to coastline flooding and understand why ice sheet co Formula: \text{SLR}_{\text{total}} = \Delta H_{\text{thermal}} + \Delta H_{\text{glaciers}} + \Delta H_{\text{ice sheets}}. 4 parameters.

### Earth Science (2 simulations)
- [Coriolis Effect & Weather Patterns Simulator](/lab/earth-science/coriolis-effect): Watch how Earth's rotation deflects moving objects — explore the Coriolis effect that creates hurricanes, trade winds, and the great ocean circulation patterns Formula: f = 2Ω sin(φ) (Coriolis parameter). 4 parameters.
- [Seismic Wave Propagation Simulator](/lab/earth-science/earthquake-waves): Trace P-waves and S-waves through Earth's layered interior — watch how seismic waves reveal the planet's hidden structure from crust to core Formula: M₀ = μAD (seismic moment: rigidity × fault area × slip distance). 4 parameters.

### Oceanography & Marine Science (5 simulations)
- [Coral Bleaching & Temperature Stress Simulator](/lab/oceanography/coral-bleaching): Model coral bleaching events — adjust sea surface temperature, duration, and recovery time to see threshold effects on reef health Formula: DHW = Σ max(SST - MMM - threshold_offset, 0) × Δt (°C-weeks). 4 parameters.
- [Global Ocean Circulation Patterns](/lab/oceanography/ocean-currents): Visualize global ocean currents driven by wind, Coriolis force, and temperature gradients — watch gyres form and heat move from equator to poles Formula: Ekman transport: τ = ρ · Cd · U²_wind (surface wind stress). 4 parameters.
- [Lunar Tidal Forces & Cycles](/lab/oceanography/tidal-forces-moon): Visualize how the Moon's gravity creates tidal bulges on Earth — explore spring tides, neap tides, and the 12.42-hour tidal cycle Formula: Tidal force: F_tidal = 2GMm·r / d³ (differential gravity). 4 parameters.
- [Ocean Wave Dynamics & Dispersion](/lab/oceanography/wave-dynamics): Model ocean wave propagation — adjust wind speed, fetch, and depth to see how waves form, grow, disperse, and break approaching shore Formula: Deep water phase speed: c = g·T/(2π) ≈ 1.56·T (m/s). 4 parameters.
- [Thermohaline Conveyor Belt Simulator](/lab/oceanography/thermohaline-circulation): Trace the global thermohaline circulation — watch dense cold water sink in the North Atlantic and travel the ocean floor for a thousand-year journey Formula: Seawater density: ρ(T,S,p) via UNESCO-80 equation of state. 4 parameters.

## earth-sciences (23 topics, 115 simulations)

### Biogeochemistry & Nutrient Cycles (5 simulations)
- [Global Carbon Cycle Model](/lab/biogeochemistry/carbon-cycle): Model Earth's carbon cycle — explore how fossil fuel emissions, photosynthesis, ocean absorption, and volcanism control atmospheric CO₂ and climate over decades to millennia Formula: dC_atm/dt = E_ff + E_def − F_ocean − F_land (carbon budget). 5 parameters.
- [Nitrogen Cycle & Fixation Model](/lab/biogeochemistry/nitrogen-cycle): Model the nitrogen cycle — explore biological fixation, nitrification, denitrification, and anthropogenic inputs that drive eutrophication and N₂O emissions Formula: N₂ + 8H⁺ + 8e⁻ + 16ATP → 2NH₃ + H₂ (biological fixation). 4 parameters.
- [Ocean Biological Carbon Pump](/lab/biogeochemistry/ocean-carbon-pump): Model the ocean biological carbon pump — explore how phytoplankton productivity, sinking particle flux, and deep ocean circulation sequester carbon from the atmosphere Formula: F_export = PP × e-ratio (export flux below mixed layer). 4 parameters.
- [Phosphorus Cycle & Weathering](/lab/biogeochemistry/phosphorus-cycle): Model the phosphorus cycle — explore weathering inputs, biological uptake, sedimentation, and anthropogenic mining that control phosphorus availability and eutrophication Formula: Ca₅(PO₄)₃(OH) → 5Ca²⁺ + 3PO₄³⁻ + OH⁻ (apatite weathering). 4 parameters.
- [Sulfur Cycle & Redox Chemistry](/lab/biogeochemistry/sulfur-cycle): Model the sulfur cycle — explore volcanic emissions, biological reduction, acid rain formation, and ocean sulfate dynamics across oxidation states from S²⁻ to SO₄²⁻ Formula: SO₂ + H₂O + ½O₂ → H₂SO₄ (acid rain formation). 4 parameters.

### Dendrochronology & Tree-Ring Science (5 simulations)
- [Annual Ring Width & Climate Proxy](/lab/dendrochronology/ring-width): Simulate annual tree-ring growth as a function of temperature, precipitation, and tree age — explore how climate signals are encoded in ring widths Formula: RW = G_max × min(f(T), f(P)) × exp(-k × age) × σ. 4 parameters.
- [Crossdating & Master Chronology](/lab/dendrochronology/crossdating): Simulate crossdating by sliding two ring-width series to find the optimal match position using correlation analysis Formula: r = Σ(xi - x̄)(yi - ȳ) / √(Σ(xi - x̄)² × Σ(yi - ȳ)²). 4 parameters.
- [Fire History from Scar Analysis](/lab/dendrochronology/fire-scar): Simulate fire scar frequency analysis — explore how fire return intervals, drought cycles, and suppression affect wildfire regimes recorded in tree rings Formula: MFI = Σ(intervals) / (number of intervals). 4 parameters.
- [Radiocarbon Calibration with Tree Rings](/lab/dendrochronology/radiocarbon-calibration): Simulate the calibration of radiocarbon dates using the tree-ring calibration curve to convert 14C years to calendar years Formula: A(t) = A₀ × exp(-λt), where λ = ln(2) / 5568. 4 parameters.
- [Temperature Reconstruction from Ring-Width Index](/lab/dendrochronology/climate-reconstruction): Simulate paleoclimate temperature reconstruction using linear regression transfer functions between ring-width indices and instrumental temperature Formula: T_recon = β₀ + β₁ × RWI (linear transfer function). 4 parameters.

### Environmental Science & Pollution (5 simulations)
- [Air Quality Index (AQI) Simulator](/lab/environmental-science/air-quality): Model air quality under varying pollutant concentrations — compute AQI from PM2.5, ozone, NO₂, and SO₂ levels and visualize health impact zones Formula: AQI_p = (I_hi - I_lo) / (BP_hi - BP_lo) × (C_p - BP_lo) + I_lo. 4 parameters.
- [Biodiversity Index Calculator](/lab/environmental-science/biodiversity-index): Calculate Shannon, Simpson, and species richness indices — visualize species abundance distributions and compare ecosystem health metrics Formula: H' = -Σᵢ pᵢ ln(pᵢ) (Shannon diversity index). 4 parameters.
- [Carbon Cycle & CO₂ Flux Simulator](/lab/environmental-science/carbon-cycle): Model the global carbon cycle — adjust emission rates, deforestation, and ocean uptake to see how atmospheric CO₂ evolves over decades Formula: dC_atm/dt = E_fossil + E_land - S_ocean - S_land. 4 parameters.
- [Soil Contamination & Plume Transport](/lab/environmental-science/soil-contamination): Simulate contaminant transport through soil — model advection, dispersion, and adsorption of pollutant plumes in groundwater systems Formula: R × ∂C/∂t = D × ∂²C/∂x² - v × ∂C/∂x (advection-dispersion with retardation). 4 parameters.
- [Water Treatment Process Simulator](/lab/environmental-science/water-treatment): Design a multi-stage water treatment plant — adjust coagulation, filtration, and disinfection parameters to see removal efficiency for turbidity, pathogens, and contaminants Formula: η_coag = 1 - exp(-k × dose / NTU₀) (coagulation efficiency). 4 parameters.

### Fluvial Geomorphology & River Dynamics (5 simulations)
- [Alluvial Fan Deposition & Morphology](/lab/fluvial-geomorphology/alluvial-fan): Simulate alluvial fan formation — explore how sediment supply, discharge, and slope control fan radius, gradient, and depositional patterns Formula: R ∝ (Q × Qs)^0.5 / S (fan radius scaling). 4 parameters.
- [Hydraulic Geometry & Channel Scaling](/lab/fluvial-geomorphology/hydraulic-geometry): Simulate hydraulic geometry relationships — explore how discharge controls channel width, depth, and velocity through power-law scaling Formula: w = a × Q^b (width-discharge relation). 4 parameters.
- [Knickpoint Retreat & Bedrock Incision](/lab/fluvial-geomorphology/knickpoint-retreat): Simulate knickpoint retreat — explore how bedrock strength, stream power, and drainage area control the upstream migration of waterfalls and river knickpoints Formula: E = K × A^m × S^n (stream power incision model). 4 parameters.
- [Meander Migration & Channel Evolution](/lab/fluvial-geomorphology/meander-migration): Simulate river meander migration — explore how flow velocity, bank erodibility, and channel curvature drive lateral channel movement and oxbow lake formation Formula: M = K × V × κ (lateral migration rate). 4 parameters.
- [Stream Power & Erosion Potential](/lab/fluvial-geomorphology/stream-power): Simulate stream power — explore how discharge, channel slope, and width determine a river's erosive capacity and sediment transport potential Formula: Ω = ρgQS (total stream power, W/m). 4 parameters.

### Geomorphology & Landscape Evolution (5 simulations)
- [Alluvial Fan Formation & Sediment Deposition](/lab/geomorphology/alluvial-fan): Simulate alluvial fan growth — explore how sediment supply, water discharge, and slope control fan morphology, radius, and depositional patterns Formula: R = (3V / (π·tan(S)))^(1/3)  [Fan radius from volume]. 4 parameters.
- [Coastal Erosion & Cliff Retreat](/lab/geomorphology/coastal-erosion): Simulate wave-driven coastal erosion — explore how wave energy, rock strength, and sea level control cliff retreat rates and shore platform development Formula: P = ρg²H²T / (32π)  [Wave energy flux, W/m]. 4 parameters.
- [Glacial Erosion & U-Valley Formation](/lab/geomorphology/glacial-landforms): Simulate glacial valley carving — explore how ice thickness, sliding velocity, and bedrock hardness produce characteristic U-shaped valley cross-sections Formula: E = K_g × u_b^l × (ρ_i·g·H·sinα)^m  [Glacial erosion law]. 4 parameters.
- [Hillslope Diffusion & Soil Creep](/lab/geomorphology/hillslope-diffusion): Simulate hillslope evolution via diffusive soil creep — explore how diffusivity, slope, and uplift shape hill profiles over geological time Formula: ∂z/∂t = κ·∂²z/∂x² + U  [Hillslope diffusion equation]. 4 parameters.
- [River Erosion & Channel Incision](/lab/geomorphology/river-erosion): Simulate stream power erosion — explore how river discharge, slope, and rock erodibility control channel incision rates and valley formation Formula: E = K × A^m × S^n  [Stream power incision law]. 4 parameters.

### Geophysics & Earth Structure (5 simulations)
- [Earth's Magnetic Field & Geodynamo](/lab/geophysics/magnetic-field): Visualize Earth's dipolar magnetic field, explore how core convection speed, core radius, and conductivity affect field strength, and observe field line geometry from pole to equator. Formula: B_r = (2μ₀ M cosθ) / (4π r³), B_θ = (μ₀ M sinθ) / (4π r³). 4 parameters.
- [Gravity Anomaly Modeling](/lab/geophysics/gravity-anomaly): Model Bouguer and free-air gravity anomalies over subsurface density contrasts, visualizing how buried structures like ore bodies, basins, and plutons create measurable gravitational signatures. Formula: Δg = (4/3)π G Δρ R³ z / (x² + z²)^(3/2) (buried sphere). 4 parameters.
- [Isostatic Equilibrium](/lab/geophysics/isostasy): Explore Airy and Pratt isostasy models showing how continental and oceanic crust float on the denser mantle, with adjustable crustal thickness, density, and surface topography. Formula: h = T(ρ_m − ρ_c)/ρ_m − T_ref(ρ_m − ρ_c)/ρ_m (Airy elevation). 4 parameters.
- [Mantle Convection Simulation](/lab/geophysics/mantle-convection): Simulate thermally driven convection in Earth's mantle, exploring how Rayleigh number, viscosity, and internal heating control convection cell geometry, surface heat flow, and plate velocities. Formula: Ra = α g ΔT D³ / (κ ν). 4 parameters.
- [Seismic Wave Propagation](/lab/geophysics/seismic-waves): Visualize P-wave and S-wave propagation through Earth's layered interior, showing refraction, reflection, and shadow zones created by the liquid outer core. Formula: V_P = sqrt((K + 4G/3) / ρ), V_S = sqrt(G / ρ). 4 parameters.

### Glaciology & Ice Dynamics (5 simulations)
- [Glacier Flow & Ice Deformation](/lab/glaciology/glacier-flow): Simulate glacier flow dynamics — explore how ice thickness, slope, temperature, and Glen's flow law govern the creep and sliding of glaciers through mountain valleys Formula: u_s = (2A/(n+1)) × (ρg sinα)^n × H^{n+1}, n = 3. 4 parameters.
- [Ice Core Paleoclimate Analysis](/lab/glaciology/ice-core-analysis): Simulate ice core analysis — explore how trapped gas bubbles and isotope ratios in Antarctic and Greenland ice cores reveal 800,000 years of climate history Formula: age(z) ≈ (H/A) × ln(H/(H-z)) (Nye model). 4 parameters.
- [Iceberg Calving Dynamics](/lab/glaciology/iceberg-calving): Simulate iceberg calving from tidewater glaciers — explore how ice cliff height, water depth, crevasse penetration, and ocean temperature control the fracture and release of icebergs Formula: c = k × H_w × (d_c / H_total) (empirical calving law). 4 parameters.
- [Permafrost Thaw & Carbon Release](/lab/glaciology/permafrost-thaw): Simulate permafrost thaw dynamics — explore how rising temperatures, active layer deepening, and thermokarst formation drive carbon and methane release from frozen ground Formula: ALT = k × √(DDT), DDT = ∑max(T_daily - 0, 0). 4 parameters.
- [Sea Ice Extent & Polar Feedback](/lab/glaciology/sea-ice-extent): Simulate Arctic and Antarctic sea ice extent — explore how temperature, albedo feedback, ocean heat transport, and seasonal cycles govern polar ice coverage Formula: E(T,m) = E_0 × (1 - kΔT) × [1 + A_s×cos(2π(m-3)/12)]. 4 parameters.

### Hydrology & Water Systems (5 simulations)
- [Evapotranspiration & Water Budget](/lab/hydrology/evapotranspiration): Simulate evapotranspiration rates using the Penman-Monteith method — explore temperature, solar radiation, wind speed, and humidity to estimate water loss from vegetated surfaces Formula: ET₀ = [0.408Δ(Rn - G) + γ(900/(T+273))u₂(es - ea)] / [Δ + γ(1 + 0.34u₂)]. 4 parameters.
- [Flood Wave Routing & Attenuation](/lab/hydrology/flood-routing): Simulate flood wave propagation through a river channel — explore inflow peak, channel storage, travel time, and weighting factor to predict downstream hydrograph attenuation Formula: S = K × [X × I + (1 - X) × O] (Muskingum storage equation). 4 parameters.
- [Groundwater Flow & Aquifer Dynamics](/lab/hydrology/groundwater-flow): Simulate groundwater flow through porous media — explore hydraulic conductivity, head gradient, porosity, and aquifer thickness to visualize Darcy flow and drawdown cones Formula: q = K × i (Darcy velocity, m/day). 4 parameters.
- [Integrated Watershed Model](/lab/hydrology/watershed-model): Simulate an integrated watershed water balance — explore precipitation, land use, soil type, and slope to track partitioning between runoff, infiltration, ET, and baseflow Formula: P = Q + ET + ΔS (water balance). 4 parameters.
- [Rainfall-Runoff Response Model](/lab/hydrology/rainfall-runoff): Simulate how precipitation transforms into surface runoff — explore rainfall intensity, soil infiltration capacity, curve number, and basin lag to predict hydrograph shape Formula: S = (25400 / CN) - 254 (maximum retention, mm). 4 parameters.

### Isotope Geochemistry (5 simulations)
- [Cosmogenic Nuclide Exposure Dating](/lab/isotope-geochemistry/cosmogenic-nuclide): Simulate cosmogenic nuclide production and exposure age calculation — explore how cosmic ray flux, altitude, latitude, and erosion rate control surface exposure dating Formula: P(z) = P₀ × exp(−ρz / Λ) (depth-dependent production). 4 parameters.
- [Isochron Method & Age Determination](/lab/isotope-geochemistry/isochron-method): Simulate the isochron method — explore how multiple samples with different Rb/Sr ratios define a line whose slope reveals the rock's age Formula: ⁸⁷Sr/⁸⁶Sr = (⁸⁷Sr/⁸⁶Sr)₀ + ⁸⁷Rb/⁸⁶Sr × (e^(λt) − 1). 4 parameters.
- [Isotope Mixing Models](/lab/isotope-geochemistry/isotope-mixing): Simulate isotopic mixing between two or three end-members — explore how mixing proportions and end-member compositions produce hyperbolic mixing arrays in isotope space Formula: δ_mix = f × δ_B + (1 − f) × δ_A (linear mixing for ratios). 5 parameters.
- [Radiometric Dating & Decay Curves](/lab/isotope-geochemistry/radiometric-dating): Simulate radioactive decay and age determination — explore how half-life, parent-daughter ratios, and decay constants constrain the ages of rocks and minerals Formula: N(t) = N₀ × e^(−λt), where λ = ln(2) / t½. 4 parameters.
- [Stable Isotope Fractionation](/lab/isotope-geochemistry/stable-isotope-fractionation): Simulate equilibrium and kinetic isotope fractionation — explore how temperature, mass difference, and reaction kinetics partition stable isotopes between phases Formula: α = R_A / R_B (fractionation factor between phases). 4 parameters.

### Limnology & Freshwater Science (5 simulations)
- [Dissolved Oxygen Depth Profile](/lab/limnology/oxygen-profile): Model how dissolved oxygen varies with depth in a stratified lake, from supersaturated surface waters to anoxic hypolimnion Formula: DO(z,t) = DO_sat(T) - BOD·t for z > z_thermocline (hypolimnion depletion). 4 parameters.
- [Eutrophication & Algal Bloom Dynamics](/lab/limnology/eutrophication): Model nutrient loading from agricultural runoff and watch algal blooms develop, collapse, and create oxygen dead zones in a lake Formula: TP = L_P / (z̄ · (ρ + σ)) — Vollenweider steady-state P model. 4 parameters.
- [Lake Sediment Core Dating & Proxy Records](/lab/limnology/sediment-core): Explore how lake sediment layers record environmental history through radiometric dating, pollen, diatoms, and geochemical proxies Formula: A(z) = A₀ · exp(-λ · z / SAR) where λ(²¹⁰Pb) = 0.03114 yr⁻¹. 4 parameters.
- [Lake Thermal Stratification & Turnover](/lab/limnology/thermal-stratification): Visualize how temperature layers form in lakes through seasonal heating and wind mixing, including spring and fall turnover events Formula: ρ(T) = 999.842594 + 0.06793952T - 0.009095290T² + ... (UNESCO equation of state). 4 parameters.
- [Lake Water Balance & Residence Time](/lab/limnology/water-budget): Calculate a lake's water budget from precipitation, evaporation, inflow, and outflow — and determine how long water stays in the lake Formula: ΔV/Δt = P·A + Q_in + G_in - E·A - Q_out - G_out. 4 parameters.

### Meteorology & Weather Systems (5 simulations)
- [Atmospheric Pressure & Barometric Formula Simulator](/lab/meteorology/atmospheric-pressure): Explore how atmospheric pressure decreases with altitude using the barometric formula, and understand the physics behind weather systems Formula: Barometric formula: P(h) = P₀ × (1 - Lh/T₀)^(gM/(RL)). 4 parameters.
- [Cloud Classification & Altitude Simulator](/lab/meteorology/cloud-types): Explore the 10 cloud genera classified by altitude and formation process — from low stratus to towering cumulonimbus Formula: Lifting condensation level: LCL ≈ 125 × (T - T_d) metres. 4 parameters.
- [Cold & Warm Front Dynamics Simulator](/lab/meteorology/weather-fronts): Visualize the collision of cold and warm air masses, observe frontal lifting, and see how precipitation forms along weather fronts Formula: Frontal lifting rate: w = v_front × sin(θ) where θ is the frontal slope angle. 4 parameters.
- [Hurricane Structure & Saffir-Simpson Scale Simulator](/lab/meteorology/hurricane-structure): Explore hurricane anatomy from eye to rainbands, adjust sea-surface temperature and wind shear to see how hurricanes intensify or weaken Formula: Maximum potential intensity: V_max = sqrt((C_k/C_d) × (SST - T_outflow)/T_outflow × (h_s* - h_a)). 4 parameters.
- [Tornado Formation & Mesocyclone Simulator](/lab/meteorology/tornado-formation): Watch a mesocyclone develop from wind shear and updraft interaction, and see how tornadoes form from rotating supercell thunderstorms Formula: Maximum updraft speed: w_max = sqrt(2 × CAPE). 4 parameters.

### Ocean Acoustics & Underwater Sound (5 simulations)
- [Ocean Acoustic Tomography](/lab/ocean-acoustics/acoustic-tomography): Simulate acoustic tomography of the ocean — send sound pulses between transceiver pairs and reconstruct temperature fields from travel time perturbations Formula: t₀ = L / c₀ (baseline travel time). 4 parameters.
- [Ocean Ambient Noise Spectrum](/lab/ocean-acoustics/ambient-noise): Simulate the ocean ambient noise spectrum — explore how wind speed, shipping density, sea state, and rain contribute to the underwater noise floor across frequency Formula: NL_wind(f) = 50 + 7.5√(V_w) + 20log₁₀(f/1000) × (−17 dB/decade slope). 4 parameters.
- [SOFAR Channel & Deep Sound Propagation](/lab/ocean-acoustics/sofar-channel): Simulate the SOFAR (Sound Fixing and Ranging) channel — explore how the ocean's sound speed minimum traps acoustic energy and allows propagation over thousands of kilometers Formula: c(z) = 1449 + 4.6T − 0.055T² + 0.0003T³ + (1.39−0.012T)(S−35) + 0.017z. 4 parameters.
- [Sonar Equation & Detection Range](/lab/ocean-acoustics/sonar-equation): Simulate the sonar equation — balance source level, transmission loss, target strength, noise level, and detection threshold to determine maximum sonar detection range Formula: SE = SL − 2TL + TS − NL − DT (active sonar equation). 5 parameters.
- [Underwater Acoustic Ray Tracing](/lab/ocean-acoustics/ray-tracing): Simulate acoustic ray tracing in a layered ocean — launch sound rays at various angles and watch them refract through the sound speed profile, revealing shadow zones and convergence zones Formula: cos(θ)/c(z) = cos(θ₀)/c(z_s) = ξ (Snell's law, constant along ray). 4 parameters.

### Paleoceanography & Ocean History (5 simulations)
- [CCD Depth & Lysocline](/lab/paleoceanography/carbonate-compensation): Explore the carbonate compensation depth — the ocean floor below which calcium carbonate dissolves faster than it accumulates, controlled by CO₂, temperature, and ocean chemistry Formula: Ω = [Ca²⁺][CO₃²⁻] / Ksp(T,P) (calcite saturation state). 4 parameters.
- [Eustatic Sea Level Reconstruction](/lab/paleoceanography/sea-level-curve): Reconstruct eustatic sea-level curves from ice volume, thermal expansion, and tectonic factors — visualize how global sea level has changed over geological time Formula: ΔSL_ice = -(Vi/100 - 1) × 130 m (ice volume to sea level conversion). 4 parameters.
- [Oceanic Anoxic Events & Black Shale](/lab/paleoceanography/ocean-anoxia): Model the conditions that trigger oceanic anoxic events — explore how warming, nutrient loading, and circulation stagnation create oxygen-free oceans and deposit organic-rich black shales Formula: O₂_sat(T) = 475 - 8.5 × T µmol/kg (oxygen solubility vs temperature). 4 parameters.
- [Thermohaline Conveyor Belt & Shutdown Events](/lab/paleoceanography/thermohaline-history): Model the Atlantic thermohaline circulation and its sensitivity to freshwater forcing — explore how meltwater pulses triggered abrupt climate events like the Younger Dryas Formula: Ψ_AMOC = k × (α × ΔT - β × ΔS) (Stommel two-box model). 4 parameters.
- [δ18O & Ice Volume Proxy](/lab/paleoceanography/oxygen-isotope): Simulate how oxygen isotope ratios in foraminifera shells record past ice volume and ocean temperature — explore the relationship between δ18O, global ice sheets, and deep-sea temperatures Formula: T(°C) = 16.9 - 4.38 × (δ18O_calcite - δ18O_seawater) (Epstein equation). 4 parameters.

### Paleoclimate & Ancient Atmospheres (5 simulations)
- [CO₂ Proxy Reconstruction from Ice Cores](/lab/paleoclimate/co2-proxy): Reconstruct atmospheric CO₂ from ice core data — explore how bubble depth, temperature, and accumulation rate affect the CO₂ record over 800,000 years Formula: CO₂(t) = CO₂₀ + A·sin(2πt/100 kyr) + secondary orbital terms. 4 parameters.
- [Glacial-Interglacial Cycles & Ice Volume](/lab/paleoclimate/ice-age-cycles): Model the growth and retreat of continental ice sheets — explore how orbital forcing, CO₂ feedback, and albedo combine to produce glacial-interglacial oscillations Formula: dV/dt = -V/τ + λ·F(t) + β·CO₂(V) + α·A(V). 4 parameters.
- [Milankovitch Cycles & Orbital Forcing](/lab/paleoclimate/milankovitch-cycles): Visualize Earth's orbital variations — eccentricity, obliquity, and precession — and their combined effect on insolation patterns over hundreds of thousands of years Formula: Q ∝ S₀ / π · (1 + e·sin(ω))² / Math.pow(1 - e², 1.5). 4 parameters.
- [Thermohaline Ocean Circulation & AMOC](/lab/paleoclimate/ocean-circulation): Model the Atlantic Meridional Overturning Circulation — explore how salinity, temperature, and freshwater forcing control deep water formation and heat transport Formula: Ψ ∝ α·ΔT - β·ΔS (Stommel density-driven flow). 4 parameters.
- [Volcanic Winter & Aerosol Cooling](/lab/paleoclimate/volcanic-winter): Simulate the climate impact of major volcanic eruptions — explore how sulfur emissions, plume height, and latitude affect global temperature through stratospheric aerosol forcing Formula: ΔT ≈ -k × (SO₂/20 Mt) × (h/25 km) × cos(φ·π/180). 4 parameters.

### Paleomagnetism & Ancient Magnetic Fields (5 simulations)
- [Apparent Polar Wander Path](/lab/paleomagnetism/apparent-polar-wander): Trace apparent polar wander paths — explore how plate velocity, rotation rate, continent latitude, and geological time combine to produce the sweeping pole tracks that prove continental drift Formula: d = v × Δt (plate displacement). 4 parameters.
- [Declination & Inclination Stereographic Projection](/lab/paleomagnetism/declination-inclination): Visualize paleomagnetic field directions on a stereographic projection — explore how declination, inclination, latitude, and Fisher precision parameter define field direction distributions and confide Formula: tan I = 2 tan λ (dipole formula). 4 parameters.
- [Geomagnetic Polarity Reversal](/lab/paleomagnetism/magnetic-reversal): Simulate geomagnetic field reversals — explore how dipole strength, reversal rate, transition duration, and field asymmetry govern the polarity flip recorded in volcanic rocks and ocean sediments Formula: B = μ₀ × M / (4π × R³). 4 parameters.
- [Paleointensity & Ancient Field Strength](/lab/paleomagnetism/paleointensity): Simulate paleointensity determination — explore how NRM/TRM ratios, laboratory field strength, thermal alteration, and the Arai plot diagram reveal the strength of Earth's ancient magnetic field Formula: Banc = |slope| × Blab (Thellier equation). 4 parameters.
- [Virtual Geomagnetic Pole (VGP)](/lab/paleomagnetism/virtual-geomagnetic-pole): Compute virtual geomagnetic poles from paleomagnetic directions — explore how site latitude, declination, and inclination map to pole positions on the globe using the geocentric axial dipole hypothesi Formula: p = arctan(2 / tan I). 4 parameters.

### Paleoseismology & Ancient Earthquakes (5 simulations)
- [Earthquake Recurrence Interval Simulator](/lab/paleoseismology/recurrence-interval): Simulate earthquake recurrence statistics — explore how mean interval, coefficient of variation, elapsed time, and record length affect probability estimates and hazard forecasts Formula: P(t, W) = [F(t+W) - F(t)] / [1 - F(t)] (conditional probability). 4 parameters.
- [Fault Slip Rate Calculator & Simulator](/lab/paleoseismology/slip-rate): Simulate long-term fault slip rate determination — explore how offset features, dating uncertainty, geodetic rates, and fault geometry constrain tectonic loading and earthquake potential Formula: v = D / T (geologic slip rate, mm/yr). 4 parameters.
- [Fault Trench Stratigraphy Simulator](/lab/paleoseismology/fault-trenching): Simulate fault trench excavation — explore how displacement, sedimentation rate, number of events, and erosion affect the stratigraphic record of past earthquakes Formula: D_total = D × N (cumulative displacement). 4 parameters.
- [Liquefaction Evidence Analysis Simulator](/lab/paleoseismology/liquefaction-evidence): Simulate earthquake-induced liquefaction — explore how peak ground acceleration, soil density, water table depth, and sand layer thickness determine liquefaction potential and feature size Formula: CSR = 0.65 × (PGA/g) × (σ_v/σ'_v) × r_d. 4 parameters.
- [Paleomagnitude Estimation Simulator](/lab/paleoseismology/magnitude-estimation): Simulate paleoearthquake magnitude estimation — explore how surface displacement, rupture length, fault type, and depth constrain magnitude from geological evidence Formula: M = 6.69 + 0.74 × log₁₀(D) (Wells & Coppersmith, displacement). 4 parameters.

### Petrology & Rock Formation (5 simulations)
- [Igneous Crystallization & Bowen's Series](/lab/petrology/igneous-crystallization): Simulate magma cooling and watch minerals crystallize in Bowen's reaction series — explore how temperature, cooling rate, and silica content control igneous rock texture and composition Formula: φ = 1 - exp(-k × (T_liquidus - T) / ΔT). 4 parameters.
- [Metamorphic Grade & Facies Diagram](/lab/petrology/metamorphic-grade): Simulate pressure–temperature paths through metamorphic facies — explore how burial depth, geothermal gradient, and protolith composition determine metamorphic grade and mineral assemblages Formula: z_depth = P / (ρ_avg × g) ≈ P(GPa) × 30 km. 4 parameters.
- [Mineral Identification & Optical Properties](/lab/petrology/mineral-identification): Simulate polarized light microscopy — explore how birefringence, extinction angle, crystal system, and refractive index help identify rock-forming minerals in thin section Formula: Γ = δ × d (retardation in nm). 4 parameters.
- [Sediment Transport & Deposition](/lab/petrology/sediment-transport): Simulate sediment entrainment and deposition in a river channel — explore how flow velocity, grain size, channel slope, and water depth control erosion, transport, and sedimentation Formula: τ_bed = ρ_w × g × h × sin(S). 4 parameters.
- [The Rock Cycle: Transform & Recycle](/lab/petrology/rock-cycle): Simulate the complete rock cycle — explore how weathering rate, tectonic uplift, burial depth, and heat flow drive transformations between igneous, sedimentary, and metamorphic rocks Formula: T_burial = T_surface + (Q / k) × Z. 4 parameters.

### Plate Tectonics & Continental Drift (5 simulations)
- [Continental Drift: Plate Reconstruction & Pangaea](/lab/plate-tectonics/continental-drift): Reconstruct Earth's tectonic plates through deep time — watch continents assemble into Pangaea and drift apart to their present positions Formula: d = v × t (total displacement over time). 4 parameters.
- [Hotspot Volcanism: Mantle Plumes & Island Chains](/lab/plate-tectonics/hotspot-volcanism): Simulate a mantle plume creating a volcanic island chain as a tectonic plate drifts overhead — the Hawaiian model Formula: L = v × t (chain length from plate velocity and time). 4 parameters.
- [Mantle Convection: Rayleigh-Bénard Cells in the Mantle](/lab/plate-tectonics/mantle-convection): Simulate thermal convection in Earth's mantle — Rayleigh number, convection cell patterns, and heat transport from core to surface Formula: Ra = αgΔTD³/(κν) (Rayleigh number). 4 parameters.
- [Mid-Ocean Ridge: Seafloor Spreading & Magnetic Stripes](/lab/plate-tectonics/mid-ocean-ridge): Simulate seafloor spreading at a mid-ocean ridge — new crust formation, magnetic polarity reversals, and symmetric stripe patterns Formula: d(age) = d₀ + 350√(age) meters (depth-age relation). 4 parameters.
- [Subduction Zone: Oceanic-Continental Convergence](/lab/plate-tectonics/subduction-zone): Simulate an oceanic plate subducting beneath a continental plate — trench formation, volcanic arc, and Benioff zone seismicity Formula: h ≈ 2.32 × √(κ × age) (plate thickness from half-space cooling). 4 parameters.

### Sedimentology & Stratigraphy (5 simulations)
- [Compaction & Porosity: Athy's Law of Burial](/lab/sedimentology/compaction-porosity): Model porosity reduction with burial depth using Athy's exponential compaction law — understand how sediments transform into rock and predict reservoir quality at depth Formula: φ(z) = φ₀ × exp(-c × z) — Athy's law. 4 parameters.
- [Grain Settling: Stokes Law & Reynolds Number](/lab/sedimentology/grain-settling): Simulate particle settling through water using Stokes law — explore how grain size, density, and fluid viscosity control settling velocity and sediment sorting Formula: v_Stokes = (ρ_s - ρ_f) × g × d² / (18 × μ). 4 parameters.
- [Sediment Transport: Shields Parameter & Bedload](/lab/sedimentology/sediment-transport): Calculate the Shields parameter and critical shear stress for sediment entrainment — determine when a river or current begins to move sand, gravel, or cobbles along the bed Formula: θ = ρ_f × u*² / ((ρ_s - ρ_f) × g × d). 4 parameters.
- [Stratigraphic Column: Walther's Law & Facies](/lab/sedimentology/stratigraphic-column): Build a stratigraphic column by varying sea level and sediment supply — observe how laterally adjacent facies stack vertically according to Walther's law of facies succession Formula: Accommodation = dSL/dt + Subsidence. 4 parameters.
- [Turbidite Sequence: Bouma Divisions & Graded Bedding](/lab/sedimentology/turbidite-sequence): Model turbidite deposition showing the Bouma sequence — visualize how flow velocity decay produces graded bedding, parallel lamination, ripple cross-lamination, and hemipelagic drape Formula: U(t) = U₀ × exp(-t/τ) — velocity decay. 4 parameters.

### Seismology & Earthquake Science (5 simulations)
- [Earthquake Magnitude & Energy Calculator](/lab/seismology/earthquake-magnitude): Simulate earthquake magnitude scales — explore how seismogram amplitude, epicentral distance, depth, and frequency content determine Richter, moment, and body-wave magnitudes Formula: ML = log10(A) + 2.56 × log10(Δ) - 1.67. 4 parameters.
- [Fault Mechanics & Stress Accumulation](/lab/seismology/fault-mechanics): Simulate earthquake fault mechanics — explore how tectonic stress rate, fault friction, normal stress, and recurrence interval govern the seismic cycle and rupture potential Formula: τ_f = μ × σ_n — Coulomb failure criterion. 4 parameters.
- [Seismic Refraction & Ray Tracing](/lab/seismology/seismic-refraction): Simulate seismic wave refraction through layered Earth — explore how layer velocities, thicknesses, source offset, and interface dip control travel-time curves and head-wave arrivals Formula: θ_c = arcsin(V₁ / V₂). 4 parameters.
- [Seismogram Waveform Analysis](/lab/seismology/seismogram-analysis): Simulate a three-component seismogram — explore how epicentral distance, magnitude, focal depth, and station noise affect P-wave, S-wave, and surface wave arrivals Formula: t_P ≈ Δ / 6.0 (crustal P-wave approximation). 4 parameters.
- [Soil Liquefaction Risk Assessment](/lab/seismology/liquefaction): Simulate seismic liquefaction susceptibility — explore how earthquake magnitude, PGA, soil density, and water table depth determine factor of safety and liquefaction potential Formula: CSR = 0.65 × (σ_v / σ'_v) × a_max × r_d / MSF. 4 parameters.

### Soil Science & Pedology (5 simulations)
- [Soil Compaction Under Mechanical Load](/lab/soil-science/compaction-model): Model how applied pressure compacts soil — visualize pore space collapse, bulk density increase, and the Proctor compaction curve at varying moisture contents Formula: ρ_d = G_s · ρ_w / (1 + e). 4 parameters.
- [Soil Erosion by Water (USLE Model)](/lab/soil-science/erosion-model): Calculate annual soil loss using the Universal Soil Loss Equation — visualize how rainfall, slope, soil type, cover, and conservation practices interact to drive erosion Formula: A = R · K · LS · C · P (USLE). 4 parameters.
- [Soil Nutrient Cycle & Decomposition](/lab/soil-science/nutrient-cycle): Model the nitrogen and carbon cycles in soil — trace how organic matter decomposes, minerals are released, plants absorb nutrients, and microbes drive the whole system Formula: dC/dt = C_in - k · C_soil (first-order decay). 4 parameters.
- [Soil Texture Classification (USDA Triangle)](/lab/soil-science/soil-classification): Plot sand, silt, and clay percentages on the USDA texture triangle — instantly classify soil type and estimate hydraulic properties Formula: Sand% + Silt% + Clay% = 100%. 4 parameters.
- [Water Percolation & Infiltration Rate](/lab/soil-science/percolation-rate): Model water infiltration through soil layers — visualize the wetting front advancing through porous media with Darcy's law and the Green-Ampt model Formula: f(t) = K_s · (1 + ψ_f · Δθ / F(t)) (Green-Ampt). 4 parameters.

### Volcanology & Magma Dynamics (5 simulations)
- [Eruption Column & Plume Height](/lab/volcanology/eruption-column): Model volcanic eruption columns — explore how mass discharge rate, vent radius, exit velocity, and atmospheric conditions control plume height and umbrella cloud spread Formula: H = 0.236 × Q^0.25 (Sparks 1986 plume height). 4 parameters.
- [Lava Flow Advance & Cooling](/lab/volcanology/lava-flow): Simulate lava flow advance down a slope — model flow velocity, cooling rate, crust formation, and maximum runout distance as functions of effusion rate, viscosity, slope angle, and ambient conditions Formula: v = (ρ × g × sin(α) × h²) / (3 × η) (Jeffreys equation). 4 parameters.
- [Magma Viscosity & Rheology](/lab/volcanology/magma-viscosity): Simulate magma viscosity as a function of temperature, silica content, water content, and crystal fraction — visualize flow behavior from basaltic to rhyolitic compositions Formula: log₁₀(η) = A + B / (T − C), where A, B, C depend on composition (GRD 2008 model). 4 parameters.
- [Pyroclastic Density Currents](/lab/volcanology/pyroclastic-flow): Simulate pyroclastic density currents — model flow velocity, runout distance, dynamic pressure, and temperature as functions of collapse height, particle concentration, slope, and entrainment Formula: v_max = sqrt(2 × g × H₀ × Δρ/ρ_ambient) (energy conservation). 4 parameters.
- [Volcanic Explosivity Index (VEI)](/lab/volcanology/volcanic-explosivity): Calculate the Volcanic Explosivity Index from eruption volume, column height, duration, and mass discharge rate — compare with historical eruptions from Strómboli to Yellowstone Formula: VEI ≈ log₁₀(V_km³) + 4.02 (volume-based estimate). 4 parameters.

### Wildfire Science & Fire Behavior (5 simulations)
- [Crown Fire Transition & Torching Index](/lab/wildfire-science/crown-fire): Simulate crown fire initiation — explore how canopy base height, foliar moisture, and fireline intensity determine the transition from surface to crown fire Formula: I_crit = (0.010 × CBH × (460 + 25.9 × FMC))^1.5. 4 parameters.
- [Dead Fuel Moisture & Ignition Threshold](/lab/wildfire-science/fuel-moisture): Simulate dead fuel moisture response to temperature and humidity — explore how timelag classes and ignition thresholds determine fire danger Formula: EMC = 0.942 × RH^0.679 + 0.000499 × exp(0.1 × RH) + 0.18 × (21.1 − T) (Simard approximation). 4 parameters.
- [Fire Weather Index System](/lab/wildfire-science/fire-weather): Simulate the Canadian Fire Weather Index — explore how temperature, humidity, wind, and rain combine into composite fire danger ratings Formula: FFMC = 59.5 × (250 − m) / (147.2 + m) where m = fine fuel EMC. 4 parameters.
- [Gaussian Plume Smoke Transport](/lab/wildfire-science/smoke-dispersion): Simulate smoke dispersion from wildfire using a Gaussian plume model — explore how wind speed, atmospheric stability, and source strength shape downwind concentrations Formula: C(x,y,0) = Q / (2π × U × σ_y × σ_z) × exp(−y²/2σ_y²) × exp(−H²/2σ_z²). 4 parameters.
- [Rothermel Fire Spread Model](/lab/wildfire-science/fire-spread): Simulate wildfire spread rate using the Rothermel model — explore how wind speed, fuel load, moisture, and slope drive fire behavior Formula: R = IR × ξ × (1 + φ_w + φ_s) / (ρ_b × ε × Q_ig). 4 parameters.

## economics (2 topics, 10 simulations)

### Behavioral Economics (5 simulations)
- [Endowment Effect & WTP vs WTA](/lab/behavioral-economics/endowment-effect): Explore why owning something makes it feel more valuable — visualize the gap between willingness-to-pay and willingness-to-accept in Thaler's classic mug experiment Formula: WTA = item_value * loss_aversion * (1 - market_experience * 0.5) * (1 + ownership_duration * 0.01). 4 parameters.
- [Loss Aversion Experiment](/lab/behavioral-economics/loss-aversion): Experience Kahneman & Tversky's prospect theory viscerally — watch how losses loom larger than gains and fair gambles feel negative Formula: Prospect theory value: v(x) = x^alpha if x >= 0, -lambda * (-x)^alpha if x < 0. 4 parameters.
- [Mental Accounting](/lab/behavioral-economics/mental-accounting): See how identical dollars are treated differently based on their source — windfall vs earned money flows to different mental accounts in predictably irrational ways Formula: Windfall marginal propensity to consume (MPC): typically 0.6-0.9. 4 parameters.
- [Nudge Architecture](/lab/behavioral-economics/nudge-effects): See how default options, framing, and social proof dramatically shift behavior without restricting choice — the power of nudge architecture Formula: Default effect: P(opt-out) - P(opt-in) ≈ 40-60 percentage points. 4 parameters.
- [Social Proof & Information Cascades](/lab/behavioral-economics/social-proof-cascade): Watch how rational individuals ignoring their own information creates cascading herd behavior — sometimes converging on the wrong answer Formula: Bayesian update: P(A|signal,history) = P(signal|A) * P(A|history) / P(signal). 4 parameters.

### Finance & Markets (5 simulations)
- [Black-Scholes Option Pricing Model](/lab/finance/black-scholes): Calculate fair option prices using the Black-Scholes model — the formula that earned a Nobel Prize and transformed derivatives trading forever Formula: d1 = [ln(S/K) + (r + σ²/2)T] / (σ√T). 5 parameters.
- [Bond Yield Curve & Term Structure](/lab/finance/bond-yield-curve): Visualize and model the yield curve — how interest rates vary across bond maturities and what the shape signals about the economy Formula: Slope = Y(10Y) − Y(3M). 5 parameters.
- [Cryptocurrency Mining Profitability](/lab/finance/crypto-mining): Calculate whether crypto mining is profitable — factoring in hash rate, electricity costs, block rewards, difficulty, and hardware efficiency Formula: Daily Revenue = (hash_rate / network_hash_rate) × blocks_per_day × block_reward × price. 5 parameters.
- [Monte Carlo Simulation for Option Pricing](/lab/finance/monte-carlo-pricing): Price options by simulating thousands of random stock price paths — the brute-force method that can handle any payoff structure Formula: S(t+dt) = S(t) · exp((r − σ²/2)dt + σ√dt · Z). 5 parameters.
- [Value at Risk (VaR) Portfolio Analysis](/lab/finance/portfolio-var): Measure portfolio risk with Value at Risk — quantify the maximum expected loss at a given confidence level over a time horizon Formula: Parametric VaR = V · σ · z_α · √(T/252). 5 parameters.

## engineering (53 topics, 265 simulations)

### Additive Manufacturing & 3D Printing (5 simulations)
- [Binder Jetting & Saturation Control](/lab/additive-manufacturing/binder-jetting): Simulate binder jetting — control binder saturation, droplet spacing, layer thickness, and powder packing to optimize green part density before sintering Formula: S = V_binder / V_void × 100% (binder saturation). 4 parameters.
- [FDM Extrusion & Layer Deposition](/lab/additive-manufacturing/fdm-printing): Simulate fused deposition modeling — control nozzle temperature, layer height, print speed, and extrusion width to visualize thermoplastic filament deposition and interlayer bonding Formula: Q = h × w × v (volumetric flow rate, mm³/s). 4 parameters.
- [SLA Photopolymer Curing](/lab/additive-manufacturing/sla-curing): Model stereolithography UV curing — control laser power, exposure time, resin absorption, and layer thickness to optimize photopolymerization depth and resolution Formula: Cd = Dp × ln(E / Ec) (Jacobs cure depth equation). 4 parameters.
- [SLM Melt Pool Dynamics](/lab/additive-manufacturing/slm-melting): Model selective laser melting of metal powder — visualize melt pool formation, solidification, and how laser power, scan speed, and hatch spacing control density and microstructure Formula: E_v = P / (v × d × t) (volumetric energy density, J/mm³). 4 parameters.
- [Support Structure Optimization](/lab/additive-manufacturing/support-optimization): Optimize support structures for 3D printing — balance overhang angle, support density, material usage, and removal difficulty for any geometry Formula: Self-support angle: θ_critical ≈ arctan(layer_height / layer_width). 4 parameters.

### Aerospace Engineering & Flight (5 simulations)
- [Airfoil Lift & Drag Simulator](/lab/aerospace/lift-drag): Visualize lift and drag forces on an airfoil — adjust angle of attack, airspeed, and air density to see force vectors and the lift-to-drag ratio Formula: Lift: L = ½ × ρ × V² × S × Cl. 4 parameters.
- [Atmospheric Re-entry Heating Simulator](/lab/aerospace/reentry-heating): Model the extreme heating during atmospheric re-entry — adjust velocity, altitude, nose radius, and ballistic coefficient to see heat flux and temperature profiles Formula: Sutton-Graves convective heating: q = k × √(ρ∞ / Rn) × V³. 4 parameters.
- [Rocket Staging & Delta-V Calculator](/lab/aerospace/rocket-staging): Simulate multi-stage rocket performance — adjust exhaust velocity, mass ratios, and stage count to see delta-v accumulation and payload fraction Formula: Tsiolkovsky equation: Δv = Ve × ln(M0 / Mf). 4 parameters.
- [Supersonic Flow & Shock Wave Simulator](/lab/aerospace/supersonic-flow): Visualize supersonic flow past a wedge — adjust Mach number, wedge angle, and altitude to see oblique shocks, expansion fans, and downstream conditions Formula: θ-β-M relation: tan θ = 2 cot β × (M²sin²β − 1) / (M²(γ + cos 2β) + 2). 4 parameters.
- [Wing Planform & Aspect Ratio Designer](/lab/aerospace/wing-design): Design wing planforms — adjust span, sweep, taper, and aspect ratio to see how geometry affects lift distribution, induced drag, and structural loads Formula: Aspect ratio: AR = b² / S. 4 parameters.

### Antenna Design & Radiation Patterns (5 simulations)
- [Half-Wave Dipole Radiation Pattern](/lab/antenna-design/dipole-antenna): Simulate a half-wave dipole antenna — explore how frequency, length, and orientation shape the radiation pattern Formula: λ = c / f (wavelength from frequency). 4 parameters.
- [Horn Antenna: Aperture & Beamwidth](/lab/antenna-design/horn-antenna): Simulate a pyramidal horn antenna — explore how aperture dimensions and flare angle control beamwidth and gain Formula: G = 10 log₁₀(4π × ε_ap × A_w × A_h / λ²) dBi. 4 parameters.
- [Linear Antenna Array & Beam Steering](/lab/antenna-design/antenna-array): Simulate a linear phased array — explore how element count, spacing, and progressive phase shift steer and shape the beam Formula: AF = sin(Nψ/2) / (N × sin(ψ/2)), ψ = kd cosθ + β. 4 parameters.
- [Microstrip Patch Antenna & Bandwidth](/lab/antenna-design/patch-antenna): Simulate a rectangular microstrip patch antenna — explore how dimensions, substrate, and feed position control resonance and bandwidth Formula: W = c / (2f₀) × √(2 / (ε_r + 1)). 4 parameters.
- [Yagi-Uda Array: Gain & Directivity](/lab/antenna-design/yagi-uda): Simulate a Yagi-Uda antenna array — explore how element count, spacing, and length ratios control gain and front-to-back ratio Formula: G ≈ 10 × log₁₀(4.25 × N × d/λ) dBi (Kraus approximation). 4 parameters.

### Architecture & Structural Design (5 simulations)
- [Arch & Dome Force Analysis](/lab/architecture/arch-forces): Visualize how arches and domes redirect gravity loads into compression — adjust span, rise, and load to see thrust lines and force vectors Formula: Horizontal thrust H = w × L² / (8 × f) for parabolic arch under UDL. 4 parameters.
- [Building Solar Orientation & Daylighting](/lab/architecture/solar-orientation): Optimize building orientation for solar gain and daylighting — simulate sun paths across seasons and latitudes to maximize natural light and energy efficiency Formula: Solar declination δ = 23.44° × sin(360° × (284 + day) / 365). 4 parameters.
- [Golden Section in Architecture](/lab/architecture/golden-section): Explore the golden ratio (1.618) in architectural proportions — overlay golden rectangles, spirals, and Le Corbusier's Modulor on building facades Formula: φ = (1 + √5) / 2 ≈ 1.6180339887. 4 parameters.
- [Room Acoustics & Reverberation Simulator](/lab/architecture/acoustic-design): Calculate reverberation time (RT60) using the Sabine equation — adjust room dimensions, materials, and absorption to design spaces that sound right Formula: RT60 = 0.161 × V / A (Sabine equation, metric). 4 parameters.
- [Structural Load Distribution Simulator](/lab/architecture/structural-loads): Visualize how loads distribute through beams, columns, and supports — adjust span, load position, and material to see shear force and bending moment diagrams Formula: R_left = P × (L − a) / L for simply supported beam. 4 parameters.

### Astrodynamics & Spacecraft Orbits (5 simulations)
- [Gravity Assist & Planetary Slingshot](/lab/astrodynamics/gravity-assist): Simulate a gravity assist flyby — explore how approach velocity, periapsis distance, planet mass, and deflection angle determine the free delta-v gained from a planetary encounter Formula: δ = 2 × arcsin(1 / (1 + r_p × v∞² / μ_planet)). 4 parameters.
- [Hohmann Transfer Orbit](/lab/astrodynamics/hohmann-transfer): Simulate a Hohmann transfer between two circular orbits — explore how orbital radii, central body mass, and burn timing determine delta-v and transfer time Formula: Δv₁ = √(μ/r₁) × (√(2r₂/(r₁+r₂)) - 1). 4 parameters.
- [Lagrange Points & Three-Body Equilibrium](/lab/astrodynamics/lagrange-points): Simulate the five Lagrange points in a two-body gravitational system — explore how mass ratio, orbital radius, and perturbation velocity affect stability and halo orbits Formula: r_L1 ≈ R × (μ/3)^(1/3). 4 parameters.
- [Orbital Decay & Atmospheric Drag](/lab/astrodynamics/orbital-decay): Simulate orbital decay from atmospheric drag — explore how altitude, ballistic coefficient, solar activity, and spacecraft area determine orbital lifetime and re-entry timeline Formula: F_drag = ½ × ρ × v² × C_d × A. 4 parameters.
- [Orbital Rendezvous & Docking](/lab/astrodynamics/orbital-rendezvous): Simulate orbital rendezvous between two spacecraft — explore how phase angle, altitude difference, thrust timing, and relative velocity determine approach trajectory and docking conditions Formula: ω = √(μ / r³) [orbital angular rate]. 4 parameters.

### Biomass Energy & Biofuel Technology (5 simulations)
- [Anaerobic Digestion & Methane Yield](/lab/biomass-energy/biogas-digestion): Model anaerobic digestion of organic waste — explore how temperature, retention time, and feedstock composition affect biogas production and methane concentration Formula: Y_biogas = BMP × (1 - e^(-k × HRT)) (first-order kinetics). 4 parameters.
- [Biomass Pyrolysis & Biochar Production](/lab/biomass-energy/pyrolysis): Simulate thermal decomposition of biomass in the absence of oxygen — explore how temperature, heating rate, and residence time determine the yields of biochar, bio-oil, and syngas Formula: k(T) = A × exp(-Ea / RT) (Arrhenius pyrolysis kinetics). 4 parameters.
- [Cellulosic Ethanol Conversion](/lab/biomass-energy/bioethanol): Model cellulosic bioethanol production — explore how enzyme loading, fermentation temperature, and feedstock composition affect sugar yield and ethanol conversion efficiency Formula: Y_glucose = cellulose × 1.11 × hydrolysis_efficiency (mass factor from water). 4 parameters.
- [Syngas Gasification & Equivalence Ratio](/lab/biomass-energy/gasification): Model biomass gasification — explore how equivalence ratio, temperature, and steam-to-biomass ratio control syngas composition (H₂, CO, CO₂, CH₄) and cold gas efficiency Formula: C + ½O₂ → CO, ΔH = -111 kJ/mol (partial oxidation). 4 parameters.
- [Wood Pellet Boiler Efficiency](/lab/biomass-energy/pellet-combustion): Model wood pellet combustion in automated boilers — explore how excess air, pellet moisture, and boiler design affect thermal efficiency, emissions, and heat output Formula: NCV = HHV × (1 - MC) - 2.44 × MC (net calorific value, MJ/kg). 4 parameters.

### Biomedical Imaging & Diagnostics (5 simulations)
- [CT Back-Projection & Hounsfield Units](/lab/biomedical-imaging/ct-reconstruction): Simulate filtered back-projection in computed tomography — explore how projection count, filter kernel, and tissue density map to Hounsfield units Formula: HU = 1000 × (μ_tissue − μ_water) / μ_water. 4 parameters.
- [MRI Spin Echo & Relaxation Times](/lab/biomedical-imaging/mri-signal): Simulate MRI spin-echo signal formation — explore how T1, T2, TR, and TE control tissue contrast in magnetic resonance imaging Formula: S = M₀ × (1 − e^(−TR/T1)) × e^(−TE/T2). 4 parameters.
- [PET Tracer Kinetics & SUV](/lab/biomedical-imaging/pet-tracer): Simulate PET tracer uptake and clearance — explore how injection dose, body weight, uptake rate, and decay constant determine standardized uptake values Formula: SUV = (tissue activity [kBq/mL] × body weight [kg]) / injected dose [kBq]. 4 parameters.
- [Ultrasound Pulse-Echo & B-Mode Resolution](/lab/biomedical-imaging/ultrasound-imaging): Simulate ultrasound pulse-echo imaging — explore how frequency, depth, and tissue impedance affect B-mode resolution and penetration Formula: λ = c / f (wavelength in tissue, c ≈ 1540 m/s). 4 parameters.
- [X-ray Beer-Lambert Attenuation & Contrast](/lab/biomedical-imaging/xray-attenuation): Simulate X-ray attenuation through tissue layers — explore how photon energy, tissue thickness, and material density affect transmitted intensity and image contrast Formula: I = I₀ × e^(−μx) (Beer-Lambert exponential attenuation). 4 parameters.

### Chemical Engineering & Process Design (5 simulations)
- [Chemical Reaction Kinetics Simulator](/lab/chemical-engineering/reaction-kinetics): Visualize concentration vs. time profiles for a reversible reaction with Arrhenius temperature dependence in a CSTR or batch reactor Formula: k = A * exp(-Ea / (R * T)). 4 parameters.
- [Distillation Column Simulator](/lab/chemical-engineering/distillation): Simulate a binary distillation column with adjustable reflux ratio, feed composition, and tray count to visualize separation efficiency on a McCabe-Thiele diagram Formula: y = alpha * x / (1 + (alpha - 1) * x). 4 parameters.
- [Fluid Mixing & Stirred Tank Simulator](/lab/chemical-engineering/fluid-mixing): Visualize turbulent mixing in a stirred tank reactor with adjustable impeller speed, fluid viscosity, and baffle configuration Formula: Re = rho * N * D^2 / mu. 4 parameters.
- [Heterogeneous Catalyst Design Simulator](/lab/chemical-engineering/catalyst-design): Explore how catalyst particle size, porosity, surface area, and temperature affect reaction rate through effectiveness factor and Thiele modulus analysis Formula: phi = R * sqrt(k_s / D_eff). 4 parameters.
- [Shell-and-Tube Heat Exchanger Simulator](/lab/chemical-engineering/heat-exchanger): Design and visualize a counter-flow shell-and-tube heat exchanger with real-time temperature profiles and heat duty calculations Formula: epsilon = (1 - exp(-NTU*(1-Cr))) / (1 - Cr*exp(-NTU*(1-Cr))). 4 parameters.

### Combustion & Flame Dynamics (5 simulations)
- [Combustion Efficiency Analysis](/lab/combustion/combustion-efficiency): Simulate combustion efficiency — explore how equivalence ratio, excess air, fuel type, and exhaust temperature determine thermal losses and overall efficiency Formula: η_c = (Q_fuel − Q_stack − Q_unburnt) / Q_fuel. 4 parameters.
- [Detonation Wave Structure](/lab/combustion/detonation-wave): Simulate detonation wave propagation — explore Chapman-Jouguet velocity, pressure ratios, and cellular detonation structure in reactive gas mixtures Formula: D_CJ = √(2(γ²−1)·q + a₁²) (Chapman-Jouguet velocity). 4 parameters.
- [Flammability Limits & Explosion Envelopes](/lab/combustion/flammability-limits): Simulate flammability limits — explore how fuel concentration, temperature, pressure, and diluent gases define the boundaries where ignition is possible Formula: LFL(T) = LFL₂₅ × [1 − 0.000721 × (T − 298)]. 4 parameters.
- [Laminar Flame Speed Propagation](/lab/combustion/flame-speed): Simulate laminar flame speed — explore how equivalence ratio, initial temperature, pressure, and fuel type affect flame propagation velocity Formula: S_L = S_L0 × (T/T_ref)^α × (P/P_ref)^β. 4 parameters.
- [Thermodynamic Engine Cycle (Otto & Diesel)](/lab/combustion/engine-cycle): Simulate the Otto and Diesel thermodynamic engine cycles — explore how compression ratio, heat addition, and working fluid properties determine thermal efficiency and work output Formula: η_Otto = 1 − 1/r^(γ−1). 4 parameters.

### Control Systems & Feedback (5 simulations)
- [Bode Plot Analyzer](/lab/control-systems/bode-plot): Visualize frequency response of transfer functions — plot magnitude and phase Bode diagrams to analyze gain margin, phase margin, and bandwidth Formula: G(jω) = Kωn² / ((jω)² + 2ζωn(jω) + ωn²). 4 parameters.
- [Nyquist Stability Diagram](/lab/control-systems/nyquist-stability): Plot the Nyquist contour of an open-loop transfer function — count encirclements of the critical point to determine closed-loop stability Formula: G(jω) = K / ((1+jωτ₁)(1+jωτ₂)) · e^(-jωTd). 4 parameters.
- [PID Controller Tuner](/lab/control-systems/pid-controller): Tune a PID controller in real time — adjust proportional, integral, and derivative gains to minimize overshoot, settling time, and steady-state error Formula: u(t) = Kp·e(t) + Ki·∫e(τ)dτ + Kd·de(t)/dt. 4 parameters.
- [Root Locus Explorer](/lab/control-systems/root-locus): Trace how closed-loop poles move in the complex plane as gain varies — visualize stability boundaries, breakaway points, and asymptotic angles Formula: 1 + K·G(s)H(s) = 0 (characteristic equation). 4 parameters.
- [State-Space Simulator](/lab/control-systems/state-space): Explore state-space representations of dynamic systems — visualize state trajectories, eigenvalue placement, and controllability in the phase plane Formula: ẋ = Ax + Bu (state equation). 4 parameters.

### Corrosion Science & Protection (5 simulations)
- [Cathodic Protection System Design](/lab/corrosion-science/cathodic-protection): Simulate cathodic protection — explore how anode material, current density, soil resistivity, and structure surface area determine protection coverage and anode lifetime Formula: I_total = i_prot × A_structure. 4 parameters.
- [Corrosion Rate Prediction](/lab/corrosion-science/corrosion-rate): Simulate corrosion rate — explore how oxygen concentration, temperature, pH, and flow velocity affect uniform dissolution rate using the Butler-Volmer equation Formula: CR(mm/yr) = 3.27 × 10⁻³ × i_corr × M / (n × ρ). 4 parameters.
- [Galvanic Corrosion Between Dissimilar Metals](/lab/corrosion-science/galvanic-corrosion): Simulate galvanic corrosion — explore how electrode potential difference, electrolyte conductivity, area ratio, and temperature drive corrosion current between coupled metals Formula: i_corr = ΔE × σ / (1 + Ac/Aa) × k_T. 4 parameters.
- [Passivation Film Growth & Breakdown](/lab/corrosion-science/passivation-film): Simulate passive oxide film formation — explore how chromium content, pH, applied potential, and chloride concentration affect film stability and breakdown Formula: d = k × ln(1 + E/E₀) × (Cr%/12)  [High-field model]. 4 parameters.
- [Pitting Corrosion Initiation & Growth](/lab/corrosion-science/pitting-corrosion): Simulate pitting corrosion — explore how chloride concentration, temperature, alloy PREN, and applied potential control pit initiation probability and growth rate Formula: PREN = %Cr + 3.3 × %Mo + 16 × %N. 4 parameters.

### Desalination & Water Purification (5 simulations)
- [Electrodialysis Ion Transport Simulator](/lab/desalination/electrodialysis): Simulate electrodialysis desalination — explore how voltage, membrane pairs, feed salinity, and flow rate control ion removal and energy consumption Formula: i = κ × V / (N × d) (current density, A/m²). 4 parameters.
- [Forward Osmosis Draw Solution Simulator](/lab/desalination/forward-osmosis): Simulate forward osmosis desalination — explore how draw solution concentration, membrane properties, feed salinity, and temperature affect water flux and dilution Formula: J_w = A_w × (π_draw - π_feed) (water flux, simplified). 4 parameters.
- [Multi-Stage Flash Distillation Simulator](/lab/desalination/msf-distillation): Simulate MSF distillation — explore how number of stages, top brine temperature, feed salinity, and steam ratio affect freshwater yield and thermal efficiency Formula: GOR = m_distillate / m_steam. 4 parameters.
- [Reverse Osmosis Membrane Simulator](/lab/desalination/reverse-osmosis): Simulate reverse osmosis desalination — explore how feed pressure, salinity, membrane permeability, and recovery ratio affect freshwater production and energy consumption Formula: π = 0.7 × TDS/1000 (bar, approximation for NaCl). 4 parameters.
- [Solar Still Evaporation Simulator](/lab/desalination/solar-still): Simulate passive solar still desalination — explore how solar irradiance, basin depth, glass angle, and ambient temperature affect freshwater yield and thermal efficiency Formula: q_evap = h_evap × (P_w - P_g) (evaporative heat transfer). 4 parameters.

### Distillation & Separation Processes (5 simulations)
- [Azeotropic Distillation & Entrainer](/lab/distillation/azeotrope): Investigate azeotropic behavior in binary systems — visualize how positive deviations from Raoult's law create minimum-boiling azeotropes that limit ordinary distillation, and explore entrainer-based  Formula: γ_i = exp(A₁₂ · x_j²) (one-suffix Margules equation). 4 parameters.
- [Isothermal Flash Drum Calculation](/lab/distillation/flash-drum): Solve the isothermal flash problem for a binary mixture — determine vapor fraction, liquid and vapor compositions using the Rachford-Rice equation with animated phase envelope visualization Formula: Σ z_i(K_i − 1) / (1 + ψ(K_i − 1)) = 0 (Rachford-Rice). 4 parameters.
- [McCabe-Thiele Binary Distillation](/lab/distillation/mccabe-thiele): Construct a McCabe-Thiele diagram for binary distillation — step off theoretical stages between operating lines and the equilibrium curve to determine the number of ideal plates required for a given s Formula: y = (R·x + x_D) / (R + 1) (rectifying operating line). 4 parameters.
- [Raoult's Law & Ideal Solutions](/lab/distillation/raoults-law): Explore Raoult's law for ideal binary mixtures — visualize partial pressures, total pressure, and deviation from ideality as you adjust pure-component vapor pressures and composition Formula: p_i = x_i · P_i^sat (Raoult's law for ideal solutions). 4 parameters.
- [Vapor-Liquid Equilibrium T-xy Diagram](/lab/distillation/vle-diagram): Generate T-xy phase diagrams for binary mixtures using Antoine equation vapor pressures — visualize bubble-point and dew-point curves, tie lines, and the two-phase region Formula: log₁₀(P^sat) = A − B/(C + T) (Antoine equation). 4 parameters.

### Electrical Engineering (5 simulations)
- [Ohm's Law Circuit Simulator](/lab/electrical-engineering/ohms-law): Explore the fundamental relationship between voltage, current, and resistance — adjust any parameter and watch the others respond in real time Formula: I = V / R (Ohm's law). 4 parameters.
- [Operational Amplifier Configurations](/lab/electrical-engineering/op-amp): Explore inverting, non-inverting, and differential op-amp configurations — adjust feedback resistors to set precise gain and see the output waveform Formula: Av_inverting = -Rf / Rin. 5 parameters.
- [RC Circuit Charging & Discharging](/lab/electrical-engineering/rc-circuit): Watch a capacitor charge and discharge through a resistor — visualize the exponential time constant τ = RC that governs every timing circuit Formula: V_charge(t) = V₀ × (1 - e^(-t/RC)). 4 parameters.
- [RL Circuit Transient Response](/lab/electrical-engineering/rl-circuit): Simulate an inductor-resistor circuit to visualize current buildup, back-EMF, and the L/R time constant that governs inductive transients Formula: I_energize(t) = (V₀/R) × (1 - e^(-Rt/L)). 4 parameters.
- [Transistor Amplifier Gain](/lab/electrical-engineering/transistor-amplifier): Design a common-emitter BJT amplifier — adjust bias resistors and load to see voltage gain, operating point, and signal clipping in real time Formula: Av ≈ -Rc / Re (voltage gain with emitter degeneration). 5 parameters.

### Energy Systems & Power Engineering (5 simulations)
- [Battery Discharge Curves & Capacity](/lab/energy-systems/battery-discharge): Simulate battery voltage discharge curves under different loads — explore C-rates, Peukert's law, and how discharge rate affects usable capacity Formula: Peukert's law: C = I^k × t (k = Peukert exponent). 4 parameters.
- [Nuclear Reactor Control Rods & Criticality](/lab/energy-systems/nuclear-reactor): Simulate nuclear reactor control — adjust control rod insertion to maintain criticality and balance neutron multiplication for stable power output Formula: Multiplication factor: k = η × f × p × ε × P_NL × P_TNL (six-factor formula). 4 parameters.
- [Power Grid Frequency Stability](/lab/energy-systems/grid-frequency): Simulate power grid frequency regulation — balance generation and demand to keep the grid at 50/60 Hz and prevent blackouts Formula: Swing equation: 2H × df/dt = (P_supply - P_demand) / S_base. 4 parameters.
- [Solar Panel Efficiency & Angle Optimization](/lab/energy-systems/solar-panel-efficiency): Optimize solar panel tilt angle for maximum energy harvest — simulate irradiance, temperature effects, and seasonal variation across latitudes Formula: Irradiance on tilted surface: I = I₀ × cos(θ), where θ = angle of incidence. 4 parameters.
- [Wind Turbine Power & Betz Limit](/lab/energy-systems/wind-turbine): Simulate wind turbine power output and explore the Betz limit — the theoretical maximum of 59.3% energy extraction from wind Formula: Available wind power: P = ½ρAv³. 4 parameters.

### Ergonomics & Human Factors (5 simulations)
- [Ergonomic Workstation Design](/lab/ergonomics/workstation-design): Simulate workstation ergonomics — explore how user height, desk height, monitor distance, and chair settings affect posture angles and musculoskeletal comfort Formula: Optimal desk height = 0.41 × user_height. 4 parameters.
- [NIOSH Lifting Equation Calculator](/lab/ergonomics/lifting-equation): Simulate the NIOSH lifting equation — explore how load weight, horizontal distance, vertical height, and asymmetry angle determine the recommended weight limit and lifting index Formula: RWL = 23 × (25/H) × (1 - 0.003|V-75|) × (1 - 0.0032A). 4 parameters.
- [Repetitive Strain Injury Risk Assessment](/lab/ergonomics/repetitive-strain): Simulate repetitive strain injury risk — explore how repetition rate, force level, posture deviation, and recovery time determine cumulative trauma disorder probability Formula: SI = ISI × DSI × ESI × PSI × SSI × DDI. 4 parameters.
- [Thermal Comfort & PMV Model](/lab/ergonomics/thermal-comfort): Simulate thermal comfort using Fanger's PMV-PPD model — explore how air temperature, humidity, air velocity, and metabolic rate determine comfort perception and dissatisfaction percentage Formula: PMV = [0.303 × exp(-0.036M) + 0.028] × L. 4 parameters.
- [Visual Ergonomics & Display Optimization](/lab/ergonomics/visual-ergonomics): Simulate visual ergonomics — explore how viewing distance, text size, screen luminance, and ambient lighting affect readability, eye strain, and visual comfort Formula: Visual angle (arcmin) = 2 × arctan(h / 2D) × 3438. 4 parameters.

### Fiber Optics & Optical Communication (5 simulations)
- [EDFA Gain & Noise Figure](/lab/fiber-optics/fiber-amplifier): Simulate an erbium-doped fiber amplifier — explore gain, noise figure, and ASE noise as pump power and signal wavelength change Formula: G(dB) = 10 × log10(P_out / P_in). 4 parameters.
- [Fiber Attenuation & OTDR Trace](/lab/fiber-optics/fiber-attenuation): Simulate optical power loss along a fiber link — build loss budgets and visualize OTDR backscatter traces with splice and connector events Formula: P_out = P_in × 10^(-αL/10). 4 parameters.
- [Modal Dispersion & Pulse Broadening](/lab/fiber-optics/modal-dispersion): Simulate multimode pulse broadening — watch how different guided modes spread optical pulses over distance and limit bandwidth Formula: ΔT = (L × n₁ / c) × (n₁ - n₂) / n₂. 4 parameters.
- [Total Internal Reflection & Numerical Aperture](/lab/fiber-optics/total-internal-reflection): Simulate total internal reflection inside an optical fiber — explore critical angle, acceptance cone, and numerical aperture as core and cladding indices change Formula: θ_c = arcsin(n_clad / n_core). 4 parameters.
- [Wavelength-Division Multiplexing (WDM)](/lab/fiber-optics/wavelength-division): Simulate WDM channel packing — explore how channel spacing, channel count, and per-channel data rate determine total fiber capacity Formula: C_total = N_channels × R_per_channel. 4 parameters.

### Fire Engineering & Life Safety (5 simulations)
- [Axisymmetric Fire Plume Model](/lab/fire-engineering/fire-plume): Simulate buoyant fire plumes — explore how heat release rate, fire diameter, and ceiling height determine plume temperature, velocity, and entrainment using Heskestad correlations Formula: L = 0.235 Q̇^(2/5) − 1.02D (flame height, m). 4 parameters.
- [Evacuation Egress Time Calculator](/lab/fire-engineering/egress-time): Simulate building evacuation — explore how occupant load, exit width, travel distance, and pre-movement delay determine Required Safe Egress Time (RSET) Formula: RSET = t_detection + t_pre-movement + max(t_travel, t_flow). 4 parameters.
- [Flashover Prediction Model](/lab/fire-engineering/flashover-model): Simulate flashover conditions in a compartment — explore how ventilation, fuel load, and room geometry determine the critical heat release rate for flashover using Thomas and Babrauskas correlations Formula: Q̇_fo = 7.8Aₜ + 378Aᵥ√Hᵥ (Thomas correlation, kW). 4 parameters.
- [Smoke Layer Descent Model](/lab/fire-engineering/smoke-layer): Simulate smoke filling in a compartment — explore how fire size, room geometry, and ventilation determine the rate of hot smoke layer descent and available safe egress time Formula: ṁ_p = 0.071 Q_c^(1/3) z^(5/3) + 0.0018 Q_c (plume mass flow). 4 parameters.
- [Sprinkler Activation Response Model](/lab/fire-engineering/sprinkler-activation): Simulate automatic sprinkler activation — explore how RTI, ceiling jet temperature, sprinkler spacing, and fire growth rate determine activation time using the lumped-mass thermal model Formula: dTe/dt = (u^½ / RTI)(Tg − Te) (lumped mass model). 5 parameters.

### Geotechnical Engineering & Soil Mechanics (5 simulations)
- [1D Terzaghi Consolidation Settlement](/lab/geotechnical-engineering/consolidation): Simulate one-dimensional consolidation of clay layers under load — visualize pore pressure dissipation, settlement over time, and the degree of consolidation using Terzaghi's theory Formula: S = C_c · H · log₁₀((σ₀' + Δσ) / σ₀') / (1 + e₀). 4 parameters.
- [Axial Pile Load Capacity](/lab/geotechnical-engineering/pile-capacity): Estimate the axial load capacity of driven piles using static analysis — visualize tip resistance, skin friction distribution, and load transfer along the pile shaft Formula: Q_u = Q_b + Q_s (ultimate pile capacity). 4 parameters.
- [Rankine Earth Pressure on Retaining Walls](/lab/geotechnical-engineering/retaining-wall): Calculate active and passive earth pressures using Rankine theory — visualize pressure distributions, resultant forces, and the effect of soil properties on retaining wall design Formula: K_a = tan²(45° − φ/2). 4 parameters.
- [Soil Liquefaction & Cyclic Stress Analysis](/lab/geotechnical-engineering/liquefaction): Evaluate soil liquefaction potential under seismic loading — calculate the cyclic stress ratio (CSR), cyclic resistance ratio (CRR), and factor of safety against liquefaction using the simplified proc Formula: CSR = 0.65 · (a_max/g) · (σ_v/σ'_v) · r_d. 4 parameters.
- [Terzaghi Bearing Capacity of Shallow Foundations](/lab/geotechnical-engineering/bearing-capacity): Calculate the ultimate bearing capacity of shallow foundations using Terzaghi's classical equation — visualize failure surfaces, pressure bulbs, and the contributions of cohesion, surcharge, and soil  Formula: q_u = cN_c + γD_fN_q + 0.5γBN_γ. 4 parameters.

### Geothermal Energy Engineering (5 simulations)
- [Binary-Cycle ORC Power Plant](/lab/geothermal-energy/binary-cycle): Simulate a binary-cycle organic Rankine cycle plant — explore how brine temperature, working fluid, and condenser conditions affect power output and efficiency Formula: η_Carnot = 1 - (T_cold + 273) / (T_hot + 273). 4 parameters.
- [Enhanced Geothermal System (EGS) Stimulation](/lab/geothermal-energy/egs-stimulation): Simulate hydraulic stimulation of hot dry rock — explore how injection pressure, flow rate, rock stress, and fracture density create engineered geothermal reservoirs Formula: P_breakdown = σ₃ + T_tensile (fracture initiation). 4 parameters.
- [Geothermal Reservoir Thermal Model](/lab/geothermal-energy/reservoir-modeling): Simulate subsurface reservoir temperature — explore how depth, geothermal gradient, rock thermal conductivity, and fluid flow affect reservoir thermal state Formula: T_reservoir = T_surface + G × D. 4 parameters.
- [Heat Extraction Optimization](/lab/geothermal-energy/heat-extraction): Simulate geothermal heat extraction — explore how flow rate, reservoir permeability, well spacing, and injection temperature affect thermal output and reservoir life Formula: P_thermal = Q × ρ_w × c_pw × (T_prod - T_inj). 4 parameters.
- [Wellbore Heat Loss Model](/lab/geothermal-energy/wellbore-heat-loss): Simulate heat loss in geothermal wellbores — explore how depth, flow velocity, insulation, and casing design affect the temperature of produced fluid at the surface Formula: T(z) = T_res - (T_res - T_surface) × (1 - exp(-U×π×d×z / (m×c_p))). 4 parameters.

### Haptics Engineering (5 simulations)
- [Electrostatic Friction Display](/lab/haptics-engineering/electrostatic-friction): Simulate electrostatic friction modulation — explore how applied voltage, frequency, dielectric thickness, and finger contact mechanics control friction forces on touchscreen haptic displays Formula: F_electrostatic = ε₀ × εᵣ × A × V² / (2 × d²). 4 parameters.
- [Force Feedback Control Loop](/lab/haptics-engineering/force-feedback): Simulate a force-feedback haptic device — explore how stiffness, damping, virtual wall position, and control loop gain affect stability and force rendering fidelity Formula: F = K × (x - x_wall) + B × ẋ (virtual wall impedance). 4 parameters.
- [Haptic Texture Rendering](/lab/haptics-engineering/texture-rendering): Simulate haptic texture rendering — explore how spatial frequency, amplitude, scanning velocity, and friction modulation create the sensation of touching different surface textures Formula: f_temporal = v_scan / λ_spatial (Hz). 4 parameters.
- [Thermal Haptic Display](/lab/haptics-engineering/thermal-display): Simulate thermal haptic displays — explore how contact conductance, skin thermal properties, temperature difference, and contact area determine the perception of hot, cold, and material identity Formula: Contact temperature: T_c = (T₁e₁ + T₂e₂) / (e₁ + e₂). 4 parameters.
- [Vibrotactile Feedback Design](/lab/haptics-engineering/vibrotactile-feedback): Simulate vibrotactile waveform design — explore how frequency, amplitude, envelope shape, and duty cycle create distinct tactile sensations in haptic actuators Formula: Displacement: x = A / (2πf)² (from acceleration amplitude). 4 parameters.

### Heat Transfer & Thermal Analysis (5 simulations)
- [Convective Heat Transfer Coefficient Calculator](/lab/heat-transfer/convection-coefficient): Calculate convective heat transfer coefficients for forced and natural convection — explore how fluid velocity, surface geometry, and fluid properties determine cooling rates Formula: q = h × (T_surface − T_fluid) (Newton’s law of cooling). 4 parameters.
- [Extended Surface Fin Efficiency Analyzer](/lab/heat-transfer/fin-efficiency): Analyze fin performance — explore how fin length, thickness, thermal conductivity, and convection coefficient determine efficiency and total heat dissipation Formula: m = sqrt(2 × h / (k × t)) (fin parameter for rectangular fin). 4 parameters.
- [Heat Pipe Thermal Performance Simulator](/lab/heat-transfer/heat-pipe): Simulate heat pipe operation — explore how working fluid, wick structure, pipe dimensions, and heat load determine effective thermal conductivity and transport limits Formula: k_eff = Q × L_eff / (A_cross × ΔT) (effective thermal conductivity). 4 parameters.
- [Radiative Heat Exchange Between Surfaces](/lab/heat-transfer/radiation-exchange): Model thermal radiation exchange between two surfaces — explore how emissivity, temperature, and view factors determine radiative heat transfer rates Formula: E = ε × σ × T⁴ (Stefan-Boltzmann law for grey body). 4 parameters.
- [Steady-State Conduction Through a Composite Wall](/lab/heat-transfer/conduction-wall): Simulate heat conduction through a multi-layer wall — explore how thermal conductivity, wall thickness, and boundary temperatures create linear temperature profiles Formula: q = k × (T_hot − T_cold) / L (Fourier’s law, 1D steady-state). 4 parameters.

### Hydraulics & Pipe Flow Systems (5 simulations)
- [Bernoulli Equation & Venturi Effect](/lab/hydraulics/bernoulli-theorem): Simulate Bernoulli's energy equation along a streamline — explore how cross-section changes in a venturi create pressure drops, velocity increases, and flow measurement Formula: P₁ + ½ρV₁² + ρgz₁ = P₂ + ½ρV₂² + ρgz₂ (Bernoulli). 4 parameters.
- [Darcy-Weisbach Friction & Moody Chart](/lab/hydraulics/pipe-friction): Simulate pipe friction losses — explore how flow velocity, pipe diameter, roughness, and Reynolds number determine head loss via Darcy-Weisbach and the Moody chart Formula: hf = f × (L/D) × V² / (2g) (Darcy-Weisbach). 4 parameters.
- [Manning Equation & Open Channel Flow](/lab/hydraulics/open-channel): Simulate open channel flow — explore how channel slope, roughness, cross-section shape, and depth control flow velocity and discharge via Manning's equation Formula: V = (1/n) × R^(2/3) × S^(1/2) (Manning's equation). 4 parameters.
- [Pump Performance & System Curve](/lab/hydraulics/pump-curve): Simulate pump-system interaction — explore how pump head, efficiency, system resistance, and static lift determine the operating point and power consumption Formula: H_pump = H₀ × (1 − (Q/Q_r)²) (parabolic pump curve). 4 parameters.
- [Water Hammer & Joukowsky Equation](/lab/hydraulics/water-hammer): Simulate water hammer pressure transients — explore how valve closure time, flow velocity, pipe material, and wave speed generate dangerous pressure surges Formula: ΔP = ρ × a × V (Joukowsky pressure rise). 4 parameters.

### Hydrometallurgy & Aqueous Processing (5 simulations)
- [Electrowinning Cell Design](/lab/hydrometallurgy/electrowinning): Simulate an electrowinning cell for metal deposition — optimize current density, voltage, electrolyte composition, and energy consumption for copper, zinc, or nickel recovery Formula: m = J·A·η·M / (n·F) — Faraday's law deposition rate. 4 parameters.
- [Ion-Exchange Breakthrough Curves](/lab/hydrometallurgy/ion-exchange): Simulate ion-exchange column chromatography for metal recovery — model breakthrough curves, resin capacity, and selectivity for hydrometallurgical separations Formula: C/C₀ = 1 / (1 + exp(-k·(BV - BV_50))) — Thomas model breakthrough. 4 parameters.
- [Leaching Kinetics & Shrinking-Core Model](/lab/hydrometallurgy/leaching-kinetics): Simulate acid leaching of metal ores using the shrinking-core model — explore how particle size, acid concentration, temperature, and diffusion control dissolution rates Formula: τ = ρ_B · R₀² / (6 · b · D_e · C_A) — diffusion-controlled complete reaction time. 4 parameters.
- [Selective Precipitation & pH Control](/lab/hydrometallurgy/precipitation): Simulate selective metal precipitation by pH adjustment — explore solubility curves, precipitation windows, and co-precipitation risks for multi-metal solutions Formula: Ksp = [M^n+] · [OH⁻]^n — solubility product expression. 4 parameters.
- [Solvent Extraction & McCabe-Thiele](/lab/hydrometallurgy/solvent-extraction): Simulate counter-current solvent extraction for metal separation — design McCabe-Thiele diagrams, optimize organic/aqueous ratios, and predict stage efficiency Formula: E = 1 - 1/(1 + K_D · O/A)^N — extraction efficiency for N stages. 4 parameters.

### Industrial Fermentation & Bioprocess Engineering (5 simulations)
- [Bioreactor Scale-Up](/lab/industrial-fermentation/scale-up): Simulate bioreactor scale-up from bench to production — compare scale-up criteria including constant P/V, tip speed, kLa, and mixing time across scales Formula: Constant P/V: N₂ = N₁ × (D₁/D₂)^(2/3). 4 parameters.
- [Continuous Culture & Chemostat](/lab/industrial-fermentation/continuous-culture): Simulate continuous chemostat culture — explore steady-state biomass, substrate, and dilution rate effects including washout and oscillatory dynamics Formula: S_ss = Ks × D / (μ_max - D). 5 parameters.
- [Fed-Batch Fermentation Strategy](/lab/industrial-fermentation/fed-batch): Simulate fed-batch fermentation — optimize substrate feeding profiles to maximize biomass and product while avoiding overflow metabolism and substrate inhibition Formula: F(t) = (μ_set × X₀ × V₀) / (Yx/s × S_f) × exp(μ_set × t). 5 parameters.
- [Oxygen Transfer & kLa](/lab/industrial-fermentation/oxygen-transfer): Simulate oxygen mass transfer in bioreactors — calculate kLa, dissolved oxygen profiles, and the oxygen transfer rate as functions of agitation and aeration Formula: kLa = 0.026 × (P/V)^0.4 × VVM^0.5 (van't Riet). 4 parameters.
- [Stirred-Tank Bioreactor Design](/lab/industrial-fermentation/bioreactor-design): Simulate stirred-tank bioreactor geometry — optimize vessel dimensions, impeller configuration, and power input for microbial and mammalian cell culture Formula: D = (4V / (π × H/D))^(1/3). 5 parameters.

### LiDAR & Remote Sensing (5 simulations)
- [Bathymetric LiDAR & Water Penetration](/lab/lidar-remote-sensing/bathymetric-lidar): Simulate bathymetric LiDAR — explore how laser wavelength, water clarity, depth, and bottom reflectance determine subaqueous mapping capability Formula: D_max ≈ 1.5 / K_d (maximum detectable depth). 4 parameters.
- [Canopy Height Model & Forest Structure](/lab/lidar-remote-sensing/canopy-height): Simulate forest canopy height estimation from LiDAR — explore how first/last return differences, point density, and crown shape determine canopy height models Formula: CHM(x,y) = DSM(x,y) − DEM(x,y). 4 parameters.
- [LiDAR Pulse Propagation & Return](/lab/lidar-remote-sensing/lidar-pulse): Simulate LiDAR laser pulse propagation — explore how pulse energy, beam divergence, atmospheric attenuation, and surface reflectance determine return signal strength Formula: R = c × t / 2 (range from round-trip time). 4 parameters.
- [Point Density & Survey Design](/lab/lidar-remote-sensing/point-density): Simulate LiDAR survey planning — explore how pulse rate, scan angle, flying speed, and altitude determine point density and coverage for geospatial accuracy requirements Formula: ρ = PRR / (v × 2H × tan(α)) (nominal point density). 4 parameters.
- [Terrain Classification & Ground Filtering](/lab/lidar-remote-sensing/terrain-classification): Simulate LiDAR point cloud classification — explore how slope threshold, height difference, and cell size parameters affect ground vs. non-ground filtering Formula: slope = arctan(Δz / Δxy) (point-to-point slope). 4 parameters.

### Materials Fatigue & Failure Analysis (5 simulations)
- [Fatigue Crack Propagation (Paris Law)](/lab/materials-fatigue/crack-propagation): Simulate fatigue crack growth using the Paris-Erdogan law — predict how crack length evolves with loading cycles and estimate remaining component life Formula: da/dN = C × (ΔK)^m (Paris law). 5 parameters.
- [Goodman Diagram (Mean Stress Effect)](/lab/materials-fatigue/goodman-diagram): Construct modified Goodman diagrams — evaluate fatigue safety under combined alternating and mean stress for different materials and loading conditions Formula: σₐ/Se + σₘ/Sᵤ = 1/n (modified Goodman line). 5 parameters.
- [Miner's Rule (Cumulative Damage)](/lab/materials-fatigue/miners-rule): Apply Palmgren-Miner's linear damage accumulation rule — predict fatigue failure under variable amplitude loading with multiple stress levels Formula: D = Σ(nᵢ / Nᵢ) — Palmgren-Miner damage sum. 5 parameters.
- [S-N Curve (Wohler Curve) Simulator](/lab/materials-fatigue/sn-curve): Generate and analyze S-N curves — plot stress amplitude versus cycles to failure for different materials and loading conditions Formula: Sₐ = Sf' × kₛ × (2Nf)^b (Basquin equation). 4 parameters.
- [Stress Concentration Factor (Kt)](/lab/materials-fatigue/stress-concentration): Analyze stress concentrations around geometric discontinuities — visualize how holes, notches, and fillets amplify local stress and reduce fatigue life Formula: Kt = 3.0 for circular hole in infinite plate (Kirsch solution). 4 parameters.

### Materials Science & Engineering (5 simulations)
- [Binary Phase Diagram Simulator](/lab/materials-science/phase-diagram): Explore a binary alloy phase diagram — adjust composition and temperature to see which phases are stable and apply the lever rule Formula: f_L = (C_S - C₀) / (C_S - C_L) (lever rule, liquid fraction). 4 parameters.
- [Crystal Lattice Structure Simulator](/lab/materials-science/crystal-structure): Visualize and compare FCC, BCC, and HCP crystal lattice structures — see how atomic packing determines material properties Formula: APF_FCC = (4 × (4/3)π(a√2/4)³) / a³ = π/(3√2) ≈ 0.7405. 4 parameters.
- [Heat Conduction & Thermal Diffusivity Simulator](/lab/materials-science/heat-conduction): Watch heat diffuse through a 1D bar in real time — adjust thermal conductivity, density, and heat capacity to see how temperature evolves Formula: q = -k dT/dx (Fourier's law). 4 parameters.
- [Polymer Chain Dynamics & Random Coil Simulator](/lab/materials-science/polymer-chains): Watch polymer chains writhe in Brownian motion — explore how chain length, temperature, and stiffness affect the random coil conformation Formula: ⟨R²⟩ = N b² (mean-square end-to-end distance, freely-jointed chain). 4 parameters.
- [Stress-Strain Curve Simulator](/lab/materials-science/stress-strain): Stretch a virtual specimen to observe elastic deformation, yielding, strain hardening, necking, and fracture on an interactive stress-strain curve Formula: σ = F / A₀ (engineering stress). 4 parameters.

### Membrane Science & Separation Technology (5 simulations)
- [Electrodialysis & Ion Transport](/lab/membrane-science/electrodialysis): Simulate electrodialysis — explore how voltage, membrane conductivity, and current density drive ion transport and current efficiency in desalination stacks Formula: i = V / (N·(R_AEM + R_CEM + R_dilute + R_conc))  [Current density]. 4 parameters.
- [Gas Separation Membranes](/lab/membrane-science/gas-separation): Simulate gas permeation membranes — explore the permeability-selectivity tradeoff and Robeson upper bound for CO2/N2 and O2/N2 separation Formula: P = S × D  [Permeability = solubility × diffusivity]. 4 parameters.
- [Membrane Distillation](/lab/membrane-science/membrane-distillation): Simulate thermal membrane distillation — explore how temperature difference, membrane porosity, and pore size drive vapor flux across hydrophobic membranes Formula: J = C_m × (p_h − p_c)  [Vapor flux through membrane]. 4 parameters.
- [Reverse Osmosis Desalination](/lab/membrane-science/reverse-osmosis): Simulate RO desalination — explore how applied pressure, osmotic pressure, and membrane permeability govern water flux and salt rejection Formula: J_w = A × (ΔP − Δπ)  [Solution-diffusion model]. 4 parameters.
- [Ultrafiltration & Molecular Weight Cutoff](/lab/membrane-science/ultrafiltration): Simulate ultrafiltration membranes — explore molecular weight cutoff, pore size, and solute rejection curves for protein and colloid separation Formula: R(MW) = 1 − exp(−(MW/MWCO)²)  [Sigmoidal rejection model]. 4 parameters.

### Microfluidics & Lab-on-a-Chip (5 simulations)
- [Capillary-Driven Flow in Microchannels](/lab/microfluidics/capillary-flow): Simulate capillary-driven flow through rectangular microchannels — explore how contact angle, channel dimensions, and fluid viscosity govern the Washburn flow dynamics Formula: ΔP = 2γ cos θ / w (capillary pressure in slit). 4 parameters.
- [Diffusion-Based Laminar Micromixing](/lab/microfluidics/laminar-mixing): Simulate diffusion-driven mixing in laminar microchannels — explore how channel length, flow velocity, and diffusion coefficient affect mixing efficiency Formula: Pe = v × w / D (Péclet number). 4 parameters.
- [Digital Microfluidics: Electrowetting Control](/lab/microfluidics/electrowetting): Simulate electrowetting-on-dielectric droplet manipulation — control voltage, dielectric thickness, and droplet volume to explore contact angle modulation and droplet transport Formula: cos θ = cos θ₀ + ε₀εrV² / (2dγ) (Lippmann-Young). 4 parameters.
- [Fluorescence-Activated Cell Sorting (FACS)](/lab/microfluidics/cell-sorting): Simulate microfluidic fluorescence-activated cell sorting — explore how flow speed, laser power, and sorting threshold affect purity and throughput Formula: Throughput = Q × C / 60 (events per second). 4 parameters.
- [T-Junction Droplet Generation](/lab/microfluidics/droplet-generation): Simulate monodisperse droplet formation at a microfluidic T-junction — control flow rates, channel geometry, and interfacial tension to explore dripping and jetting regimes Formula: d ≈ w × (Qd/Qc)^(1/3) (squeezing regime). 4 parameters.

### Mining Engineering & Extraction (5 simulations)
- [Crushing & Grinding Energy — Bond Work Index](/lab/mining-engineering/comminution): Simulate comminution energy — explore how Bond work index, feed size, product size, and throughput affect grinding power and energy consumption Formula: W = 10 × Wi × (1/√P₈₀ - 1/√F₈₀) (Bond's equation). 4 parameters.
- [Cutoff Grade & Economic Optimization](/lab/mining-engineering/ore-grade): Simulate cutoff grade decisions — explore how metal price, processing cost, recovery rate, and mining cost affect economic ore boundaries Formula: g_co = (C_mining + C_processing) / (Price × Recovery) (breakeven cutoff). 4 parameters.
- [Drill-and-Blast Pattern & Fragmentation](/lab/mining-engineering/blast-design): Simulate drill-and-blast patterns — explore how burden, spacing, hole diameter, and charge density affect rock fragmentation and powder factor Formula: PF = ρ_e × π × D² / (4 × B × S) (powder factor). 4 parameters.
- [Mine Ventilation Network & Fan Selection](/lab/mining-engineering/ventilation-network): Simulate underground mine ventilation — explore how airway dimensions, length, roughness, and fan pressure affect airflow quantity and power consumption Formula: P = R × Q² (Atkinson square law). 4 parameters.
- [Pit Slope Stability & Factor of Safety](/lab/mining-engineering/slope-stability): Simulate pit slope stability — explore how slope angle, cohesion, friction angle, and water pressure affect factor of safety against failure Formula: FoS = (c × L + (W cos β - U) × tan φ) / (W × sin β) (planar failure). 4 parameters.

### Nanotechnology & Nanoscale Physics (5 simulations)
- [Carbon Nanotube Mechanics & Properties](/lab/nanotechnology/carbon-nanotube): Simulate carbon nanotube mechanical behavior — explore how diameter, chirality, number of walls, and applied strain govern stiffness, strength, and buckling Formula: E = (dF/dε) / A — Young's modulus from stress-strain. 4 parameters.
- [Molecular Self-Assembly & Pattern Formation](/lab/nanotechnology/self-assembly): Simulate molecular self-assembly — explore how interaction strength, concentration, temperature, and molecular geometry drive spontaneous pattern formation at the nanoscale Formula: CMC ∝ exp(-Nε / kT) — critical micelle concentration. 4 parameters.
- [Nanofiltration Membrane Transport](/lab/nanotechnology/nanofiltration): Simulate nanofiltration membrane performance — explore how pore size, pressure, solute radius, and membrane thickness govern rejection rate and permeate flux Formula: J = εrp²ΔP / (8μδ) — Hagen-Poiseuille flux through nanopores. 4 parameters.
- [Nanoparticle Synthesis & Nucleation Kinetics](/lab/nanotechnology/nanoparticle-synthesis): Simulate nanoparticle nucleation and growth — explore how supersaturation, temperature, surface energy, and reaction time govern particle size distribution Formula: r* = 2γVm / (kT ln S). 4 parameters.
- [Quantum Dot Confinement & Emission](/lab/nanotechnology/quantum-dot): Simulate quantum dot optical properties — explore how dot radius, material band gap, effective mass, and temperature control emission wavelength and color Formula: E(r) = Eg₀ + ℏ²π² / (2m*r²) — simplified Brus equation. 4 parameters.

### Navigation & Positioning Systems (5 simulations)
- [Dead Reckoning & Position Propagation](/lab/navigation/dead-reckoning): Simulate dead reckoning — explore how speed error, heading error, drift time, and update interval cause position uncertainty to grow from a known starting point Formula: x(t) = x₀ + v·cos(θ)·t;  y(t) = y₀ + v·sin(θ)·t. 4 parameters.
- [GPS Trilateration & Satellite Positioning](/lab/navigation/gps-positioning): Simulate GPS positioning — explore how satellite geometry, clock bias, atmospheric delay, and signal noise affect position accuracy through trilateration Formula: ρᵢ = |r_sat_i - r_user| + c·δt + I + T + ε. 4 parameters.
- [Inertial Navigation System (INS)](/lab/navigation/inertial-navigation): Simulate inertial navigation — explore how accelerometer bias, gyroscope drift, integration time, and initial alignment affect position and attitude errors Formula: δx ≈ ½ · b_a · g · t²  (position error from accel bias). 4 parameters.
- [Kalman Filter & Sensor Fusion](/lab/navigation/kalman-filter): Simulate a Kalman filter — explore how process noise, measurement noise, initial uncertainty, and update rate affect optimal state estimation and sensor fusion Formula: x̂⁻ = F·x̂ + B·u;  P⁻ = F·P·F' + Q  (predict). 4 parameters.
- [Map Matching & Road Snapping](/lab/navigation/map-matching): Simulate map matching — explore how GPS noise, road network density, heading constraint, and candidate search radius affect snapping GPS traces to road segments Formula: P(zₜ|rᵢ) ∝ exp(-d²/(2σ²))  (emission probability). 4 parameters.

### Nuclear Engineering & Reactor Physics (5 simulations)
- [Fuel Burnup & Isotope Evolution](/lab/nuclear-engineering/fuel-burnup): Track the depletion of U-235 and buildup of Pu-239, fission products, and xenon-135 over a multi-year fuel cycle Formula: Burnup = Power_density × time × capacity_factor / fuel_mass. 4 parameters.
- [Gamma & Neutron Shielding Attenuation](/lab/nuclear-engineering/shielding-design): Calculate gamma-ray and neutron attenuation through multi-layer shielding materials with buildup factor corrections Formula: I = B(μx, E) · I₀ · Math.pow(e, -μ·x) (attenuated intensity with buildup). 4 parameters.
- [One-Group Neutron Diffusion & Flux Profile](/lab/nuclear-engineering/neutron-diffusion): Solve the one-group neutron diffusion equation to visualize the spatial flux profile in a bare homogeneous slab reactor Formula: -D·d²φ/dx² + Σ_a·φ = νΣ_f·φ (one-group steady-state diffusion). 4 parameters.
- [Post-Shutdown Decay Heat Curve](/lab/nuclear-engineering/decay-heat): Calculate and visualize decay heat production after reactor shutdown using the ANS standard model for fission product decay Formula: P_decay/P₀ = 0.066 · [Math.pow(t, -0.2) - Math.pow(t + T_op, -0.2)]. 4 parameters.
- [Reactor Criticality & Neutron Multiplication](/lab/nuclear-engineering/reactor-criticality): Simulate the effective multiplication factor k-eff and neutron population dynamics in a thermal reactor to explore sub-critical, critical, and super-critical states Formula: k_eff = η · f · p · ε · P_NL · T_NL (six-factor formula). 4 parameters.

### Ocean Engineering & Offshore Structures (5 simulations)
- [Catenary Mooring Line Analysis](/lab/ocean-engineering/mooring-design): Simulate catenary mooring line geometry and tension — explore how line weight, water depth, and pretension affect mooring system performance Formula: y = (T₀/w) × (cosh(w×x/T₀) - 1) — catenary equation. 4 parameters.
- [Hull Resistance & Froude Number](/lab/ocean-engineering/ship-resistance): Simulate ship hull resistance as a function of speed, hull dimensions, and form — explore frictional and wave-making resistance components across Froude numbers Formula: Fn = V / √(g × L) (Froude number). 4 parameters.
- [Morison Equation & Wave Forces](/lab/ocean-engineering/wave-loading): Simulate wave forces on a cylindrical offshore structure using the Morison equation — explore drag and inertia components across wave conditions Formula: F = 0.5 × Cd × ρ × D × u|u| + Cm × ρ × (πD²/4) × ∂u/∂t. 4 parameters.
- [Oscillating Water Column Wave Energy Converter](/lab/ocean-engineering/wave-energy): Simulate an oscillating water column (OWC) wave energy converter — explore how chamber geometry, wave conditions, and turbine characteristics affect power extraction Formula: P_wave = ρ × g² × H² × T / (32π) (deep water wave power). 4 parameters.
- [Subsea Pipeline On-Bottom Stability](/lab/ocean-engineering/pipeline-stability): Simulate subsea pipeline on-bottom stability under wave and current loading — evaluate whether the pipe remains stable or requires concrete coating or trenching Formula: F_D = 0.5 × C_D × ρ × D × (U_c + U_w)². 4 parameters.

### Optoelectronics & Photonic Devices (5 simulations)
- [Electro-Optic Modulator & Mach-Zehnder](/lab/optoelectronics/optical-modulator): Simulate a Mach-Zehnder optical modulator — explore how drive voltage, bias point, extinction ratio, and modulation frequency shape the output optical signal Formula: T(V) = cos²(π × V / (2 × Vπ)). 4 parameters.
- [Laser Diode L-I Characteristics](/lab/optoelectronics/laser-diode): Simulate a semiconductor laser diode — explore how injection current, temperature, cavity length, and mirror reflectivity affect threshold, output power, and spectral behavior Formula: I_th ∝ exp(T / T₀) — temperature dependence of threshold. 4 parameters.
- [Optical Waveguide Mode Solver](/lab/optoelectronics/waveguide-design): Simulate a slab dielectric waveguide — explore how core thickness, core and cladding refractive indices, and wavelength determine guided modes and confinement Formula: NA = √(n₁² - n₂²). 4 parameters.
- [Photodetector Responsivity & Noise](/lab/optoelectronics/photodetector): Simulate a semiconductor photodetector — explore how wavelength, optical power, reverse bias, and bandwidth affect photocurrent, responsivity, NEP, and signal-to-noise ratio Formula: R = η × q × λ / (h × c). 4 parameters.
- [Photonic Optical Switch](/lab/optoelectronics/optical-switch): Simulate a directional coupler optical switch — explore how coupling length, refractive index change, wavelength, and gap spacing control light routing between waveguide ports Formula: P_cross = sin²(κ × L / cos(δ)). 4 parameters.

### Photogrammetry & 3D Reconstruction (5 simulations)
- [Camera Calibration & Lens Distortion](/lab/photogrammetry/camera-calibration): Simulate camera calibration — explore how focal length, principal point offset, and radial distortion coefficients affect image geometry Formula: x_d = x(1 + k₁r² + k₂r⁴) (radial distortion model). 4 parameters.
- [Orthorectification & Map Projection](/lab/photogrammetry/orthorectification): Simulate orthorectification — explore how terrain elevation, camera tilt, and flight altitude affect geometric correction of aerial imagery Formula: d = Δh × r / H (relief displacement). 4 parameters.
- [Point Cloud Generation & Visualization](/lab/photogrammetry/point-cloud): Generate and visualize 3D point clouds — explore how point density, noise level, and viewing angle affect reconstructed surface quality Formula: ρ = N / A (point density per unit area). 4 parameters.
- [Stereo Matching & Disparity Maps](/lab/photogrammetry/stereo-matching): Simulate stereo matching algorithms — explore how baseline distance, block size, and disparity range affect depth estimation from stereo image pairs Formula: Z = f × B / d (depth from disparity). 4 parameters.
- [Structure from Motion Pipeline](/lab/photogrammetry/structure-from-motion): Simulate the SfM pipeline — explore how image count, feature matches, and bundle adjustment iterations affect 3D reconstruction quality Formula: x = PX = K[R|t]X (projection equation). 4 parameters.

### Photovoltaics & Solar Energy (5 simulations)
- [Partial Shading & Bypass Diode Losses](/lab/photovoltaics/shading-loss): See how partial shading on even one cell can collapse an entire module's output — and how bypass diodes limit the damage by routing current around shaded cell strings Formula: Series string current limited by weakest cell: I_string = min(I_cell_i). 4 parameters.
- [Solar Cell I-V Curve & Maximum Power Point](/lab/photovoltaics/iv-curve): Trace the current-voltage characteristic of a solar cell and find the maximum power point — the operating condition where a cell extracts the most energy from sunlight Formula: Shockley diode: I = Iph - I₀ × [exp(V/(n×Vt)) - 1] - V/Rsh. 4 parameters.
- [Solar Module Layout & Array Optimization](/lab/photovoltaics/module-layout): Optimize solar module arrangement on a rooftop — balance row spacing, tilt angle, and azimuth to maximize energy density while avoiding inter-row shading Formula: Row spacing: d = L × sin(tilt) / tan(sun_elevation_min). 4 parameters.
- [Solar Spectrum & Photon Energy Distribution](/lab/photovoltaics/solar-spectrum): Visualize the solar irradiance spectrum from UV to infrared and see which wavelengths a silicon cell can actually absorb — revealing why single-junction efficiency has a hard ceiling Formula: Wien's displacement law: λ_max = 2.898 × 10⁶ / T (nm). 4 parameters.
- [Temperature Coefficient & Thermal Derating](/lab/photovoltaics/temperature-coefficient): Watch solar cell efficiency decline as temperature rises — simulate the physics of thermal derating across different cell technologies from silicon to perovskite Formula: Thermal derating: η(T) = η_STC × [1 + β × (T_cell - 25)]. 4 parameters.

### Powder Metallurgy (5 simulations)
- [Atomization Process Simulator](/lab/powder-metallurgy/atomization): Interactive model of gas and water atomization showing how process parameters control powder particle size distribution, morphology, and yield Formula: d50 = K · D_melt / (We^0.5 · GMR^n). 4 parameters.
- [Hot Isostatic Pressing (HIP) Simulator](/lab/powder-metallurgy/hot-isostatic-pressing): Interactive model of HIP densification showing how temperature, pressure, and time eliminate residual porosity through creep and diffusion mechanisms Formula: Creep rate: dρ/dt = A·(P_eff/σ₀)^n·exp(-Q/RT)·f(ρ). 4 parameters.
- [Metal Injection Molding (MIM) Simulator](/lab/powder-metallurgy/metal-injection-molding): Interactive model of the MIM process showing how powder loading, binder properties, and sintering conditions determine final part density, shrinkage, and properties Formula: Shrinkage = 1 - (φ_powder)^(1/3). 4 parameters.
- [Powder Compaction Simulator](/lab/powder-metallurgy/powder-compaction): Interactive model of uniaxial die compaction showing how applied pressure, friction, and powder properties determine green density distribution Formula: Heckel: ln(1/(1-D)) = K·P + A, K ≈ 1/(3σ_y). 4 parameters.
- [Sintering Kinetics Simulator](/lab/powder-metallurgy/sintering-kinetics): Interactive model of solid-state sintering showing neck growth, densification, and grain coarsening as functions of temperature, time, and particle size Formula: (x/a)^n = (B·D·γ·Ω·t) / (a^m·k·T). 4 parameters.

### Power Systems & Electrical Grids (5 simulations)
- [Grid Frequency Stability](/lab/power-systems/grid-stability): Simulate grid frequency response after a generation-load imbalance — visualize inertia, governor response, and frequency nadir Formula: 2H · d(Δf)/dt = ΔP_m − ΔP_e − D · Δf. 4 parameters.
- [Load Flow Analysis (Newton-Raphson)](/lab/power-systems/load-flow): Simulate power flow across a multi-bus electrical network using the Newton-Raphson method — visualize voltage profiles, line losses, and power distribution Formula: P_i = |V_i| Σ |V_j| (G_ij cos θ_ij + B_ij sin θ_ij). 4 parameters.
- [Power Factor Correction](/lab/power-systems/power-factor): Visualize the relationship between real, reactive, and apparent power — calculate capacitor bank sizing for power factor correction Formula: Q_c = P × (tan(arccos(PF_old)) − tan(arccos(PF_new))). 4 parameters.
- [Symmetrical Fault Analysis](/lab/power-systems/fault-analysis): Calculate three-phase fault currents using Thevenin equivalent circuits — visualize fault propagation and protective relay coordination Formula: I_fault = V_prefault / (Z_thevenin + Z_fault). 4 parameters.
- [Transformer Design & Efficiency](/lab/power-systems/transformer-design): Design a power transformer by specifying turns ratio, core material, and loading — visualize efficiency curves, losses, and voltage regulation Formula: η = S × load × PF / (S × load × PF + P_core + P_copper × load²). 4 parameters.

### Propulsion Systems & Thrust (5 simulations)
- [Brayton Cycle & Turbojet Thrust](/lab/propulsion/turbojet-cycle): Model the Brayton thermodynamic cycle — adjust compression ratio, turbine inlet temperature, and airspeed to compute turbojet thrust and thermal efficiency Formula: Ideal thermal efficiency: η_th = 1 − 1/π_c^((γ−1)/γ). 4 parameters.
- [Ion Thruster & Specific Impulse](/lab/propulsion/ion-thruster): Simulate electrostatic ion propulsion — adjust beam voltage, propellant flow, and grid geometry to see thrust, Isp, and power requirements Formula: Exhaust velocity: V_e = √(2·q·V_b / m_ion). 4 parameters.
- [Propeller Thrust & Blade Element Theory](/lab/propulsion/propeller-thrust): Calculate propeller thrust using blade element theory — adjust blade count, pitch, RPM, and airspeed to visualize the advance ratio and efficiency curve Formula: Advance ratio: J = V∞ / (n × D). 4 parameters.
- [Ramjet Performance & Supersonic Inlet](/lab/propulsion/ramjet-performance): Model ram compression and combustion in a ramjet — adjust Mach number, combustion temperature, and altitude to compute thrust and propulsive efficiency Formula: Ram pressure ratio: (1 + (γ−1)/2 · M²)^(γ/(γ−1)). 4 parameters.
- [Tsiolkovsky Rocket Equation & Delta-V](/lab/propulsion/rocket-equation): Calculate delta-v from exhaust velocity and mass ratio — visualize the exponential propellant cost of velocity change Formula: Δv = Isp × g₀ × ln(M₀ / Mf). 4 parameters.

### Radar Systems & Detection (5 simulations)
- [Doppler Radar & Velocity Measurement](/lab/radar-systems/doppler-radar): Simulate Doppler radar — explore how target radial velocity creates frequency shifts used for velocity measurement, clutter rejection, and weather detection Formula: f_d = 2·v·f₀ / c (Doppler shift). 4 parameters.
- [Phased Array Beam Steering & Antenna Patterns](/lab/radar-systems/phased-array): Simulate a phased-array antenna — explore how element spacing, phase shifts, and array size control beam direction, sidelobe levels, and angular resolution Formula: AF(θ) = Σ aₙ·exp(j·n·2π·d/λ·(sin(θ) - sin(θ₀))). 4 parameters.
- [Pulse Compression & Chirp Radar](/lab/radar-systems/pulse-compression): Simulate pulse compression radar — explore how linear frequency modulation (chirp) achieves fine range resolution without sacrificing detection energy Formula: ΔR = c / (2·B) (compressed range resolution). 4 parameters.
- [Radar Cross Section & Target Detectability](/lab/radar-systems/radar-cross-section): Simulate radar cross section — explore how target size, shape, material, and frequency determine RCS and stealth characteristics Formula: σ = lim(4π·R²·|E_s|²/|E_i|²) as R→∞. 4 parameters.
- [Radar Range Equation & SNR](/lab/radar-systems/radar-range): Simulate the radar range equation — explore how transmit power, antenna gain, target cross-section, and noise determine maximum detection range Formula: R_max = [Pt·G²·λ²·σ / ((4π)³·k·T·B·SNR_min)]^(1/4). 4 parameters.

### Renewable Energy & Clean Power (5 simulations)
- [Geothermal Power Plant Simulator](/lab/renewable-energy/geothermal): Model a geothermal power plant with adjustable reservoir temperature, well depth, flow rate, and plant type to calculate electrical output and thermal efficiency Formula: Q_thermal = m_dot * cp * (T_reservoir - T_reject). 4 parameters.
- [Grid Energy Storage Simulator](/lab/renewable-energy/energy-storage): Compare battery energy storage systems by modeling charge/discharge cycles, round-trip efficiency, degradation, and levelized cost for lithium-ion, flow, and compressed air technologies Formula: Duration = E_capacity / P_rating. 4 parameters.
- [Hydrogen Fuel Cell Simulator](/lab/renewable-energy/hydrogen-fuel-cell): Model a PEM hydrogen fuel cell with adjustable current density, temperature, pressure, and membrane thickness to visualize the voltage-current characteristic and efficiency Formula: E_rev = 1.229 - 0.000846*(T-298) + (RT/2F)*ln(P_H2*sqrt(P_O2)). 4 parameters.
- [Tidal Power Generation Simulator](/lab/renewable-energy/tidal-power): Model tidal stream energy extraction with variable current speed, turbine diameter, tidal range, and channel geometry to calculate power output over a tidal cycle Formula: P = 0.5 * rho_water * Cp * A * V^3. 4 parameters.
- [Wind Turbine Power Simulator](/lab/renewable-energy/wind-turbine): Simulate wind turbine power output with adjustable wind speed, blade pitch, rotor diameter, and air density to explore the Betz limit and power curves Formula: P_wind = 0.5 * rho * A * V^3. 4 parameters.

### Robotics & Control Systems (5 simulations)
- [PID Controller Tuning Simulator](/lab/robotics/pid-controller): Tune a PID controller in real time — adjust proportional, integral, and derivative gains to minimize overshoot and settling time Formula: u(t) = Kp·e(t) + Ki·∫e(τ)dτ + Kd·de(t)/dt. 4 parameters.
- [Robot Arm Forward Kinematics Simulator](/lab/robotics/robot-arm): Manipulate joint angles of a multi-link robot arm and watch the end-effector trace its path — visualizing forward kinematics and the Denavit-Hartenberg convention Formula: x = Σᵢ Lᵢ cos(θ₁ + θ₂ + ... + θᵢ) (planar FK, x-component). 4 parameters.
- [Robot Arm Inverse Kinematics Simulator](/lab/robotics/inverse-kinematics): Drag a target point and watch a 2-link robot arm solve inverse kinematics in real time — see the elbow-up and elbow-down solutions Formula: cos(θ₂) = (x² + y² - L₁² - L₂²) / (2 L₁ L₂). 4 parameters.
- [Robot Path Planning (RRT/PRM) Simulator](/lab/robotics/path-planning): Compare RRT and PRM path-planning algorithms on a randomized obstacle map — watch the tree grow and find a collision-free path Formula: x_new = x_nearest + step_size × (x_rand - x_nearest) / |x_rand - x_nearest|. 4 parameters.
- [Swarm Robotics Emergent Behavior Simulator](/lab/robotics/swarm-robotics): Watch simple local rules produce complex emergent swarm behavior — adjust cohesion, alignment, and separation to see flocking, clustering, and dispersion Formula: v_cohesion = (centroid_neighbors - position) × k_cohesion. 4 parameters.

### Signal Processing (5 simulations)
- [AM/FM Modulation & Demodulation](/lab/signal-processing/am-fm-modulation): Modulate and demodulate carrier waves using amplitude and frequency modulation — see how information rides on radio waves Formula: AM: s(t) = [1 + m·cos(2πf_m·t)] · cos(2πf_c·t). 5 parameters.
- [FFT Spectrum Analyzer](/lab/signal-processing/fft-spectrum): Decompose any composite signal into its frequency components using the Fast Fourier Transform — see the spectrum update in real time as you change frequencies and amplitudes Formula: X[k] = Σ x[n] · e^(−j2πkn/N) for n = 0 to N−1. 6 parameters.
- [Nyquist-Shannon Sampling Theorem](/lab/signal-processing/sampling-theorem): Visualize aliasing and perfect reconstruction — see why sampling rate must exceed twice the signal frequency to avoid information loss Formula: f_Nyquist = f_s / 2 (maximum recoverable frequency). 4 parameters.
- [Signal Convolution Visualizer](/lab/signal-processing/convolution): Watch convolution in action — slide one signal over another and see how the output is built point by point, the foundation of filtering and system response Formula: (f * g)(t) = ∫ f(τ) · g(t − τ) dτ. 5 parameters.
- [Signal Noise Filtering](/lab/signal-processing/noise-filtering): Apply low-pass and high-pass filters to noisy signals — visualize how cutoff frequency and filter order affect signal clarity in real time Formula: H(f) = 1 / (1 + (f/fc)^(2n)) for Butterworth low-pass of order n. 5 parameters.

### Structural Engineering & Load Analysis (5 simulations)
- [Beam Bending & Deflection Analysis](/lab/structural-engineering/beam-bending): Simulate beam bending under distributed and point loads — visualize shear force diagrams, bending moment curves, and deflection shapes for simply supported and cantilever beams Formula: M_max = qL²/8 (simply supported, uniform load). 4 parameters.
- [Column Buckling & Euler Critical Load](/lab/structural-engineering/column-buckling): Simulate Euler column buckling — explore how length, end conditions, cross-section, and material properties determine the critical load at which a column becomes unstable Formula: P_cr = π²EI/(KL)² (Euler critical load). 4 parameters.
- [Reinforced Concrete Section Design](/lab/structural-engineering/concrete-reinforcement): Design reinforced concrete beam sections — calculate required steel area, check moment capacity, visualize the stress block, and verify ductility requirements Formula: Mn = As·fy·(d - a/2) (nominal moment capacity). 4 parameters.
- [Seismic Response Spectrum Analysis](/lab/structural-engineering/seismic-design): Simulate earthquake response spectra — explore how natural period, damping ratio, soil type, and peak ground acceleration determine a structure's seismic demand Formula: Sa(T) = PGA · S · 2.5 · (Tc/T) for T > Tc. 4 parameters.
- [Truss Force Analysis & Member Sizing](/lab/structural-engineering/truss-analysis): Analyze planar truss structures — compute member forces using the method of joints, visualize tension and compression, and identify critical members under various loading conditions Formula: F_chord = M/H = PL/(4H) at midspan. 4 parameters.

### Telecommunications & Signal Transmission (5 simulations)
- [Antenna Radiation Pattern Simulator](/lab/telecommunications/antenna-radiation): Simulate antenna radiation patterns — explore how element count, spacing, and frequency shape directional gain in phased array antennas Formula: AF(θ) = Σ exp(j·n·(k·d·sinθ - β)). 4 parameters.
- [Digital Modulation Constellation Viewer](/lab/telecommunications/modulation-schemes): Visualize digital modulation constellations — compare BPSK, QPSK, 16-QAM, and 64-QAM under varying noise and observe bit error rates Formula: b = log₂(M)  bits per symbol. 4 parameters.
- [Forward Error Correction Simulator](/lab/telecommunications/error-correction): Simulate forward error correction — visualize how redundancy bits detect and correct transmission errors using Hamming, convolutional, and LDPC codes Formula: R = k/n  (code rate, k data bits, n total bits). 4 parameters.
- [Satellite Link Budget Analyzer](/lab/telecommunications/link-budget): Analyze satellite link budgets — compute path loss, received power, and link margin for LEO, MEO, and GEO communication links Formula: FSPL = 20·log₁₀(4π·d·f/c)  dB. 4 parameters.
- [Shannon Channel Capacity Calculator](/lab/telecommunications/channel-capacity): Compute the Shannon-Hartley channel capacity — explore how bandwidth, signal power, and noise determine the theoretical maximum data rate Formula: C = B × log₂(1 + S/N). 4 parameters.

### Textile Engineering & Fiber Science (5 simulations)
- [Dyeing Kinetics & Diffusion](/lab/textile-engineering/dyeing-kinetics): Simulate textile dyeing kinetics — explore how diffusion coefficient, fiber radius, dye bath concentration, and temperature control dye uptake over time Formula: C(r,t)/C0 = 1 - (4/π²)Σ (1/n²)exp(-D·n²π²t/a²) (Crank, cylindrical). 4 parameters.
- [Fabric Drape & Bending Rigidity](/lab/textile-engineering/fabric-drape): Simulate fabric drape behavior — explore how bending rigidity, fabric weight, drape radius, and shear modulus affect the drape coefficient and fold formation Formula: DC = (A_shadow - A_disk) / (A_sample - A_disk) × 100%. 4 parameters.
- [Fiber Tensile Strength & Stress-Strain](/lab/textile-engineering/fiber-strength): Simulate fiber tensile testing — explore how Young's modulus, yield strength, elongation at break, and fiber diameter determine mechanical performance Formula: σ = E × ε (Hooke's law, elastic region). 4 parameters.
- [Weave Pattern Geometry & Structure](/lab/textile-engineering/weave-patterns): Simulate fundamental weave patterns — explore how warp density, weft density, float length, and weave type create distinct fabric structures Formula: Cover factor: K = n1×d1 + n2×d2 - n1×n2×d1×d2. 4 parameters.
- [Yarn Twist & Strength Optimization](/lab/textile-engineering/yarn-twist): Simulate yarn twist mechanics — explore how twist level, fiber length, fiber strength, and yarn count affect yarn tensile strength and the critical twist angle Formula: α = arctan(π × d × T) — surface helix angle. 4 parameters.

### Tissue Engineering & Regenerative Medicine (5 simulations)
- [Bioreactor Design & Perfusion Flow](/lab/tissue-engineering/bioreactor-design): Design a perfusion bioreactor by tuning flow rate, chamber geometry, and scaffold permeability to optimize shear stress and nutrient delivery Formula: v = Q / (π(D/2)²) — superficial velocity. 4 parameters.
- [Cell Proliferation & Growth Kinetics](/lab/tissue-engineering/cell-proliferation): Model cell population growth with logistic kinetics, doubling time, and growth factor concentration effects on tissue constructs Formula: N(t) = K / (1 + (K/N₀ − 1)exp(−μt)) — logistic growth. 4 parameters.
- [Oxygen Diffusion & Tissue Viability](/lab/tissue-engineering/oxygen-diffusion): Visualize oxygen concentration gradients in tissue constructs and predict the critical thickness beyond which necrosis occurs Formula: C(x) = C₀ − (Q/2D)x(L−x) — 1D diffusion-reaction, slab geometry. 4 parameters.
- [Scaffold Degradation & Tissue Ingrowth](/lab/tissue-engineering/degradation-rate): Balance scaffold degradation kinetics with new tissue formation to maintain mechanical integrity throughout the regeneration process Formula: S(t) = S₀ × exp(−0.693t/t½) — first-order scaffold mass loss. 4 parameters.
- [Scaffold Porosity & Cell Infiltration](/lab/tissue-engineering/scaffold-porosity): Model scaffold pore architecture and simulate cell infiltration depth as a function of pore size, porosity, and interconnectivity Formula: k = ε³d² / (150(1−ε)²) — Kozeny-Carman permeability. 4 parameters.

### Transportation & Traffic Science (5 simulations)
- [Autonomous Emergency Braking Simulator](/lab/transportation/autonomous-braking): Simulate autonomous emergency braking (AEB) systems — see how sensor delay, vehicle speed, road friction, and deceleration affect total stopping distance Formula: d_reaction = v × t_delay. 4 parameters.
- [Queueing Theory (M/M/1) Simulator](/lab/transportation/queue-theory): Simulate an M/M/1 queue to explore how arrival rate and service rate determine waiting times, queue length, and system utilization Formula: ρ = λ / µ — server utilization. 4 parameters.
- [Route Optimization (TSP) Simulator](/lab/transportation/route-optimization): Visualize the traveling salesman problem — compare nearest-neighbor, 2-opt, and simulated annealing algorithms to find short routes through randomly placed cities Formula: Total permutations = (n-1)! / 2 for symmetric TSP. 4 parameters.
- [Traffic Flow Simulator](/lab/transportation/traffic-flow): Explore the fundamental diagram of traffic flow — adjust vehicle density to observe the transition from free flow to congested traffic and the emergence of phantom jams Formula: q = k × v — fundamental relation (flow = density × speed). 4 parameters.
- [Vehicle Fuel Efficiency Simulator](/lab/transportation/fuel-efficiency): Model how speed, drag coefficient, vehicle mass, and rolling resistance affect fuel consumption — find the optimal cruising speed for maximum efficiency Formula: F_drag = 0.5 × ρ_air × Cd × A × v². 4 parameters.

### Tribology & Friction Science (5 simulations)
- [Friction Coefficient & Contact Mechanics](/lab/tribology/friction-coefficient): Calculate static and kinetic friction coefficients for various material pairs — explore the effects of normal load, surface energy, contact area, and sliding velocity on friction behavior Formula: μ = τ_s / H (Bowden-Tabor adhesion model, simplified). 4 parameters.
- [Journal Bearing Design & Optimization](/lab/tribology/bearing-design): Design and optimize hydrodynamic journal bearings — calculate film thickness, eccentricity, power loss, and temperature rise using Sommerfeld number analysis Formula: S = (η × N × L × D) / (W × (c/R)²) (Sommerfeld number). 4 parameters.
- [Lubrication Regimes & Stribeck Curve](/lab/tribology/lubrication-regimes): Explore the Stribeck curve — simulate the transition from boundary to mixed to hydrodynamic lubrication as a function of viscosity, speed, load, and surface roughness Formula: S = η × U / W (Hersey/Sommerfeld number). 4 parameters.
- [Surface Roughness & Texture Analysis](/lab/tribology/surface-roughness): Analyze surface roughness parameters — simulate Ra, Rq, Rsk, and Rku for different manufacturing processes and explore their effect on contact mechanics and functional performance Formula: R_a = (1/L) × ∫|z(x)| dx (arithmetic mean deviation). 4 parameters.
- [Wear Rate & Archard Equation](/lab/tribology/wear-rate): Predict material wear rate using the Archard wear model — explore effects of normal load, sliding distance, hardness, and wear coefficient on volume loss and component lifetime Formula: V = K × F_N × L / H (Archard wear equation). 4 parameters.

### Water Treatment & Purification (5 simulations)
- [Activated Carbon Adsorption & Breakthrough](/lab/water-treatment/activated-carbon): Model activated carbon adsorption using the Freundlich isotherm — adjust carbon dose, influent concentration, and flow rate to see breakthrough curves Formula: Freundlich isotherm: q_e = K_F × C_e^(1/n). 4 parameters.
- [Chlorination & CT Disinfection Kinetics](/lab/water-treatment/chlorination): Model chlorine disinfection using the CT concept — adjust dose, contact time, pH, and temperature to see log inactivation of pathogens Formula: CT = C × t (mg·min/L). 4 parameters.
- [Coagulation, Jar Test & Zeta Potential](/lab/water-treatment/coagulation): Simulate coagulation chemistry — adjust coagulant dose, pH, and mixing intensity to model zeta potential reduction and floc formation Formula: Zeta potential: ζ = ζ₀ × exp(−k × dose). 4 parameters.
- [Sedimentation & Stokes Settling](/lab/water-treatment/sedimentation): Model particle settling using Stokes' law — adjust particle size, density, and basin geometry to calculate settling velocity and removal efficiency Formula: Stokes settling: v_s = g·d²·(ρ_p − ρ_w) / (18·μ). 4 parameters.
- [UV Disinfection & Dose-Response](/lab/water-treatment/uv-disinfection): Model UV disinfection dose-response — adjust UV intensity, exposure time, and transmittance to calculate log inactivation of pathogens Formula: UV dose: D = I × t (mJ/cm²). 4 parameters.

### Wind Engineering & Structural Aerodynamics (5 simulations)
- [Atmospheric Boundary Layer Wind Profile](/lab/wind-engineering/wind-profile): Visualize how wind speed increases with height using power law and logarithmic atmospheric boundary layer profiles for different terrain categories Formula: V(z) = V_ref × (z / z_ref)^α (power law). 4 parameters.
- [Bridge Deck Aeroelastic Flutter Analysis](/lab/wind-engineering/flutter-analysis): Determine the critical flutter speed of a bridge deck by analyzing the coupling between torsional and vertical vibration modes under aerodynamic forcing Formula: V_cr = f_α × B × C × √(μ × r² × (1 − (f_h/f_α)²)) (Selberg). 4 parameters.
- [Building Pressure Coefficient (Cp) Visualizer](/lab/wind-engineering/pressure-coefficient): Visualize pressure coefficient distribution around a rectangular building cross-section as wind angle changes Formula: Cp = (p − p₀) / q where q = 0.5 × ρ × V². 4 parameters.
- [Pedestrian Wind Comfort Criteria Simulator](/lab/wind-engineering/wind-comfort): Assess pedestrian wind comfort at ground level using Lawson and Davenport criteria for walking, sitting, and standing activities Formula: V_gust = V̄ × (1 + g_p × I_u) where g_p ≈ 3.5. 4 parameters.
- [Vortex Shedding & Strouhal Number Simulator](/lab/wind-engineering/vortex-shedding): Observe vortex shedding from a bluff body and the dangerous lock-in phenomenon when shedding frequency matches structural natural frequency Formula: f_s = St × V / D (shedding frequency). 4 parameters.

## humanities (2 topics, 10 simulations)

### Linguistics & Language (5 simulations)
- [Language Family Tree & Divergence Simulator](/lab/linguistics/language-tree): Watch languages evolve and diverge from common ancestors in real time — explore how Proto-Indo-European split into hundreds of modern languages over thousands of years Formula: Lexical retention rate: R(t) = R_0 * Math.pow(1 - drift/100, t) per generation. 4 parameters.
- [Markov Chain Text Generator](/lab/linguistics/markov-text): Generate surprisingly coherent text from simple probability chains — see how statistical patterns in language create the illusion of understanding without any grammar rules Formula: P(w_t | w_{t-n}, ..., w_{t-1}) = count(w_{t-n}...w_t) / count(w_{t-n}...w_{t-1}). 4 parameters.
- [Phonetic Vowel Space Simulator](/lab/linguistics/phonetic-vowel-space): Explore the IPA vowel chart as an interactive acoustic space — see how tongue height, backness, and rounding map to formant frequencies across the world's languages Formula: Euclidean distance: d(v1, v2) = sqrt(Math.pow(F1_1 - F1_2, 2) + Math.pow(F2_1 - F2_2, 2)). 4 parameters.
- [Word Embedding Vector Space Simulator](/lab/linguistics/word-embedding): Explore the geometric space where words become vectors — see how king minus man plus woman equals queen, and why meaning has direction Formula: Cosine similarity: cos(A, B) = (A · B) / (|A| * |B|). 4 parameters.
- [Zipf's Law Word Frequency Simulator](/lab/linguistics/zipfs-law): Visualize how word frequencies in natural language follow a power law — the nth most common word appears proportional to 1/n, a pattern universal across all human languages Formula: f(r) = C / r^s where r = rank, s = exponent, C = normalization constant. 4 parameters.

### Philosophy & Thought Experiments (5 simulations)
- [Nozick's Experience Machine Thought Experiment](/lab/philosophy/experience-machine): Would you plug into a machine that gives you any experience you want? Simulate the trade-offs between pleasure, authenticity, and agency Formula: Machine utility = pleasure_level × duration. 4 parameters.
- [Rawls' Veil of Ignorance Resource Allocation](/lab/philosophy/veil-of-ignorance): Design a just society from behind the veil of ignorance — allocate resources when you don't know which position you'll occupy Formula: Maximin = max(min(welfare_i)) across all possible social arrangements. 4 parameters.
- [Ship of Theseus Identity Paradox](/lab/philosophy/ship-of-theseus): Watch the Ship of Theseus transform plank by plank — at what percentage of replacement does identity break? Formula: Identity continuity = (original_parts_remaining / total_parts) × 100%. 4 parameters.
- [Trolley Problem Ethical Dilemma Simulator](/lab/philosophy/trolley-problem): Explore the classic trolley problem and its variants — simulate utilitarian vs deontological outcomes across thousands of moral scenarios Formula: Net utility = lives_on_main_track - lives_on_spur_track. 4 parameters.
- [Utility Monster & Utilitarian Paradox](/lab/philosophy/utility-monster): Simulate the Utility Monster paradox — when one being derives vastly more utility from resources, should everyone else sacrifice for it? Formula: Monster utility = monster_multiplier × resources_allocated. 4 parameters.

## interdisciplinary (1 topics, 5 simulations)

### Complexity Science (5 simulations)
- [Ant Colony Optimization](/lab/complexity-science/ant-colony): Watch virtual ants solve the traveling salesman problem using pheromone trails — emergent intelligence from simple insects finding near-optimal routes Formula: P(i→j) = τ(i,j)^α · η(i,j)^β / Σ τ(i,k)^α · η(i,k)^β. 4 parameters.
- [Boids Flocking Algorithm](/lab/complexity-science/boids-flocking): Watch emergent flocking behavior arise from three simple rules — separation, alignment, and cohesion — with no leader and no central control Formula: v_sep = -Σ (neighbor_pos - boid_pos) / |distance|². 5 parameters.
- [Edge of Chaos (1D Cellular Automata)](/lab/complexity-science/edge-of-chaos): Explore Wolfram's 256 elementary cellular automata — from dead uniformity through complex computation to pure chaos — and find the edge where Rule 110 achieves Turing completeness Formula: λ = (k^n − n_q) / k^n (Langton's lambda). 4 parameters.
- [Self-Organized Criticality (Abelian Sandpile)](/lab/complexity-science/sandpile): Watch a sandpile self-organize to criticality — where adding a single grain can trigger avalanches of any size, following a power law that explains earthquakes and market crashes Formula: P(s) ~ s^(-τ), τ ≈ 1.1 (avalanche size distribution). 4 parameters.
- [Turing Morphogenesis Patterns](/lab/complexity-science/turing-patterns): Simulate Alan Turing's activator-inhibitor model of biological pattern formation — watch spots, stripes, and labyrinths emerge from random noise Formula: ∂a/∂t = Da·∇²a + f(a,h) — activator equation. 4 parameters.

## life-sciences (18 topics, 90 simulations)

### Aquaculture & Fish Farming Science (5 simulations)
- [Ammonia-Nitrite-Nitrate Nitrogen Cycle](/lab/aquaculture/water-quality): Simulate the nitrification cycle in a fish pond showing ammonia production, bacterial conversion to nitrite and nitrate, and toxicity thresholds Formula: NH₃_fraction = 1 / (1 + Math.pow(10, pKa - pH)), where pKa ≈ 9.25 at 25°C. 4 parameters.
- [Dissolved Oxygen Dynamics & Aeration](/lab/aquaculture/dissolved-oxygen): Simulate dissolved oxygen levels in a fish pond under varying temperature, biomass load, and aeration rates to prevent hypoxic events Formula: DO_sat = 14.62 - 0.3898·T + 0.006969·T² - 5.897e-5·T³. 4 parameters.
- [Feed Conversion Ratio & Growth Modeling](/lab/aquaculture/feed-conversion): Model fish growth curves and feed conversion ratios under varying feeding rates, protein levels, and temperature regimes Formula: FCR = Total feed consumed / Total weight gain. 4 parameters.
- [Recirculating Aquaculture System (RAS) Biofilter Performance](/lab/aquaculture/recirculating-system): Design and simulate a RAS biofilter system showing water recirculation, ammonia removal efficiency, and energy requirements Formula: TAN_removal_rate = media_SSA × media_volume × k_nitr × TAN / (K_m + TAN). 4 parameters.
- [Stocking Density & Carrying Capacity](/lab/aquaculture/stocking-density): Model the relationship between fish stocking density, biomass growth, carrying capacity, and mortality to optimize production without overcrowding Formula: Biomass(t) = K / (1 + ((K - B₀)/B₀) · Math.pow(e, -r·t)) (logistic growth). 4 parameters.

### Astrobiology & Origin of Life (5 simulations)
- [Circumstellar Habitable Zone Calculator](/lab/astrobiology/habitable-zone): Calculate the inner and outer edges of the habitable zone for any star based on luminosity, effective temperature, and albedo feedback Formula: r_inner = √(L★ / 1.107) AU (runaway greenhouse). 4 parameters.
- [Extremophile Survival Boundaries](/lab/astrobiology/extremophile-limits): Explore the temperature, pH, radiation, and pressure limits of extremophile organisms and map their survival envelopes Formula: S(T) = exp(-(T - T_opt)² / 2σ_T²) — thermal survival curve. 4 parameters.
- [Habitable Worlds Probability Estimator](/lab/astrobiology/drake-parameters): Estimate the probability of habitable worlds using stellar formation rates, planetary occurrence, and habitability constraints Formula: N_habitable = R★ × f_p × n_hz × f_h × T_galaxy. 4 parameters.
- [Miller-Urey Prebiotic Chemistry Simulation](/lab/astrobiology/miller-urey): Simulate the Miller-Urey experiment — model amino acid synthesis rates from primitive atmospheric gases under electrical discharge Formula: Y = k × [CH₄]^a × [NH₃]^b × exp(-Ea / RT). 4 parameters.
- [Spectral Biosignature Analysis](/lab/astrobiology/biosignature-detection): Analyze exoplanet transmission spectra for biosignature gases — oxygen, methane, ozone, and water vapor absorption features Formula: τ(λ) = σ(λ) × N × L — optical depth from cross-section, column density, path length. 4 parameters.

### Bioinformatics & Computational Biology (5 simulations)
- [De Bruijn Graph Genome Assembly](/lab/bioinformatics/genome-assembly): Simulate genome assembly using de Bruijn graphs — explore how k-mer size, read length, and coverage depth affect assembly contiguity and correctness Formula: k-mers per read = L - k + 1. 4 parameters.
- [Needleman-Wunsch Global Sequence Alignment](/lab/bioinformatics/sequence-alignment): Simulate the Needleman-Wunsch dynamic programming algorithm for global sequence alignment — explore how scoring parameters affect optimal alignments Formula: F(i,j) = max(F(i-1,j-1) + s(xi,yj), F(i-1,j) + g, F(i,j-1) + g). 4 parameters.
- [Protein Folding Energy Landscape](/lab/bioinformatics/protein-folding): Simulate a protein folding energy landscape — explore how temperature, hydrophobic strength, and chain length affect the folding funnel and native state stability Formula: ΔG = ΔH - TΔS (Gibbs free energy of folding). 4 parameters.
- [RNA-seq Differential Gene Expression](/lab/bioinformatics/gene-expression): Simulate RNA-seq differential expression analysis — explore how fold change, sample size, and read depth affect statistical power to detect differentially expressed genes Formula: Var(Y) = μ + φμ² (negative binomial variance). 4 parameters.
- [UPGMA Phylogenetic Tree Construction](/lab/bioinformatics/phylogenetic-tree): Simulate UPGMA hierarchical clustering to build phylogenetic trees from distance matrices — explore how sequence divergence maps to evolutionary relationships Formula: d(A,B) = Σ differences / alignment_length (p-distance). 4 parameters.

### Circadian Biology & Chronobiology (5 simulations)
- [Clock Gene Network Oscillator](/lab/circadian-biology/clock-gene-network): Simulate the molecular clock gene feedback loop — explore how CLOCK/BMAL1 activation, PER/CRY inhibition, and degradation rates generate ~24-hour oscillations Formula: dM/dt = k_t · K^n / (K^n + P^n) - k_dm · M — mRNA dynamics with Hill repression. 4 parameters.
- [Jet Lag Recovery Model](/lab/circadian-biology/jet-lag-model): Simulate jet lag recovery dynamics — model how time zone shifts, travel direction, light exposure, and individual chronotype affect circadian re-entrainment speed Formula: φ(n) = φ(0) + n·(τ-24) + Σ Δφ_light(i) — daily phase accumulation. 4 parameters.
- [Light Entrainment & Zeitgeber Strength](/lab/circadian-biology/light-entrainment): Simulate circadian entrainment to the light-dark cycle — explore how photoperiod, light intensity, spectral composition, and intrinsic period determine phase angle and stability Formula: dφ/dt = (τ - T)/τ + (Z/τ)·PRC(φ) — Winfree entrainment equation. 4 parameters.
- [Phase Response Curve (PRC)](/lab/circadian-biology/phase-response-curve): Simulate the circadian phase response curve to light pulses — explore how timing, intensity, and duration of light exposure advance or delay the biological clock Formula: Δφ(t) = A·sin(2π·t/τ) — Type 1 (weak) PRC sinusoidal approximation. 4 parameters.
- [Social Jet Lag & Health Impact](/lab/circadian-biology/social-jet-lag): Simulate social jet lag — quantify the mismatch between biological and social clocks, and explore its effects on metabolic health, cognitive performance, and disease risk Formula: SJL = |MSF - MSW| — social jet lag in hours. 4 parameters.

### Cognitive Neuroscience (5 simulations)
- [Attention Network Selection Model](/lab/cognitive-neuroscience/attention-network): Simulate attentional filtering — explore how signal strength, distractor count, and top-down bias shape the selection of relevant stimuli Formula: acc = Φ((S × (1 + β) - μ_d) / (σ × √N)). 4 parameters.
- [Mirror Neuron Activation Model](/lab/cognitive-neuroscience/mirror-neuron): Simulate mirror neuron firing — explore how action observation, motor familiarity, and social context modulate the mirror system response Formula: FR = FR₀ × (S_a / 100) × log₁₀(E + 1) × (V / 100). 4 parameters.
- [Neural Oscillation & Brainwave Synchrony](/lab/cognitive-neuroscience/neural-oscillation): Simulate cortical oscillations — explore how frequency bands, phase coupling, and amplitude modulation produce EEG rhythms linked to cognition Formula: x(t) = A_α × sin(2π × f_α × t) + A_γ × (1 + κ × sin(2π × f_θ × t)) × sin(2π × f_γ × t). 5 parameters.
- [Neuroplasticity & Synaptic Strength](/lab/cognitive-neuroscience/neuroplasticity): Simulate experience-dependent neuroplasticity — explore how learning rate, spike timing, and repetition drive long-term synaptic potentiation and depression Formula: Δw = η × A₊ × exp(-Δt/τ₊) if Δt > 0 (LTP). 4 parameters.
- [Working Memory Capacity Model](/lab/cognitive-neuroscience/working-memory): Simulate working memory capacity — explore how chunk size, rehearsal rate, and decay time limit the number of items held in conscious awareness Formula: C_eff = k × R × τ × (1 - A/100). 4 parameters.

### Comparative Anatomy & Morphology (5 simulations)
- [Allometric Scaling Laws](/lab/comparative-anatomy/allometric-scaling): Explore how body mass drives organ size, metabolic rate, and lifespan across species using power-law scaling relationships Formula: Y = a × M^b (general allometric equation). 4 parameters.
- [Cephalization Index & Brain Allometry](/lab/comparative-anatomy/cephalization-index): Compare brain-to-body mass ratios across species — quantify encephalization quotients and explore what drives cognitive complexity in the animal kingdom Formula: EQ = Brain_mass / (0.12 × Body_mass^0.76). 4 parameters.
- [Convergent Evolution & Analogous Forms](/lab/comparative-anatomy/convergent-evolution): Visualize how unrelated lineages independently evolve similar body plans — streamlined shapes in sharks, dolphins, and ichthyosaurs under identical hydrodynamic pressures Formula: F_drag = 0.5 × ρ × V² × Cd × A (drag equation). 4 parameters.
- [Homologous Structures & Skeletal Mapping](/lab/comparative-anatomy/homologous-structures): Map homologous bones across vertebrate forelimbs — compare human arms, bat wings, whale flippers, and horse legs to reveal the shared ancestral blueprint Formula: SI = 1 - (1/N) × Σ |L_bone_A - L_bone_B| / max(L_bone_A, L_bone_B). 4 parameters.
- [Vestigial Organs & Evolutionary Regression](/lab/comparative-anatomy/vestigial-organs): Model the regression of vestigial structures over evolutionary time — trace how functional organs become rudimentary under relaxed selection pressure Formula: S(t) = S₀ × exp(s × t) (organ size decay). 4 parameters.

### Endocrinology & Hormone Signaling (5 simulations)
- [Circadian Cortisol Rhythm Simulator](/lab/endocrinology/circadian-cortisol): Visualize the diurnal cortisol rhythm driven by the suprachiasmatic nucleus, showing peak morning levels and nocturnal nadir with adjustable circadian parameters Formula: C(t) = mesor + amplitude * cos(2 * pi * (t - acrophase) / 24). 4 parameters.
- [Glucose-Insulin Homeostasis Simulator](/lab/endocrinology/insulin-glucose): Interactive model of glucose-insulin dynamics showing how beta-cell secretion and tissue sensitivity maintain blood sugar within physiological range Formula: dG/dt = glucose_input + hepatic_output - insulin_sensitivity * I * G / G_ref. 4 parameters.
- [Hormone-Receptor Binding Kinetics Simulator](/lab/endocrinology/hormone-receptor): Interactive model of ligand-receptor binding equilibrium showing how hormone concentration, receptor density, and affinity determine cellular signaling response Formula: Fractional occupancy = [L]^nH / (Kd^nH + [L]^nH). 4 parameters.
- [HPA Axis Feedback Simulator](/lab/endocrinology/hpa-axis): Model the hypothalamic-pituitary-adrenal axis negative feedback loop showing how CRH, ACTH, and cortisol levels respond to stress and regulate each other Formula: dCortisol/dt = ACTH_sensitivity * ACTH - clearance * Cortisol. 4 parameters.
- [Thyroid Regulation Simulator](/lab/endocrinology/thyroid-regulation): Model the TSH-T3-T4 feedback loop showing how the hypothalamic-pituitary-thyroid axis maintains metabolic homeostasis Formula: TSH = TRH_drive / (1 + feedback_sensitivity * T4). 4 parameters.

### Entomology & Insect Science (5 simulations)
- [Caste Ratio: Eusocial Colony Caste Allocation](/lab/entomology/caste-ratio): Optimize eusocial colony fitness by adjusting worker-soldier caste ratios — explore the trade-off between foraging capacity and colony defense Formula: Workers = N - floor(N × S%/100) - 1. 4 parameters.
- [Insect Flight: Wing Kinematics & Lift Generation](/lab/entomology/insect-flight): Simulate insect wing kinematics — explore how stroke frequency, wing span, and stroke amplitude generate aerodynamic lift at low Reynolds numbers Formula: F_lift = 0.5 × ρ × v_tip² × S_wing × C_L. 4 parameters.
- [Insect Population Cycle: Outbreak Dynamics & Voltinism](/lab/entomology/population-cycle): Model insect population outbreaks using discrete logistic growth — explore how growth rate, carrying capacity, and generation count drive boom-bust cycles Formula: N(t+1) = r × N(t) × (1 - N(t) / K). 4 parameters.
- [Metamorphosis Timing: Degree-Day Development Model](/lab/entomology/metamorphosis-timing): Predict insect development timing using degree-day accumulation — model how temperature drives progression through egg, larval, pupal, and adult stages Formula: DD_daily = max(0, T_mean - T_base). 4 parameters.
- [Pollination Efficiency: Flower Visit Rate & Pollen Transfer](/lab/entomology/pollination-efficiency): Model pollination efficiency — explore how visit rate, transfer efficiency, flower density, and foraging time determine fruit set in pollinator-dependent crops Formula: Fruit set = min(1, V × T × η / D) × 100%. 4 parameters.

### Enzyme Kinetics & Catalysis (5 simulations)
- [Allosteric Regulation & Hill Equation](/lab/enzyme-kinetics/allosteric-regulation): Simulate allosteric cooperativity — explore how the Hill coefficient shapes the sigmoidal response curve and enables switch-like metabolic regulation Formula: v = Vmax·[S]ⁿ / (K₀.₅ⁿ + [S]ⁿ) (Hill equation). 4 parameters.
- [Competitive Inhibition](/lab/enzyme-kinetics/competitive-inhibition): Simulate competitive enzyme inhibition — explore how inhibitor concentration and Ki alter apparent Km while Vmax remains unchanged Formula: v = Vmax·[S] / (Km·(1 + [I]/Ki) + [S]). 4 parameters.
- [Enzyme Temperature Response & Arrhenius Kinetics](/lab/enzyme-kinetics/enzyme-temperature): Simulate enzyme activity vs temperature — explore Arrhenius activation energy and thermal denaturation that create an optimal temperature peak Formula: k = A·exp(−Ea/(R·T)) (Arrhenius equation). 4 parameters.
- [Michaelis-Menten Kinetics & Lineweaver-Burk](/lab/enzyme-kinetics/michaelis-menten): Simulate Michaelis-Menten enzyme kinetics — explore how Vmax, Km, and substrate concentration determine reaction velocity with Lineweaver-Burk linearization Formula: v = Vmax·[S] / (Km + [S]). 4 parameters.
- [Multi-Substrate Mechanisms](/lab/enzyme-kinetics/multi-substrate): Simulate ping-pong and sequential bi-substrate enzyme mechanisms — explore how two substrates interact with the enzyme in ordered and random pathways Formula: v_PP = Vmax·[A]·[B] / ([A]·Km_B + [B]·Km_A + [A]·[B]). 4 parameters.

### Ethology & Animal Behavior (5 simulations)
- [Fixed Action Pattern Simulator](/lab/ethology/fixed-action-pattern): Interactive model of innate behavioral sequences triggered by sign stimuli, showing how stimulus intensity and threshold determine the probability and completeness of fixed action patterns Formula: P(trigger) = 1 / (1 + exp(-(stimulus - threshold) / k)). 4 parameters.
- [Honeybee Waggle Dance Simulator](/lab/ethology/waggle-dance): Interactive visualization of Karl von Frisch's waggle dance communication system, showing how bees encode food source distance and direction through dance angle and duration Formula: Waggle duration (s) ≈ distance (m) / 750. 4 parameters.
- [Imprinting & Critical Period Simulator](/lab/ethology/imprinting): Interactive model of Lorenz's filial imprinting showing how sensitive periods, stimulus exposure timing, and irreversibility shape early behavioral attachment Formula: Imprint strength = salience × overlap(exposure, critical_period) × log(1 + duration). 4 parameters.
- [Kin Selection & Hamilton's Rule Simulator](/lab/ethology/kin-selection): Interactive model of Hamilton's rule (rB > C) showing how genetic relatedness drives the evolution of altruistic behavior among relatives Formula: Hamilton's rule: rB > C. 5 parameters.
- [Optimal Foraging Theory Simulator](/lab/ethology/optimal-foraging): Interactive model of the marginal value theorem showing how animals optimize time allocation across food patches of varying quality Formula: E(t) = Q × (1 - exp(-d × t)). 4 parameters.

### Forensic Genetics & DNA Profiling (5 simulations)
- [DNA Degradation Kinetics & Fragment Size Distribution](/lab/forensic-genetics/degradation-kinetics): Model time-dependent DNA degradation under environmental conditions and predict recoverable fragment sizes for forensic STR analysis Formula: k = A × exp(-Ea / (R × T)) × (1 + α × RH/100) × (1 + β × UV) — Arrhenius with environmental modifiers. 5 parameters.
- [DNA Mixture Deconvolution & Contributor Estimation](/lab/forensic-genetics/mixture-analysis): Analyze mixed DNA profiles by estimating contributor ratios, stochastic thresholds, and mixture likelihood ratios across STR loci Formula: Major_RFU = Total_RFU × R / (R + 1) where R is the mixture ratio. 5 parameters.
- [Forensic Population Statistics & Match Probability](/lab/forensic-genetics/population-statistics): Calculate random match probabilities, likelihood ratios, and database search statistics using Hardy-Weinberg expectations and population correction factors Formula: RMP = ∏ P(genotype_i) for i = 1 to n loci. 4 parameters.
- [Paternity Index & Likelihood Ratio](/lab/forensic-genetics/paternity-index): Calculate the combined paternity index and probability of paternity from STR genotype data using likelihood ratios across multiple loci Formula: Per-locus PI: PI_i = P(child genotype | alleged father is father) / P(child genotype | random man is father). 4 parameters.
- [STR Loci & Allele Frequency Analysis](/lab/forensic-genetics/str-profiling): Visualize Short Tandem Repeat allele frequency distributions across forensic loci and calculate per-locus heterozygosity and discrimination power Formula: Heterozygosity: H = 1 - Σ(p_i²) where p_i is frequency of allele i. 4 parameters.

### Mycology & Fungal Biology (5 simulations)
- [Decomposition Rate: Lignocellulose Decay & Carbon Cycling](/lab/mycology/decomposition-rate): Model fungal decomposition of lignocellulose — explore how moisture, temperature, and lignin content determine decay rate, half-life, and carbon flux Formula: M(t) = M₀ × exp(-k × t). 4 parameters.
- [Fungal Pharmacology: Bioactive Compound Dose-Response](/lab/mycology/fungal-pharmacology): Model dose-response curves for fungal bioactive compounds — explore how dose, EC50, Hill coefficient, and maximum efficacy determine pharmacological response Formula: E = E_max × Dⁿ / (EC₅₀ⁿ + Dⁿ) [Hill equation]. 4 parameters.
- [Mycelial Growth: Radial Colony Expansion & Nutrient Uptake](/lab/mycology/mycelial-growth): Model radial mycelial colony expansion — explore how growth rate, nutrient concentration, temperature, and humidity drive hyphal extension and biomass accumulation Formula: K_eff = Kᵣ × exp(-(T-25)²/100) × min(1, RH/95) × C/(C+5). 4 parameters.
- [Spore Dispersal: Wind-Borne Spore Transport Model](/lab/mycology/spore-dispersal): Simulate wind-borne fungal spore dispersal — model how wind speed, release height, spore diameter, and turbulence determine transport distance and deposition patterns Formula: V_settle = 2ρ g r² / (9μ) [Stokes' law]. 4 parameters.
- [Symbiosis Model: Mycorrhizal Nutrient Exchange](/lab/mycology/symbiosis-model): Model mycorrhizal symbiosis — explore how carbon allocation, phosphorus availability, root density, and fungal cover determine nutrient exchange and net plant benefit Formula: P_uptake = P_soil × colonization × efficiency × root_factor. 4 parameters.

### Paleoanthropology & Human Evolution (5 simulations)
- [Brain Size Evolution Over Time](/lab/paleoanthropology/cranial-capacity): Simulate cranial capacity evolution — explore how encephalization rate, dietary quality, and social group size correlate with brain volume across hominin species Formula: EQ = brain_mass / (0.12 × body_mass^0.67) (encephalization quotient). 4 parameters.
- [K-Ar & U-Pb Radiometric Age Determination](/lab/paleoanthropology/radiometric-dating): Simulate radiometric dating — explore how parent isotope ratio, half-life, decay constant, and measurement precision determine age calculations for geological and fossil samples Formula: t = (1/λ) × ln(1 + D/P) (radiometric age equation). 4 parameters.
- [Lithic Technology & Cognitive Evolution](/lab/paleoanthropology/tool-complexity): Simulate stone tool complexity — explore how reduction steps, planning depth, technique variety, and symmetry requirements reflect cognitive evolution in hominins Formula: C = N × log₂(Tk+1) × (1 + Dp/10) (complexity index). 4 parameters.
- [Out-of-Africa Dispersal Simulation](/lab/paleoanthropology/migration-model): Simulate hominin dispersal from Africa — explore how migration speed, population growth, carrying capacity, and climate barriers shape expansion patterns Formula: t_arrival = distance / (v_migration × generation_time) (simple dispersal time). 4 parameters.
- [Stable Isotope & Dental Diet Analysis](/lab/paleoanthropology/dietary-reconstruction): Simulate dietary reconstruction — explore how carbon-13, nitrogen-15 isotope ratios, dental microwear, and tooth morphology reveal ancient hominin diets Formula: δ¹³C = ((¹³C/¹²C)_sample / (¹³C/¹²C)_standard - 1) × 1000 (delta notation). 4 parameters.

### Paleobotany & Fossil Plants (5 simulations)
- [Fossil Wood Growth Rings & Paleoclimate](/lab/paleobotany/wood-anatomy): Analyze growth ring patterns in petrified wood to reconstruct paleoclimate seasonality, temperature, and precipitation Formula: MS = (1/n) * sum(|2(R_i+1 - R_i) / (R_i+1 + R_i)|) (mean sensitivity). 4 parameters.
- [Leaf Margin Analysis & Paleotemperature](/lab/paleobotany/leaf-margin): Estimate mean annual temperature from the percentage of entire-margined (smooth-edged) leaves in a fossil flora using leaf margin analysis Formula: MAT = a + b * P (Wolfe linear regression, P = fraction of entire-margined species). 4 parameters.
- [Phytolith Morphology & Habitat Reconstruction](/lab/paleobotany/phytolith-analysis): Classify silica phytolith morphotypes from sediment samples to reconstruct grassland versus forest habitats through geological time Formula: D/P = n_forest / (n_forest + n_grass) (tree cover index). 4 parameters.
- [Pollen Percentage Diagram & Vegetation Reconstruction](/lab/paleobotany/pollen-diagram): Build a pollen percentage diagram from simulated core data and reconstruct shifting vegetation zones through time Formula: P_i% = (n_i / N_total) * 100 (pollen percentage for taxon i). 4 parameters.
- [Stomatal Index & Paleo-CO2 Reconstruction](/lab/paleobotany/stomatal-density): Reconstruct ancient atmospheric CO2 concentrations from the stomatal index of fossil leaves using the inverse relationship between stomatal density and CO2 Formula: SI = (S / (S + E)) * 100 (stomatal index, S = stomata, E = epidermal cells). 4 parameters.

### Paleontology & Deep Time (5 simulations)
- [Continental Drift & Plate Tectonics Simulator](/lab/paleontology/plate-tectonics-evolution): Watch continents drift across the globe over 300 million years — from Pangaea to today's configuration and into the future Formula: Plate displacement: d = v × t where v ≈ 2-10 cm/yr. 4 parameters.
- [Dinosaur Body Size Scaling Laws Simulator](/lab/paleontology/dinosaur-size): Explore allometric scaling laws that governed dinosaur body size — from metabolic rate to bone strength and the physics of gigantism Formula: Kleiber's law: BMR = 70 × M^0.75 watts (endotherm) or 10 × M^0.75 (ectotherm). 4 parameters.
- [Mass Extinction Events Timeline Simulator](/lab/paleontology/mass-extinction): Explore the Big Five mass extinctions across 500 million years — visualize species loss, recovery time, and the causes behind each catastrophe Formula: Extinction magnitude: E = 1 - (S_after / S_before) where S = number of species. 4 parameters.
- [Phylogenetic Tree Reconstruction Simulator](/lab/paleontology/phylogenetic-tree): Build phylogenetic trees from trait matrices using distance-based and parsimony methods — explore how evolutionary relationships are inferred Formula: Number of unrooted trees for n taxa: T(n) = (2n-5)!! = (2n-5)!/(2^(n-3) × (n-3)!). 4 parameters.
- [Radiometric Fossil Dating Methods Simulator](/lab/paleontology/fossil-dating): Explore how radioactive decay curves are used to determine the age of fossils and rocks — from carbon-14 to uranium-lead dating Formula: Radioactive decay: N(t) = N₀ × (1/2)^(t/t½). 4 parameters.

### Parasitology & Host-Parasite Dynamics (5 simulations)
- [Anderson-May Host-Parasite Model](/lab/parasitology/host-parasite-model): Simulate the Anderson-May SIR framework with parasite load, showing how transmission rate, host recovery, and parasite-induced mortality shape endemic equilibria Formula: R0 = beta / (gamma + alpha). 4 parameters.
- [Helminth Burden Distribution Simulator](/lab/parasitology/helminth-burden): Visualize the overdispersed distribution of worm burdens across a host population using the negative binomial model with adjustable aggregation Formula: P(X=j) = C(j+k-1, j) * (mu/(mu+k))^j * (k/(mu+k))^k. 4 parameters.
- [Mosquito Vectorial Capacity Simulator](/lab/parasitology/vector-capacity): Calculate vectorial capacity for mosquito-borne disease transmission incorporating biting rate, survival probability, vector competence, and extrinsic incubation period Formula: C = m * a^2 * p^n / (-ln(p)). 4 parameters.
- [Parasite Drug Resistance Evolution Simulator](/lab/parasitology/drug-resistance): Model the evolutionary dynamics of drug resistance in parasite populations showing how treatment pressure, fitness costs, and mutation rates drive resistance spread Formula: s = treatment_coverage * drug_efficacy - fitness_cost. 4 parameters.
- [Plasmodium Erythrocytic Cycle Simulator](/lab/parasitology/malaria-lifecycle): Model the intra-erythrocytic lifecycle of Plasmodium falciparum showing merozoite invasion, replication, and red blood cell lysis dynamics Formula: Multiplication rate R = merozoites * invasion_efficiency * (1 - immune_clearance). 4 parameters.

### Speech Science & Phonetics (5 simulations)
- [Articulatory Speech Synthesis](/lab/speech-science/speech-synthesis): Synthesize vowel sounds from articulatory parameters — control tongue position, jaw opening, and lip rounding to shape the vocal tract and generate formant patterns Formula: F1 ≈ k₁ × (jaw_opening + (1 - tongue_height)). 5 parameters.
- [Pitch Tracking & Intonation](/lab/speech-science/pitch-tracking): Visualize fundamental frequency contours — explore how pitch patterns encode questions, statements, emotions, and tonal distinctions in speech Formula: Semitones = 12 × log₂(f₁ / f₂). 4 parameters.
- [Speech Spectrogram Viewer](/lab/speech-science/spectrogram-viewer): Generate and read spectrograms of synthetic speech signals — visualize how frequency content evolves over time for different vowels and consonants Formula: Δf × Δt ≥ 1 (time-frequency uncertainty). 5 parameters.
- [Voice Onset Time (VOT) Analyzer](/lab/speech-science/voice-onset-time): Visualize voice onset time — the critical timing difference between voiced and voiceless stop consonants across languages Formula: VOT = t_voicing_onset - t_burst_release. 4 parameters.
- [Vowel Formant Analysis](/lab/speech-science/formant-analysis): Explore how the first two formant frequencies (F1 and F2) define vowel identity — map any vowel on the acoustic vowel space in real time Formula: F_n ≈ (2n - 1) × c / (4L) for a uniform tube model. 4 parameters.

### Xenobiology & Alien Life (5 simulations)
- [Alternative Biochemistry Designer](/lab/xenobiology/alternative-biochemistry): Design non-carbon biochemistries — explore how solvent polarity, backbone element, and temperature range constrain the possibility space of alien life Formula: S = E_bond / (k_B × T) — thermodynamic stability ratio. 4 parameters.
- [Extremophile Survival Limits](/lab/xenobiology/extremophile-limits): Push extremophiles to their limits — model how temperature, pH, radiation dose, and salinity define the boundary of habitable space for life Formula: P_survival = P(T) × P(pH) × P(D) × P(S) — multi-stress survival. 4 parameters.
- [Shadow Biosphere Detector](/lab/xenobiology/shadow-biosphere): Simulate detection of a hypothetical shadow biosphere — organisms with non-standard biochemistry hiding in plain sight on Earth Formula: P(shadow|detected) = (σ × f_s) / (σ × f_s + FPR × (1 − f_s)). 4 parameters.
- [Silicon-Based Life Simulator](/lab/xenobiology/silicon-life): Model silicon-based metabolism — explore how temperature, pressure, and solvent conditions determine whether silicon chemistry can support living systems Formula: τ_SiSi = exp(226000 / (R × T)) — Si-Si bond lifetime. 4 parameters.
- [XNA Polymer Designer](/lab/xenobiology/xna-polymers): Design xeno-nucleic acid polymers — explore how sugar backbone modifications affect base-pairing stability, replication fidelity, and information density Formula: I = bp × log₂(A) — total information capacity in bits. 4 parameters.

## mathematics (17 topics, 87 simulations)

### Biostatistics & Clinical Trials (5 simulations)
- [Adaptive Clinical Trial Design](/lab/biostatistics/clinical-trial-design): Simulate a multi-arm adaptive clinical trial with interim analyses — visualize efficacy boundaries, futility stopping, and arm-dropping decisions Formula: α*(t) = 2 − 2Φ(z_{α/2} / √t) (O'Brien-Fleming spending). 4 parameters.
- [Kaplan-Meier Survival Analysis](/lab/biostatistics/survival-analysis): Generate and compare Kaplan-Meier survival curves for treatment vs. control groups — visualize censoring, confidence intervals, and hazard ratios Formula: S(t) = ∏ (1 − d_i / n_i) for all t_i ≤ t. 4 parameters.
- [Meta-Analysis Forest Plot](/lab/biostatistics/meta-analysis): Combine effect estimates from multiple studies into a pooled estimate — visualize forest plots, heterogeneity, and the impact of individual study weights Formula: θ̂ = Σ w_i · θ_i / Σ w_i  where  w_i = 1 / (σ²_i + τ²). 4 parameters.
- [Regression Diagnostics & Residual Analysis](/lab/biostatistics/regression-diagnostics): Fit a regression model and diagnose violations — visualize residual plots, leverage, Cook's distance, and normality checks Formula: R² = 1 − Σ(y_i − ŷ_i)² / Σ(y_i − ȳ)². 4 parameters.
- [Sample Size & Power Calculator](/lab/biostatistics/sample-size): Calculate the sample size needed to detect a treatment effect with specified power — visualize the relationship between effect size, sample size, and Type I/II error Formula: n = ((z_{α/2} + z_β) / d)² per group. 4 parameters.

### Computational Geometry (5 simulations)
- [Convex Hull Algorithm Visualizer](/lab/computational-geometry/convex-hull): Visualize convex hull construction — watch gift-wrapping and Graham scan algorithms build the tightest boundary around a point set step by step Formula: A = ½|Σᵢ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)| — shoelace formula for area. 4 parameters.
- [Delaunay Triangulation Builder](/lab/computational-geometry/delaunay-triangulation): Construct Delaunay triangulations interactively — observe the empty circumcircle property, flip operations, and mesh quality metrics in real time Formula: F = 2n − 2 − b — triangle count (b = boundary vertices). 4 parameters.
- [Line Sweep Intersection Detector](/lab/computational-geometry/line-sweep): Visualize the sweep-line algorithm for detecting segment intersections — watch the event queue and status structure in real time Formula: Bentley-Ottmann: O((n + k) log n) time, O(n + k) space. 4 parameters.
- [Polygon Clipping Visualizer](/lab/computational-geometry/polygon-clipping): Visualize polygon clipping operations — see Sutherland-Hodgman and Weiler-Atherton algorithms clip subject polygons against convex and concave windows Formula: Sutherland-Hodgman: O(Sₛ × Sᶜ) — subject edges × clip edges. 4 parameters.
- [Voronoi Diagram Generator](/lab/computational-geometry/voronoi-diagram): Generate interactive Voronoi diagrams — explore how seed point placement, distance metrics, and perturbation affect spatial partitioning Formula: V(pᵢ) = {x : d(x, pᵢ) ≤ d(x, pⱼ) ∀j≠i} — Voronoi cell definition. 4 parameters.

### Decision Theory (5 simulations)
- [Auction Theory Simulator](/lab/decision-theory/optimal-auction): Compare English, Dutch, first-price sealed, and Vickrey auctions — see revenue equivalence, bidding strategies, and efficiency in action Formula: Expected revenue (n uniform bidders): E[Revenue] = (n-1)/(n+1) · v_max. 4 parameters.
- [Kelly Criterion Betting](/lab/decision-theory/kelly-criterion): Visualize optimal bet sizing with the Kelly criterion — compare wealth trajectories for Kelly, over-betting, under-betting, and all-in strategies Formula: Kelly fraction: f* = p/b - q/a = (p·a - q·b) / (a·b). 5 parameters.
- [Markowitz Portfolio Frontier](/lab/decision-theory/portfolio-optimization): Explore mean-variance portfolio optimization — visualize the efficient frontier, tangency portfolio, and diversification benefits Formula: Portfolio return: Rp = Σ wᵢ · Rᵢ. 4 parameters.
- [Multi-Armed Bandit (Explore vs Exploit)](/lab/decision-theory/multi-armed-bandit): Compare exploration strategies in the multi-armed bandit problem — random, epsilon-greedy, UCB1, and Thompson sampling Formula: UCB1 selection: aₜ = argmax[x̄ᵢ + √(2·ln(t)/nᵢ)]. 4 parameters.
- [Prospect Theory & Loss Aversion](/lab/decision-theory/prospect-theory-gambles): Visualize the S-shaped value function of prospect theory — see how loss aversion, diminishing sensitivity, and reference dependence distort rational decision-making Formula: Value function: v(x) = x^α for x ≥ 0; v(x) = -λ·(-x)^β for x < 0. 4 parameters.

### Game Theory (5 simulations)
- [Auction Theory Simulator](/lab/game-theory/auction-theory): Compare first-price, second-price (Vickrey), and English auctions — visualize bidding strategies, revenue, and efficiency with configurable bidder populations Formula: Vickrey: winner pays second-highest bid; dominant strategy = bid true value. 5 parameters.
- [Evolutionary Stable Strategy (ESS) Simulator](/lab/game-theory/evolutionary-stable-strategy): Simulate the Hawk-Dove game with replicator dynamics — watch population frequencies converge to the evolutionarily stable strategy predicted by V/C Formula: Hawk vs Hawk: (V-C)/2 each. 4 parameters.
- [Iterated Prisoner's Dilemma Simulator](/lab/game-theory/prisoners-dilemma): Simulate Axelrod-style tournaments with Tit-for-Tat, Always Defect, and Always Cooperate strategies competing across hundreds of rounds with configurable noise Formula: Payoff matrix: T=5, R=3, P=1, S=0 with T > R > P > S and 2R > T + S. 5 parameters.
- [Nash Equilibrium Visualizer](/lab/game-theory/nash-equilibrium): Explore mixed-strategy Nash equilibria in 2x2 bimatrix games with interactive best-response diagrams Formula: P1 mixes: q* = (d1 - b1) / (a1 - c1 + d1 - b1). 4 parameters.
- [Tragedy of the Commons Simulator](/lab/game-theory/tragedy-of-commons): Model resource depletion dynamics with competing herders — watch how cooperation percentage determines whether a shared resource collapses or sustains Formula: R(t+1) = R(t) + r·R(t)·(1 - R(t)/K) - N·harvest_rate·R(t). 5 parameters.

### Geometry & Spatial Structures (5 simulations)
- [Hyperbolic Geometry & Poincaré Disk Tiling](/lab/geometry/hyperbolic-tiling): Explore hyperbolic tilings in the Poincaré disk model — visualize non-Euclidean geometry where parallel lines diverge and triangles have angle sums less than 180° Formula: Hyperbolic condition: (p-2)(q-2) > 4. 4 parameters.
- [Platonic Solids Wireframe Rotation](/lab/geometry/platonic-solids): Rotate all five Platonic solids in 3D wireframe — tetrahedron, cube, octahedron, dodecahedron, and icosahedron — and explore their dual relationships Formula: Euler's formula: V - E + F = 2. 4 parameters.
- [Projective Geometry & Perspective](/lab/geometry/projective-geometry): Explore projective transformations, vanishing points, and cross-ratios — the geometry behind perspective drawing, camera calibration, and algebraic geometry Formula: Cross-ratio: (AC·BD)/(BC·AD) — invariant under projection. 4 parameters.
- [Tessellation & Tiling Patterns](/lab/geometry/tessellation): Create tessellations with different polygon types and symmetry groups — explore how shapes tile the plane without gaps or overlaps Formula: Regular tiling condition: interior angle × k = 360° for integer k. 4 parameters.
- [Voronoi Diagram & Delaunay Triangulation](/lab/geometry/voronoi-diagram): Generate Voronoi diagrams and their dual Delaunay triangulations from seed points — explore spatial partitioning, nearest-neighbor regions, and computational geometry Formula: Euler's formula: V - E + F = 2. 4 parameters.

### Geostatistics & Spatial Analysis (5 simulations)
- [Indicator Kriging Probability Mapper](/lab/geostatistics/indicator-kriging): Map the probability of exceeding a critical cutoff grade using indicator kriging with binary-transformed spatial data Formula: I(x; z_c) = 1 if Z(x) ≥ z_c, 0 otherwise. 4 parameters.
- [Moran's I & LISA Spatial Clustering Detector](/lab/geostatistics/spatial-clustering): Detect and visualize spatial clusters and outliers using global Moran's I statistic and Local Indicators of Spatial Association Formula: I = (n/W) Σᵢ Σⱼ wᵢⱼ(xᵢ−x̄)(xⱼ−x̄) / Σ(xᵢ−x̄)². 4 parameters.
- [Ordinary Kriging Interpolation Simulator](/lab/geostatistics/kriging): Watch ordinary kriging build optimal interpolated surfaces from scattered sample points using spatial correlation structure Formula: Z*(x₀) = Σ λᵢ Z(xᵢ) where Σ λᵢ = 1. 4 parameters.
- [Ore Body Block Estimation Simulator](/lab/geostatistics/block-model): Estimate ore grades in a 2D block model grid using kriging and classify blocks as ore or waste based on economic cutoff Formula: Z*_B = Σ λᵢ Z(xᵢ) (block kriging estimate). 4 parameters.
- [Semivariogram Modeling & Range Estimation](/lab/geostatistics/variogram): Explore how spatial correlation decays with distance by fitting spherical, exponential, and Gaussian variogram models to scattered data Formula: γ(h) = C₀ + C[1.5(h/a) − 0.5(h/a)³] for h ≤ a (spherical). 4 parameters.

### Graph Theory & Networks (5 simulations)
- [Dijkstra's Shortest Path Simulator](/lab/graph-theory/shortest-path): Watch Dijkstra's algorithm explore a weighted graph step by step, revealing the shortest path from source to destination in real time Formula: d(v) = min(d(v), d(u) + w(u, v)) — edge relaxation. 4 parameters.
- [Eulerian & Hamiltonian Path Explorer](/lab/graph-theory/eulerian-path): Trace paths that visit every edge or every vertex exactly once — discover why Euler paths are easy to find but Hamiltonian paths are NP-hard Formula: Euler circuit exists ⟺ every vertex has even degree (connected graph). 4 parameters.
- [Graph Coloring & Chromatic Number](/lab/graph-theory/graph-coloring): Explore the graph coloring problem — assign colors to vertices so that no two adjacent vertices share a color and discover the chromatic number Formula: χ(G) ≤ Δ(G) + 1 — greedy upper bound (Brook's theorem: χ ≤ Δ for connected graphs that are not complete or odd cycles). 4 parameters.
- [Max Flow / Min Cut Simulator](/lab/graph-theory/network-flow): Push flow through a network from source to sink and discover the max-flow min-cut theorem — the duality that connects maximum throughput to minimum bottleneck Formula: max flow(s, t) = min cut(S, T) — max-flow min-cut theorem. 4 parameters.
- [Minimum Spanning Tree Simulator](/lab/graph-theory/minimum-spanning-tree): Build the cheapest network connecting all nodes — watch Kruskal's algorithm sort edges and grow a minimum spanning tree step by step Formula: MST edges = V - 1 for a connected graph with V vertices. 4 parameters.

### Information Theory (4 simulations)
- [Channel Capacity Explorer](/lab/information-theory/channel-capacity): Visualize Shannon's channel capacity theorem — the ultimate speed limit of communication. Explore how bandwidth, signal-to-noise ratio, and modulation scheme determine the maximum data rate, and see c Formula: C = B \cdot \log_2(1 + \text{SNR}). 3 parameters.
- [Data Compression Simulator](/lab/information-theory/data-compression): Visualize how Huffman coding compresses data by exploiting symbol frequency imbalance. Compare original size, compressed size, and the theoretical Shannon limit across different source types — from En Formula: \text{Compressed size} \geq N \cdot H \text{ bits}. 3 parameters.
- [Error Correction Simulator](/lab/information-theory/error-correction): Watch error correction codes detect and fix transmission errors in real time. Compare uncoded transmission, triple repetition, and Hamming(7,4) codes to see how redundancy trades bandwidth for reliabi Formula: \text{Hamming}(7,4): \text{syndrome} = H \cdot r^T. 3 parameters.
- [Shannon Entropy Calculator](/lab/information-theory/shannon-entropy): Explore the foundational measure of information: Shannon entropy. Adjust symbol probabilities to see how uncertainty, redundancy, and optimal code lengths change. Discover why English text carries abo Formula: H = -\sum_{i=1}^{N} p_i \cdot \log_2(p_i). 4 parameters.

### Mathematical Art (5 simulations)
- [L-System Fractal Plants](/lab/art-mathematics/l-system-plants): Grow fractal trees, ferns, bushes, and flowers using Lindenmayer systems — simple rewriting rules that produce stunningly realistic plant structures Formula: F → F[+F]F[-F]F (example production rule). 4 parameters.
- [Lissajous Figures](/lab/art-mathematics/lissajous): Draw beautiful Lissajous curves from two perpendicular oscillations — explore frequency ratios, phase shifts, and the figures used to calibrate oscilloscopes Formula: x(t) = A · sin(f_x · t + φ). 4 parameters.
- [Phyllotaxis / Sunflower Spiral](/lab/art-mathematics/phyllotaxis): Explore how the golden angle creates the optimal seed packing found in sunflowers, pinecones, and pineapples — and watch Fibonacci spirals emerge from pure mathematics Formula: θ_n = n × α (divergence angle for seed n). 4 parameters.
- [Reaction-Diffusion Patterns (Gray-Scott)](/lab/art-mathematics/reaction-diffusion): Watch organic patterns emerge from chemical equations — spots, stripes, spirals, and chaos generated by the Gray-Scott reaction-diffusion model that explains animal coat patterns Formula: ∂A/∂t = Da·∇²A − AB² + F(1 − A). 4 parameters.
- [Spirograph / Hypotrochoid Curves](/lab/art-mathematics/spirograph): Generate mesmerizing spirograph curves by rolling one circle inside another — explore the geometry of hypotrochoid and epicycloid curves through interactive animation Formula: x(t) = (R − r)cos(t) + d·cos((R − r)t / r). 5 parameters.

### Mathematical Biology & Population Dynamics (5 simulations)
- [Fisher-KPP Traveling Wave Equation](/lab/mathematical-biology/fisher-wave): Simulate Fisher-KPP traveling wavefronts — explore how diffusion and logistic growth combine to produce advancing population fronts at a minimum speed Formula: ∂u/∂t = D ∂²u/∂x² + ru(1 − u/K). 4 parameters.
- [Galton-Watson Branching Process](/lab/mathematical-biology/branching-process): Simulate stochastic branching processes — explore how offspring distribution determines extinction probability, population growth, and genealogical tree structure Formula: E[Zₙ] = μⁿ (expected population at generation n). 4 parameters.
- [Lotka-Volterra Predator-Prey Dynamics](/lab/mathematical-biology/lotka-volterra): Simulate classic predator-prey oscillations — explore how birth rates, predation efficiency, and mortality shape population cycles in ecological systems Formula: dx/dt = αx − βxy (prey growth minus predation). 4 parameters.
- [SIR Compartment Epidemic Model](/lab/mathematical-biology/sir-compartment): Simulate epidemic dynamics with the SIR model — explore how transmission rate, recovery rate, and population size determine outbreak size and herd immunity thresholds Formula: dS/dt = −βSI/N (susceptible depletion). 4 parameters.
- [Turing Reaction-Diffusion Patterns](/lab/mathematical-biology/turing-pattern): Simulate Turing morphogenesis — explore how activator-inhibitor diffusion ratios generate spots, stripes, and labyrinthine patterns from uniform initial conditions Formula: ∂u/∂t = Dᵤ∇²u − uv² + f(1 − u). 4 parameters.

### Mathematics (8 simulations)
- [2D Random Walk](/lab/mathematics/random-walk): Watch particles take random steps in two dimensions. Discover the square-root law of diffusion as walkers drift from the origin. Formula: E[|displacement|] ~ step_size * sqrt(n). 4 parameters.
- [Central Limit Theorem Demo](/lab/mathematics/central-limit-theorem): See the Central Limit Theorem in action: sample from any distribution and watch the distribution of means converge to a perfect bell curve. Formula: Standard Error = sigma / sqrt(n). 3 parameters.
- [Fourier Series Visualizer](/lab/mathematics/fourier-series): Decompose any waveform into sine and cosine harmonics. Watch rotating circles (epicycles) construct square waves, sawtooth waves, and more. Formula: f(x) = a0/2 + sum(n=1..N) [an*cos(nx) + bn*sin(nx)]. 4 parameters.
- [Fractal Tree Generator](/lab/mathematics/fractal-tree): Grow recursive fractal trees by adjusting branch angles, depth, and length ratios. Explore self-similarity and fractal dimensions in botanical structures. Formula: Total branches = 2^depth - 1. 4 parameters.
- [Golden Ratio & Fibonacci Spiral](/lab/mathematics/golden-ratio): Explore the golden ratio through interactive Fibonacci spirals and golden rectangle subdivisions. Watch how the ratio of consecutive Fibonacci numbers converges to phi. Formula: phi = (1 + sqrt(5)) / 2 = 1.6180339887.... 4 parameters.
- [Monte Carlo Pi Estimation](/lab/mathematics/pi-estimation): Estimate pi by throwing random darts at a square with an inscribed circle. Watch the approximation improve as more points accumulate. Formula: pi ~ 4 * N_inside / N_total. 3 parameters.
- [Normal Distribution / Bell Curve](/lab/mathematics/normal-distribution): Visualize the Gaussian bell curve and explore the 68-95-99.7 rule. Adjust mean, standard deviation, and sample size to see how the distribution shapes data. Formula: f(x) = (1 / (sigma * sqrt(2*pi))) * exp(-(x - mu)^2 / (2 * sigma^2)). 4 parameters.
- [Sieve of Eratosthenes](/lab/mathematics/prime-sieve): Watch the ancient algorithm discover prime numbers in real time. Composite numbers are crossed off one by one as each new prime is found. Formula: pi(N) ~ N / ln(N) (Prime Number Theorem). 3 parameters.

### Network Science (5 simulations)
- [Percolation Threshold Simulator](/lab/network-science/percolation-threshold): Interactive site percolation on a 2D square lattice demonstrating the critical phase transition at p_c where a giant connected cluster suddenly emerges Formula: p_c approx 0.5927 (2D square lattice, site percolation). 3 parameters.
- [Power-Law Distribution Explorer](/lab/network-science/power-law-distribution): Interactive exploration of power-law distributions and Zipf's law across city sizes, word frequencies, wealth distribution, and earthquake magnitudes Formula: P(x) ~ x^(-alpha). 3 parameters.
- [Scale-Free Networks Simulator](/lab/network-science/scale-free-networks): Interactive Barabasi-Albert preferential attachment model showing how hub-dominated scale-free networks emerge and respond to targeted vs random attacks Formula: P(k) ~ k^(-gamma). 5 parameters.
- [SIR Epidemic Spreading Model](/lab/network-science/epidemic-spreading): Interactive SIR compartmental model showing how infectious diseases spread through populations, with vaccination and herd immunity dynamics Formula: dS/dt = -beta * S * I / N. 4 parameters.
- [Small-World Network Model](/lab/network-science/small-world): Interactive Watts-Strogatz model demonstrating how a small amount of random rewiring creates the 'small-world' phenomenon with short paths and high clustering Formula: L(p) drops sharply for small p. 4 parameters.

### Number Theory & Prime Numbers (5 simulations)
- [Continued Fractions Explorer](/lab/number-theory/continued-fractions): Expand any real number into its continued fraction representation — visualize convergents approaching the target and discover best rational approximations Formula: x = a₀ + 1/(a₁ + 1/(a₂ + 1/(...))). 4 parameters.
- [Goldbach's Conjecture Verifier](/lab/number-theory/goldbach-conjecture): Test Goldbach's conjecture interactively — decompose any even number into pairs of primes and visualize the number of representations Formula: G(2k) = #{(p, q) : p + q = 2k, p ≤ q, both prime}. 4 parameters.
- [Modular Arithmetic & Clock Math](/lab/number-theory/modular-arithmetic): Explore modular arithmetic visually — compute modular exponentiation, visualize residue classes on a clock dial, and see congruence patterns Formula: a^n mod m via repeated squaring: O(log n) multiplications. 4 parameters.
- [Prime Distribution & Number Theorem](/lab/number-theory/prime-distribution): Visualize how primes thin out along the number line — compare π(x) against N/ln(N) and the logarithmic integral Li(x) in real time Formula: π(N) ~ N / ln(N) (Prime Number Theorem). 4 parameters.
- [Sieve of Eratosthenes](/lab/number-theory/prime-sieve): Visualize the Sieve of Eratosthenes in real time — watch composite numbers get eliminated as primes are discovered up to any limit Formula: π(N) ≈ N / ln(N) (Prime Number Theorem). 4 parameters.

### Operations Research & Optimization (5 simulations)
- [Economic Order Quantity & Inventory Costs](/lab/operations-research/inventory-model): Optimize inventory replenishment — explore how demand rate, ordering cost, and holding cost determine the economic order quantity and total cost Formula: Q* = √(2DS/H). 4 parameters.
- [Job Scheduling & Makespan Optimization](/lab/operations-research/scheduling-problem): Schedule jobs on parallel machines — explore how job count, machine count, and heuristic choice affect makespan and load balance Formula: Cₘₐₓ = max(completion time of machine j) for j = 1..m. 4 parameters.
- [Linear Programming & Feasible Region](/lab/operations-research/linear-programming): Visualize a 2D linear programming problem — adjust constraints and objective function to see the feasible region, corner points, and optimal solution Formula: Maximize Z = cₓ·x + cᵧ·y. 4 parameters.
- [Markov Chain & Steady-State Analysis](/lab/operations-research/markov-chain): Visualize a discrete Markov chain — adjust transition probabilities and watch the state distribution converge to its steady-state equilibrium Formula: πP = π (stationary distribution equation). 4 parameters.
- [Monte Carlo Simulation & Probability Estimation](/lab/operations-research/monte-carlo-simulation): Estimate pi and other quantities using random sampling — watch Monte Carlo convergence in real time as sample size grows Formula: π̂ = 4 × (number of points with x² + y² ≤ r²) / N. 4 parameters.

### Probability & Randomness (5 simulations)
- [Birthday Paradox Simulator](/lab/probability/birthday-paradox): Discover why only 23 people are needed for a 50% chance of a shared birthday — one of probability's most surprising results Formula: P(match) = 1 - ∏(i=0 to n-1) (D-i)/D where D = days in year. 4 parameters.
- [Galton Board (Bean Machine)](/lab/probability/galton-board): Watch balls cascade through pegs to form a perfect bell curve — the physical demonstration of the Central Limit Theorem Formula: P(k) = C(n,k) × p^k × (1-p)^(n-k) — Binomial distribution for k rights in n rows. 4 parameters.
- [Law of Large Numbers Visualizer](/lab/probability/law-of-large-numbers): Watch the running average converge to the true mean as samples accumulate — the foundation of all statistical inference Formula: P(|X̄ₙ - μ| > ε) ≤ σ²/(nε²) — Chebyshev's inequality (weak LLN). 4 parameters.
- [Monty Hall Problem Simulator](/lab/probability/monty-hall): Simulate thousands of Monty Hall games to prove why switching doors wins 2/3 of the time — the most counterintuitive result in probability Formula: P(win|switch) = (N-1)/N for N doors (host opens N-2 losing doors). 4 parameters.
- [Secretary Problem (Optimal Stopping)](/lab/probability/secretary-problem): Explore the elegant 37% rule — reject the first third of candidates, then pick the next one who's the best so far Formula: P(success) = (r/N) × ∑(i=r+1 to N) 1/(i-1), where r = ⌊N/e⌋. 4 parameters.

### Statistics & Inference (5 simulations)
- [Analysis of Variance (ANOVA)](/lab/statistics/anova): Compare means across multiple groups with one-way ANOVA — visualize between-group and within-group variance to understand the F-test Formula: F = MS_between / MS_within. 5 parameters.
- [Chi-Squared Test for Independence](/lab/statistics/chi-squared): Build contingency tables and run chi-squared tests to determine whether two categorical variables are independent or associated Formula: χ² = Σ (Oᵢ - Eᵢ)² / Eᵢ. 4 parameters.
- [Confidence Intervals & Sample Size](/lab/statistics/confidence-intervals): Visualize how confidence intervals shrink with larger samples and see why a 95% CI does not mean 95% probability the parameter is inside Formula: CI = x̄ ± z* × (σ / √n). 5 parameters.
- [Hypothesis Testing & P-Values](/lab/statistics/hypothesis-testing): Simulate hypothesis tests to understand p-values, Type I and Type II errors, and why statistical significance doesn't mean practical significance Formula: z = (x̄ - μ₀) / (σ / √n). 5 parameters.
- [Regression to the Mean](/lab/statistics/regression-to-mean): Visualize why extreme observations tend to be followed by less extreme ones — the statistical phenomenon that fools coaches, doctors, and policymakers Formula: E[X₂ | X₁] = μ + r × (X₁ - μ). 5 parameters.

### Topology & Surfaces (5 simulations)
- [Euler Characteristic V−E+F](/lab/topology/euler-characteristic): Compute the Euler characteristic for polyhedra and surfaces — the topological invariant that connects vertices, edges, and faces through V−E+F Formula: χ = V − E + F. 4 parameters.
- [Four Color Theorem Map Coloring](/lab/topology/topological-map): Color maps with the minimum number of colors so no two adjacent regions share a color — demonstrating the famous four color theorem interactively Formula: χ(G) ≤ 4 for any planar graph G (four color theorem). 4 parameters.
- [Klein Bottle Immersion in 3D](/lab/topology/klein-bottle): Visualize the Klein bottle — a closed non-orientable surface that cannot exist in 3D without self-intersection — through its classic figure-eight immersion Formula: x(u,v) = (a + cos(u/2)·sin(v) − sin(u/2)·sin(2v)) · cos(u). 4 parameters.
- [Knot Invariants & Crossings](/lab/topology/knot-theory): Explore mathematical knots — closed curves in 3D space — and compute their invariants including crossing number, writhe, and unknotting number Formula: Crossing number c(K) = min crossings over all diagrams of K. 4 parameters.
- [Möbius Strip & Non-Orientability](/lab/topology/mobius-strip): Explore the Möbius strip — a surface with only one side and one edge — and discover why non-orientability breaks our intuition about inside and outside Formula: Sides = 1 if twists mod 2 = 1; else 2. 4 parameters.

## medical-science (1 topics, 5 simulations)

### Plasma Medicine (5 simulations)
- [Atmospheric Plasma Jet Dynamics](/lab/plasma-medicine/plasma-jet): Simulate a cold atmospheric plasma jet — explore how voltage, gas flow, and nozzle diameter shape the plasma plume geometry, temperature profile, and reactive species output Formula: Plume length: L ∝ V × Q^0.5 / d. 4 parameters.
- [Cold Plasma Sterilization Kinetics](/lab/plasma-medicine/cold-plasma-sterilization): Simulate cold atmospheric plasma sterilization — explore how plasma power, exposure time, and gas composition affect microbial inactivation rates Formula: N(t) = N₀ × 10^(-k × P^0.5 × t / (1 + t/τ)). 4 parameters.
- [Plasma-Assisted Dentistry](/lab/plasma-medicine/plasma-dentistry): Simulate plasma dental treatment — explore how plasma exposure, cavity depth, and bacterial biofilm thickness affect sterilization and dentin bond strength enhancement Formula: Biofilm kill: K = 1 - exp(-k × P × t / b). 4 parameters.
- [Plasma-Assisted Wound Healing](/lab/plasma-medicine/plasma-wound-healing): Simulate plasma-accelerated wound closure — explore how treatment dose, wound area, and NO concentration drive fibroblast proliferation and tissue regeneration Formula: Closure rate: CR = k × D^0.6 × [NO]^0.3 / A^0.4. 4 parameters.
- [Reactive Oxygen & Nitrogen Species (RONS)](/lab/plasma-medicine/reactive-species): Simulate RONS generation and diffusion — explore how plasma parameters, humidity, and target distance control the therapeutic cocktail of reactive species in plasma medicine Formula: e + H₂O → OH + H + e (electron-impact dissociation). 4 parameters.

## medical-sciences (3 topics, 15 simulations)

### Epidemiological Modeling & Disease Spread (5 simulations)
- [Contact Tracing Network Visualization](/lab/epidemiological-modeling/contact-tracing): Trace infection paths through a social network — watch how identifying and isolating contacts breaks transmission chains and reduces outbreak size Formula: Tracing effectiveness: E = trace_speed × compliance / (new_cases × avg_degree). 4 parameters.
- [Pandemic Waves & Seasonal Recurrence](/lab/epidemiological-modeling/pandemic-waves): Simulate how waning immunity, seasonal forcing, and behavioral changes combine to produce the recurring waves seen in real pandemics from 1918 influenza to COVID-19 Formula: SIRS model: dS/dt = -β(t)×S×I/N + δ×R (immunity waning at rate δ). 4 parameters.
- [SIR Compartmental Model Simulation](/lab/epidemiological-modeling/sir-model): Watch a classic SIR epidemic unfold — susceptible, infected, and recovered compartments evolve according to differential equations that govern disease dynamics Formula: dS/dt = -β × S × I / N. 4 parameters.
- [Superspreader Events & Overdispersion](/lab/epidemiological-modeling/superspreader-events): Explore how overdispersed transmission — where a small fraction of cases causes most infections — fundamentally changes epidemic dynamics and control strategies Formula: Negative binomial: P(X=x) = C(x+k-1, x) × (k/(k+R₀))^k × (R₀/(k+R₀))^x. 4 parameters.
- [Vaccination & Herd Immunity Threshold](/lab/epidemiological-modeling/vaccination-threshold): Calculate the critical vaccination coverage needed to achieve herd immunity — visualize how vaccine efficacy and R₀ determine whether an outbreak can take hold Formula: Herd immunity threshold: p_c = 1 - 1/R₀. 4 parameters.

### Pharmacology & Drug Kinetics (5 simulations)
- [Bioavailability & First-Pass Effect](/lab/pharmacology/bioavailability): Simulate oral bioavailability — explore how gut absorption fraction, first-pass hepatic extraction, gut wall metabolism, and formulation factor determine systemic drug exposure Formula: F = fa × fg × fh (oral bioavailability). 4 parameters.
- [Dose-Response Curve & EC50](/lab/pharmacology/dose-response): Simulate sigmoidal dose-response relationships — explore how EC50, Hill coefficient, efficacy ceiling, and baseline activity shape the pharmacodynamic curve Formula: E = E0 + (Emax - E0) × C^n / (EC50^n + C^n). 4 parameters.
- [Drug Absorption & Plasma Concentration](/lab/pharmacology/drug-absorption): Simulate oral drug absorption — explore how absorption rate, dose, volume of distribution, and elimination rate shape the plasma concentration-time curve Formula: C(t) = (F·D·ka) / (Vd·(ka - ke)) × (e^(-ke·t) - e^(-ka·t)). 4 parameters.
- [Drug-Drug Interaction Modeling](/lab/pharmacology/drug-interaction): Simulate pharmacokinetic drug interactions — explore how enzyme inhibition, induction, and protein binding displacement alter plasma drug levels Formula: CL_effective = CL_base × inhibition_factor × induction_factor. 4 parameters.
- [Half-Life & Drug Elimination](/lab/pharmacology/half-life-elimination): Simulate drug elimination kinetics — explore how half-life, initial concentration, dosing interval, and number of doses affect steady-state accumulation Formula: C(t) = C0 × e^(-ke·t), where ke = ln(2)/t½. 4 parameters.

### Toxicology & Dose-Response (5 simulations)
- [Bioaccumulation & Biomagnification](/lab/toxicology/bioaccumulation): Simulate chemical bioaccumulation through trophic levels — explore how bioconcentration factor, trophic transfer efficiency, and elimination rate drive tissue concentration Formula: BCF = C_organism / C_water (bioconcentration factor). 4 parameters.
- [LD50 Dose-Response Estimation](/lab/toxicology/ld50-estimation): Simulate LD50 dose-response curves — explore how dose, slope factor, and organism sensitivity shape the sigmoidal mortality relationship Formula: P(D) = D^b / (LD50^b + D^b) (Hill equation for mortality). 4 parameters.
- [Multi-Route Exposure Assessment](/lab/toxicology/exposure-assessment): Simulate chemical exposure through multiple routes — explore how inhalation, ingestion, and dermal absorption contribute to total daily intake and hazard quotient Formula: ADD_inhal = C_air × IR_air × ET × EF / (BW × AT). 4 parameters.
- [Quantitative Risk Characterization](/lab/toxicology/risk-characterization): Simulate quantitative cancer and non-cancer risk assessment — explore how slope factor, exposure duration, and population size determine individual and population risk Formula: LADD = ADD × (ED / AT) (lifetime average daily dose). 4 parameters.
- [Threshold Limit Values & Workplace Safety](/lab/toxicology/threshold-limit): Simulate occupational exposure limits — explore how TLV-TWA, STEL, ceiling values, and exposure duration determine workplace safety compliance Formula: TWA = (C₁T₁ + C₂T₂ + ... + CₙTₙ) / 8 (time-weighted average). 4 parameters.

## physics (36 topics, 176 simulations)

### Acoustics & Sound Engineering (5 simulations)
- [Active Noise Cancellation](/lab/acoustics/noise-cancellation): Simulate active noise cancellation — explore how anti-phase signals, delay compensation, and filter adaptation reduce unwanted sound Formula: A_residual = sqrt(An² + Ac² + 2×An×Ac×cos(π + Δφ)) (vector sum). 4 parameters.
- [Doppler Effect & Frequency Shift](/lab/acoustics/doppler-effect): Simulate the Doppler effect — explore how source velocity, observer velocity, and medium speed produce frequency shifts in sound waves Formula: f_obs = f₀ × (c + v_observer) / (c + v_source). 4 parameters.
- [Room Acoustics & Reverberation](/lab/acoustics/room-acoustics): Simulate room acoustics and reverberation time — explore how room dimensions, surface absorption, and speaker position affect sound decay and clarity Formula: RT60 = 0.161 × V / (α × S) (Sabine equation). 4 parameters.
- [Standing Wave Resonance Modes](/lab/acoustics/resonance-modes): Simulate standing wave resonance patterns — explore how mode number, tube length, boundary conditions, and damping shape resonant frequencies and node patterns Formula: f_n = n × c / (2L) (open-open tube, all harmonics). 4 parameters.
- [Ultrasound Beam Focusing & Imaging](/lab/acoustics/ultrasound): Simulate ultrasound beam propagation — explore how frequency, aperture, focal depth, and tissue attenuation affect beam width and imaging resolution Formula: λ = c / f (where c = 1540 m/s in soft tissue). 4 parameters.

### Acousto-Optics & Light-Sound Interaction (5 simulations)
- [Acousto-Optic Deflector Resolution & Scan Range](/lab/acousto-optics/deflector-design): Design acousto-optic beam deflectors by calculating resolvable spots, scan angle, access time, and deflection efficiency across the operating bandwidth Formula: N = Δf × τ_a = Δf × D / v_s — resolvable spots. 4 parameters.
- [Acousto-Optic Modulator Rise Time & Extinction](/lab/acousto-optics/aom-modulator): Model AOM switching dynamics including rise time, modulation bandwidth, extinction ratio, and insertion loss for laser beam modulation Formula: τ_rise = d / v_s — acoustic transit across beam diameter. 4 parameters.
- [Acousto-Optic Tunable Filter (AOTF) Spectral Response](/lab/acousto-optics/tunable-filter): Model AOTF wavelength selection, spectral resolution, and tuning range for hyperspectral imaging and spectroscopic applications Formula: δλ = 0.9 λ₀² / (Δn × L) — spectral resolution (FWHM). 4 parameters.
- [Bragg Acousto-Optic Diffraction Efficiency](/lab/acousto-optics/bragg-diffraction): Calculate diffraction efficiency, Bragg angle, and bandwidth for acousto-optic Bragg regime interaction in crystalline media Formula: sin(θ_B) = λf / (2n v_s) where v_s is acoustic velocity. 4 parameters.
- [Raman-Nath Multi-Order Diffraction Pattern](/lab/acousto-optics/raman-nath): Simulate Raman-Nath acousto-optic diffraction showing multiple diffraction orders and their intensity distribution in the thin-grating regime Formula: I_m = J_m²(Δφ) where J_m is the m-th order Bessel function of the first kind. 4 parameters.

### Aeroacoustics & Flow-Generated Noise (5 simulations)
- [Aeolian Tones & Vortex Shedding](/lab/aeroacoustics/aeolian-tone): Simulate aeolian tones from vortex shedding past cylinders — explore how flow speed, cylinder diameter, and Reynolds number set the shedding frequency Formula: f = St·U/d where St ≈ 0.2. 4 parameters.
- [Rotor & Propeller Noise](/lab/aeroacoustics/propeller-noise): Simulate propeller tonal and broadband noise — explore how blade count, RPM, diameter, and advance ratio shape the acoustic signature Formula: BPF = B × N / 60. 4 parameters.
- [Sonic Boom & Shock Wave Propagation](/lab/aeroacoustics/sonic-boom): Simulate sonic boom formation — explore how Mach number, altitude, and aircraft length determine the N-wave overpressure on the ground Formula: μ = arcsin(1/M) (Mach cone half-angle). 4 parameters.
- [Turbulent Boundary Layer Noise](/lab/aeroacoustics/boundary-layer-noise): Simulate turbulent boundary layer noise — explore how freestream velocity, boundary layer thickness, and surface roughness affect wall-pressure spectra Formula: Φ_pp(ω) = (τ_w²·δ/U∞)·C_T / [(ω·δ/U∞)^0.75 + C₁]^3.7. 4 parameters.
- [Turbulent Jet Mixing Noise](/lab/aeroacoustics/jet-noise): Simulate turbulent jet mixing noise — explore how exhaust velocity, jet diameter, and temperature ratio shape the acoustic spectrum and directivity Formula: W_acoustic ∝ ρ₀·V⁸·D²/c⁵ (Lighthill V⁸ law). 4 parameters.

### Astrometry & Stellar Position Measurement (5 simulations)
- [Astrometric Binary Orbit Solver](/lab/astrometry/binary-orbit): Simulate a visual binary star system — trace the apparent elliptical orbit on the sky and derive stellar masses from the orbital period and semi-major axis using Kepler's third law Formula: M_total = a³(AU) / P²(yr)  (Kepler's third law, solar units). 5 parameters.
- [Astrometric Calibration & Error Budget](/lab/astrometry/astrometric-calibration): Simulate the astrometric calibration process — explore how atmospheric seeing, detector pixel scale, reference star count, and exposure time affect positional accuracy Formula: σ_centroid = FWHM / (2.355 × SNR) (centroid uncertainty). 4 parameters.
- [Celestial Reference Frame Builder](/lab/astrometry/reference-frame): Simulate the construction of a celestial reference frame — explore how quasar anchor points, coordinate rotations, and precession corrections establish the ICRF coordinate system Formula: Δψ = 50.29 × T  arcsec (general precession in longitude). 4 parameters.
- [Proper Motion & Stellar Kinematics](/lab/astrometry/proper-motion): Simulate stellar proper motion — watch how stars drift across the sky over decades and compute their true space velocities from angular motion and distance Formula: μ = √(μ_α² + μ_δ²) (total proper motion). 4 parameters.
- [Stellar Parallax & Distance Measurement](/lab/astrometry/parallax-measurement): Simulate stellar parallax — measure how a star's apparent position shifts against background stars as Earth orbits the Sun, and convert the parallax angle to distance in parsecs Formula: d = 1/p (distance in parsecs from parallax in arcseconds). 4 parameters.

### Chaos Theory (5 simulations)
- [Double Pendulum Simulator](/lab/chaos-theory/double-pendulum): Watch a double pendulum evolve in real time — a mechanical system that exhibits chaotic motion from simple Newtonian physics Formula: θ₁'' = [−g(2m₁+m₂)sinθ₁ − m₂g·sin(θ₁−2θ₂) − 2sin(θ₁−θ₂)·m₂·(θ₂'²L₂+θ₁'²L₁cos(θ₁−θ₂))] / [L₁(2m₁+m₂−m₂cos(2θ₁−2θ₂))]. 6 parameters.
- [Elementary Cellular Automaton](/lab/chaos-theory/cellular-automaton): Explore Wolfram's elementary cellular automata — simple local rules that generate complexity, from ordered patterns to computational universality Formula: cell(t+1, i) = rule[ cell(t, i-1) · 4 + cell(t, i) · 2 + cell(t, i+1) ]. 4 parameters.
- [Logistic Map & Bifurcation Diagram](/lab/chaos-theory/logistic-map): Explore the logistic map — a deceptively simple equation that produces period-doubling cascades, chaos, and fractal structure Formula: x(n+1) = r · x(n) · (1 − x(n)). 3 parameters.
- [Lorenz Attractor Simulator](/lab/chaos-theory/lorenz-attractor): Visualize the Lorenz attractor and observe how tiny perturbations lead to exponential divergence — the hallmark of deterministic chaos Formula: dx/dt = σ(y − x). 5 parameters.
- [Mandelbrot Set Explorer](/lab/chaos-theory/mandelbrot-explorer): Navigate the infinite fractal boundary of the Mandelbrot set — where complex iteration reveals self-similar structures at every scale Formula: z(n+1) = z(n)² + c. 4 parameters.

### Classical Mechanics (5 simulations)
- [Elastic & Inelastic Collision Simulator](/lab/classical-mechanics/momentum-collision): Smash objects together and observe conservation of momentum and energy — from perfect elastic bounces to completely inelastic collisions where objects stick together Formula: Conservation of momentum: m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'. 4 parameters.
- [Friction on Inclined Plane Simulator](/lab/classical-mechanics/friction-incline): Slide a block down a ramp with adjustable friction — explore static vs kinetic friction, critical angles, and the physics of everyday surfaces Formula: Critical angle: θ_c = arctan(μs). 4 parameters.
- [Projectile Motion with Air Resistance](/lab/classical-mechanics/projectile-motion): Launch projectiles through realistic air resistance and watch how drag transforms the perfect parabola of vacuum physics into the asymmetric arcs of the real world Formula: Vacuum range: R = v₀² * sin(2θ) / g. 4 parameters.
- [Rotational Inertia & Angular Momentum Simulator](/lab/classical-mechanics/rotational-inertia): Spin objects of different shapes and mass distributions to discover rotational inertia — why figure skaters spin faster when they pull in their arms Formula: Moment of inertia (disk): I = ½ * m * Math.pow(R, 2). 4 parameters.
- [Spring-Mass Harmonic Oscillator](/lab/classical-mechanics/spring-oscillator): Watch a spring-mass system oscillate in simple, damped, and driven modes — the fundamental model behind clocks, circuits, and quantum mechanics Formula: Equation of motion: m*x'' + b*x' + k*x = 0. 4 parameters.

### Color Science & Perception (5 simulations)
- [Additive & Subtractive Color Mixing](/lab/color-science/color-mixing): Mix colors additively (light) and subtractively (paint) — see how RGB and CMY create every color through opposite blending rules Formula: Additive: R_out = min(R1 + R2, 255) for each channel. 5 parameters.
- [Color Blindness Simulation](/lab/color-science/color-blindness): See the world through color-blind eyes — simulate protanopia, deuteranopia, and tritanopia to understand how 300 million people perceive color differently Formula: Protanopia matrix: [0.567, 0.433, 0; 0.558, 0.442, 0; 0, 0.242, 0.758]. 5 parameters.
- [Color Harmony & Complementary Palettes](/lab/color-science/color-harmony): Generate harmonious color palettes using complementary, triadic, split-complementary, and analogous rules from color theory Formula: Complementary: H₂ = (H₁ + 180) mod 360. 5 parameters.
- [Color Temperature & Black Body Radiation](/lab/color-science/color-temperature): Explore how temperature determines color — from the warm glow of candlelight at 1800K to the cool blue of a clear sky at 10000K through Planck's black body law Formula: Wien's law: λ_max = 2,897,772 / T (nm·K). 5 parameters.
- [Light Spectral Analysis & Wavelength](/lab/color-science/spectral-analysis): Decompose light into its spectral components — visualize wavelengths from 380nm violet to 700nm red and see how spectral power distributions create perceived colors Formula: X = ∫ SPD(λ) · x̄(λ) dλ, Y = ∫ SPD(λ) · ȳ(λ) dλ, Z = ∫ SPD(λ) · z̄(λ) dλ. 5 parameters.

### Cosmology & the Expanding Universe (5 simulations)
- [Big Bang Nucleosynthesis & Primordial Abundances](/lab/cosmology/big-bang-nucleosynthesis): Simulate primordial nucleosynthesis — explore how baryon density and neutron lifetime determine the abundances of hydrogen, helium, deuterium, and lithium formed in the first minutes Formula: Yp ≈ 2(n/p)_f / (1 + (n/p)_f) with (n/p)_f ≈ exp(−Δm/T_f). 4 parameters.
- [CMB Power Spectrum & Anisotropies](/lab/cosmology/cosmic-microwave): Simulate the cosmic microwave background power spectrum — explore how baryon density, dark matter, and curvature shape the acoustic peaks Formula: C_ℓ ∝ ℓ(ℓ+1) × power spectrum amplitude. 4 parameters.
- [Cosmic Inflation & Exponential Expansion](/lab/cosmology/cosmic-inflation): Simulate cosmic inflation — explore how the inflaton field drives exponential expansion, solving the flatness and horizon problems Formula: a(t) ∝ exp(Ht) during inflation, H ≈ √(V₀/(3M_Pl²)). 4 parameters.
- [Dark Energy & Cosmic Acceleration](/lab/cosmology/dark-energy): Simulate the cosmological constant and dark energy — explore how the equation of state parameter drives the accelerating expansion of the universe Formula: q₀ = Ωm/2 + ΩDE(1 + 3w)/2. 4 parameters.
- [Hubble's Law & Redshift-Distance Relation](/lab/cosmology/hubble-expansion): Simulate Hubble's law — explore how recession velocity scales with distance and how the Hubble constant determines the age of the universe Formula: v = H₀ × d (Hubble's law). 4 parameters.

### Cryogenics & Low-Temperature Physics (5 simulations)
- [Cryocooler Thermodynamic Cycle](/lab/cryogenics/cryocooler-cycle): Simulate a Gifford-McMahon cryocooler cycle — explore how pressure ratio, displacement volume, and cold-end temperature determine cooling power Formula: Q_cooling = V_d × (P_h − P_l) × frequency. 4 parameters.
- [Cryogenic Storage & Boil-Off](/lab/cryogenics/cryogenic-storage): Simulate cryogenic dewar boil-off — explore how insulation thickness, ambient temperature, liquid volume, and heat leak determine storage hold time Formula: Q_leak = k_eff × A × (T_ambient − T_boil) / d. 4 parameters.
- [Superconductivity & Critical Temperature](/lab/cryogenics/superconductivity): Simulate superconducting phase transitions — explore how temperature, magnetic field, and material type determine zero-resistance states Formula: R/Rn = 0 for T < Tc(B), 1 for T ≥ Tc(B). 4 parameters.
- [Superfluidity & Quantum Vortices](/lab/cryogenics/superfluidity): Simulate superfluid helium-4 — explore how temperature, rotation speed, and container geometry produce quantized vortices and frictionless flow Formula: ρs/ρ = 1 − (T/Tλ)^5.6. 4 parameters.
- [Thermal Contraction of Materials](/lab/cryogenics/thermal-contraction): Simulate thermal contraction at cryogenic temperatures — explore how material type, length, and temperature range determine shrinkage and thermal stress Formula: ΔL = α × L × (T₂ − T₁). 4 parameters.

### Crystallography & Crystal Structure (5 simulations)
- [Crystal Defect Structure](/lab/crystallography/defect-structure): Simulate point defects in crystals — adjust vacancy concentration, interstitial fraction, temperature, and substitutional impurity level to see how defects affect lattice energy and diffusion Formula: n_v/N = exp(-E_v / k_B T) (equilibrium vacancy fraction). 4 parameters.
- [Crystal Symmetry Operations](/lab/crystallography/crystal-symmetry): Explore crystal symmetry — apply rotation, mirror, inversion, and glide operations to a 2D motif and watch how symmetry generates the full pattern from a single asymmetric unit Formula: Multiplicity = n × (1 + m) × (1 + i) where m=mirror, i=inversion. 4 parameters.
- [Miller Indices & Crystal Planes](/lab/crystallography/miller-indices): Visualize crystal planes using Miller indices — adjust h, k, l values and lattice parameter to see how planes slice through the unit cell and calculate interplanar spacing Formula: d_hkl = a / √(h² + k² + l²) (cubic). 4 parameters.
- [Reciprocal Lattice Construction](/lab/crystallography/reciprocal-lattice): Build and explore the reciprocal lattice — adjust real-space lattice parameters and angle to see how the reciprocal lattice transforms, with Ewald sphere visualization Formula: a* = 2π / (a·sin(γ)), b* = 2π / (b·sin(γ)). 4 parameters.
- [Unit Cell Geometry Explorer](/lab/crystallography/unit-cell): Explore unit cell structures — adjust lattice parameters, crystal system type, atomic radius, and packing to visualize cubic, tetragonal, and hexagonal unit cells in real time Formula: APF = N × (4/3)πr³ / a³ (cubic systems). 4 parameters.

### Electromagnetism (5 simulations)
- [Electric Field Lines](/lab/electromagnetism/electric-field-lines): Visualize electric field lines from point charges. See how positive and negative charges create field patterns, with equipotential lines and animated test charges. Formula: F = kQ₁Q₂/r² (Coulomb's law). 4 parameters.
- [Faraday's Law of Induction](/lab/electromagnetism/electromagnetic-induction): Watch a rotating coil generate electricity through electromagnetic induction. See how changing magnetic flux produces EMF, synchronized with real-time waveform plots. Formula: ε = -NdΦ/dt (Faraday's law). 4 parameters.
- [LC Circuit Oscillator](/lab/electromagnetism/lc-circuit): Visualize energy oscillating between electric and magnetic fields in an LC circuit. Watch charge slosh between capacitor plates while current flows through the inductor. Formula: f₀ = 1/(2π√(LC)) (resonant frequency). 4 parameters.
- [Lorentz Force on Charged Particles](/lab/electromagnetism/lorentz-force): Watch charged particles spiral through magnetic fields. Visualize the Lorentz force creating circular and helical trajectories, with adjustable mass, charge, and field strength. Formula: F = qv × B (Lorentz force, magnetic component). 5 parameters.
- [Magnetic Field Around Wires](/lab/electromagnetism/magnetic-field): Visualize magnetic field lines around current-carrying wires. See circular field patterns, parallel wire interactions, and the right-hand rule in action. Formula: B = μ₀I/(2πr) (Ampere's law for long wire). 4 parameters.

### Ferroelectrics & Piezoelectric Materials (5 simulations)
- [180 and 90 Degree Domain Wall Motion](/lab/ferroelectrics/domain-switching): Watch ferroelectric domains switch under an applied electric field — visualize 180-degree and 90-degree domain wall motion in real time Formula: f(t) = 1 - exp(-(t/tau)^n) (Kolmogorov-Avrami-Ishibashi switching kinetics). 4 parameters.
- [Ferroelectric-Paraelectric Phase Transition](/lab/ferroelectrics/curie-transition): Visualize the Curie transition where a ferroelectric loses its spontaneous polarization and becomes paraelectric above the Curie temperature Formula: P_s(T) = P_0 * (1 - T/T_c)^beta (order parameter below T_c). 4 parameters.
- [Piezoelectric d33 & Coupling Factor](/lab/ferroelectrics/piezoelectric-coefficient): Calculate the piezoelectric charge coefficient d33 and electromechanical coupling factor k for various ferroelectric materials Formula: d33 = (deltaQ / A) / sigma (charge per area per stress). 4 parameters.
- [Piezoelectric Vibration Energy Harvester](/lab/ferroelectrics/energy-harvesting): Design a piezoelectric cantilever energy harvester — tune frequency, load resistance, and material properties to maximize power output from ambient vibrations Formula: P_max = (m * a)^2 / (8 * zeta_total * m * omega_n) (maximum harvestable power at resonance). 4 parameters.
- [Polarization-Electric Field Hysteresis Loop](/lab/ferroelectrics/pe-hysteresis): Trace the classic P-E hysteresis loop of a ferroelectric material and extract remnant polarization, coercive field, and saturation polarization Formula: P(E) = P_s * tanh((E +/- E_c) * sq / E_c) (phenomenological hysteresis model). 4 parameters.

### Fluid Dynamics (4 simulations)
- [Bernoulli's Principle & Lift Simulator](/lab/fluid-dynamics/bernoulli-principle): Explore how faster-moving air creates lower pressure, generating lift on an airfoil — the fundamental physics behind airplane wings and race car aerodynamics Formula: P + ½ρv² + ρgh = constant (Bernoulli's equation). 4 parameters.
- [Fick's Law of Diffusion Simulator](/lab/fluid-dynamics/diffusion-process): Watch particles spread from concentrated to dispersed through Brownian motion — visualizing Fick's law, random walks, and the square-root-of-time diffusion scaling Formula: J = -D(dC/dx) (Fick's first law). 4 parameters.
- [Laminar vs Turbulent Flow Simulator](/lab/fluid-dynamics/reynolds-number): Visualize the transition from smooth laminar flow to chaotic turbulence by adjusting velocity, pipe diameter, and fluid properties to compute the Reynolds number Formula: Re = ρvD/μ. 4 parameters.
- [Viscosity & Poiseuille Flow Simulator](/lab/fluid-dynamics/viscosity-flow): Explore how viscosity determines fluid flow rates through pipes — visualize parabolic velocity profiles and discover why small radius changes have enormous effects Formula: Q = πr⁴ΔP/(8μL) (Poiseuille's law). 4 parameters.

### Magnetism & Magnetic Materials (5 simulations)
- [B-H Hysteresis Loop & Coercivity](/lab/magnetism/hysteresis-loop): Simulate magnetic hysteresis — explore how coercivity, remanence, and saturation magnetization shape the B-H curve of ferromagnetic materials Formula: B = μ₀(H + M), where M follows hysteresis path. 4 parameters.
- [Crystal Magnetic Anisotropy & Easy Axis](/lab/magnetism/magnetic-anisotropy): Simulate magnetocrystalline anisotropy — explore how crystal symmetry, anisotropy constants, and applied field determine the easy axis and energy landscape Formula: E_uniaxial = K₁ sin²θ + K₂ sin⁴θ. 4 parameters.
- [Curie Temperature: Ferromagnetic-Paramagnetic Transition](/lab/magnetism/curie-temperature): Simulate the Curie temperature phase transition — explore how temperature destroys long-range magnetic order in ferromagnetic materials Formula: M/M_s = B_J(x), x = 3J/(J+1) × T_C/T × M/M_s (mean-field self-consistency). 4 parameters.
- [Magnetic Domain Wall Motion & Barkhausen Effect](/lab/magnetism/magnetic-domains): Simulate magnetic domain wall motion under applied field — explore how pinning sites, wall energy, and the Barkhausen effect shape magnetization Formula: γ_w = 4√(A × K₁) (180° Bloch wall energy). 4 parameters.
- [Solenoid Electromagnet Design & Force](/lab/magnetism/electromagnet-design): Design a solenoid electromagnet — explore how turns, current, core material, and geometry determine magnetic field strength and attractive force Formula: B = μ₀ × μ_r × N × I / L. 4 parameters.

### Nuclear Physics (4 simulations)
- [Nuclear Fission Chain Reaction Simulator](/lab/nuclear-physics/nuclear-fission): Simulate a nuclear fission chain reaction — watch neutrons split uranium atoms, adjust control rods and enrichment to balance criticality between sustained power and runaway explosion Formula: k_eff = k∞ · P_NL (effective multiplication factor). 4 parameters.
- [Nuclear Fusion & Ignition Simulator](/lab/nuclear-physics/nuclear-fusion): Explore the extreme conditions needed for nuclear fusion — adjust plasma temperature, density, and confinement to chase the Lawson criterion and achieve ignition Formula: nTτ > 5×10²¹ keV·s/m³ (Lawson criterion for D-T). 4 parameters.
- [Radioactive Decay & Half-Life Simulator](/lab/nuclear-physics/radioactive-decay): Watch unstable atoms decay in real time — visualize exponential decay curves, understand half-lives, and explore decay chains from parent to daughter nuclei Formula: N(t) = N₀·(½)^(t/t½). 4 parameters.
- [Radiometric Dating Calculator](/lab/nuclear-physics/half-life-dating): Calculate the age of rocks, fossils, and artifacts using radioactive isotope ratios — from Carbon-14 dating of ancient wood to Uranium-Lead dating of Earth's oldest minerals Formula: t = (t½/ln2) · ln(1/f) where f = parent fraction. 4 parameters.

### Optics & Instruments (5 simulations)
- [Fiber Optic Total Internal Reflection Simulator](/lab/optics-instruments/fiber-optics): Visualize how light propagates through optical fibers via total internal reflection, and explore how core size, refractive index, and bending affect signal transmission Formula: θ_c = arcsin(n₂ / n₁). 4 parameters.
- [Holography Interference Patterns Simulator](/lab/optics-instruments/holography): Visualize how reference and object beams interfere to create holographic fringe patterns, and explore how wavelength, angle, and film resolution affect hologram quality Formula: d = λ / (2 × sin(θ/2)). 4 parameters.
- [Laser Cavity Modes & Gain Simulator](/lab/optics-instruments/laser-cavity): Simulate longitudinal cavity modes, gain threshold, and output spectrum of a Fabry-Perot laser resonator with adjustable cavity length, reflectivity, and gain medium Formula: Δν = c / (2L). 4 parameters.
- [Microscope Resolution & Abbe Limit Simulator](/lab/optics-instruments/microscope-resolution): Explore how numerical aperture, wavelength, and immersion medium determine the resolving power of an optical microscope through the Abbe diffraction limit Formula: d_Abbe = λ / (2 × NA). 4 parameters.
- [Telescope Magnification & Resolution Simulator](/lab/optics-instruments/telescope-magnification): Simulate how objective diameter, focal length, and eyepiece choice affect a telescope's magnification, resolving power, and light-gathering ability Formula: M = f_objective / f_eyepiece. 4 parameters.

### Orbital Mechanics (4 simulations)
- [Gravity Assist Simulator](/lab/orbital-mechanics/gravity-assist): Visualize how spacecraft gain or lose speed by flying past planets, using the gravitational slingshot effect Formula: δ = 2·arcsin(1/(1+r_p·v²∞/(GM))). 4 parameters.
- [Hohmann Transfer Orbit Calculator](/lab/orbital-mechanics/hohmann-transfer): Simulate the most fuel-efficient two-impulse maneuver to transfer between circular orbits, with animated spacecraft trajectory Formula: Δv₁ = √(GM/r₁)·(√(2r₂/(r₁+r₂)) - 1). 3 parameters.
- [Kepler Orbit Simulator](/lab/orbital-mechanics/kepler-orbits): Interactive simulation of elliptical planetary orbits using Kepler's laws, showing equal-area sweeps and velocity changes Formula: r(θ) = a(1-e²)/(1+e·cos θ). 3 parameters.
- [Lagrange Points Visualizer](/lab/orbital-mechanics/lagrange-points): Explore the five equilibrium points of the restricted three-body problem with an interactive effective potential map Formula: L1: r ≈ R·(M₂/(3M₁))^(1/3). 3 parameters.

### Particle Physics (5 simulations)
- [Higgs Field & Symmetry Breaking](/lab/particle-physics/higgs-field): Visualize the Mexican hat potential and watch symmetry breaking as the Higgs field acquires a vacuum expectation value, giving mass to fundamental particles. Formula: V(φ) = -μ²|φ|² + λ|φ|⁴. 4 parameters.
- [Pair Production & Annihilation](/lab/particle-physics/pair-production): Watch matter emerge from pure energy. A high-energy photon near a nucleus creates an electron-positron pair, demonstrating E=mc² in action. Formula: E_threshold = 2m_e c² = 1.022 MeV. 4 parameters.
- [Particle Collision Event Display](/lab/particle-physics/particle-collision): Simulate particle collisions like those at the Large Hadron Collider. Watch tracks radiate from the collision point, curved by magnetic fields, with energy deposits in calorimeters. Formula: E² = (pc)² + (mc²)² (energy-momentum relation). 4 parameters.
- [Quark Confinement & Color Force](/lab/particle-physics/quark-confinement): Visualize quark confinement and the color force. Pull quarks apart and watch the gluon flux tube stretch until it snaps, creating new quark-antiquark pairs. Formula: V(r) = -4αs/(3r) + σr (Cornell potential). 4 parameters.
- [Standard Model Particle Zoo](/lab/particle-physics/standard-model): Interactive visualization of all 17 fundamental particles in the Standard Model — quarks, leptons, gauge bosons, and the Higgs boson — with their masses, charges, and spins. Formula: E = mc² (mass-energy equivalence). 4 parameters.

### Photonics & Optical Engineering (5 simulations)
- [Nonlinear Optical Effects](/lab/photonics/nonlinear-optics): Simulate self-phase modulation, four-wave mixing, and spectral broadening in optical fibers, visualizing how pulse intensity, fiber nonlinearity, and dispersion shape the output spectrum. Formula: φ_NL = γ · P₀ · L_eff. 4 parameters.
- [Optical Amplifier Gain Dynamics](/lab/photonics/optical-amplifier): Model erbium-doped fiber amplifier (EDFA) gain and noise characteristics as a function of pump power, signal wavelength, fiber length, and erbium concentration. Formula: G(dB) = 10 log₁₀(P_out/P_in) = Γ · (g · N₂ − α) · L. 4 parameters.
- [Optical Fiber Mode Propagation](/lab/photonics/optical-fiber-modes): Simulate guided mode profiles in step-index and graded-index optical fibers, visualizing how core diameter, numerical aperture, and wavelength determine single-mode vs multimode operation. Formula: V = (π · d · NA) / λ. 4 parameters.
- [Photonic Crystal Band Structure](/lab/photonics/photonic-crystal): Visualize photonic band gaps in 1D and 2D periodic dielectric structures, showing how lattice period, refractive index contrast, and fill fraction create frequency ranges where light cannot propagate. Formula: λ_mid = 2a(n₁f + n₂(1−f)). 4 parameters.
- [Waveguide Directional Coupler](/lab/photonics/waveguide-coupling): Simulate evanescent-field power transfer between two parallel optical waveguides, exploring coupling length, wavelength dependence, and cross-talk in integrated photonic circuits. Formula: η = sin²(κ L) / [1 + (Δβ/2κ)²]. 4 parameters.

### Planetary Science & Solar System (5 simulations)
- [Albedo & Planetary Energy Balance](/lab/planetary-science/albedo-temperature): Calculate equilibrium temperature from stellar luminosity, orbital distance, albedo, and greenhouse effect — understand why Venus is hotter than Mercury despite being farther from the Sun Formula: T_eq = [L_star × (1 - A) / (16 × π × σ × d²)]^(1/4). 4 parameters.
- [Impact Crater Scaling: Size vs Energy](/lab/planetary-science/crater-scaling): Calculate impact crater diameter from impactor size, velocity, and target properties — explore the physics of hypervelocity impacts that shaped planetary surfaces Formula: D_transient = 1.161 × (ρi/ρt)^0.33 × d^0.78 × v^0.44 × g^-0.22. 4 parameters.
- [Planetary Atmosphere: Scale Height & Escape Velocity](/lab/planetary-science/planetary-atmosphere): Model atmospheric structure using scale height and compare thermal velocity to escape velocity — understand why Mars lost its atmosphere while Earth kept its own Formula: H = kT / (μ × g) where g = GM/R². 4 parameters.
- [Roche Limit: Tidal Disruption Distance](/lab/planetary-science/roche-limit): Calculate the Roche limit — the critical distance where tidal forces tear apart a satellite — and visualize the disruption process that creates planetary rings Formula: d_rigid = 1.26 × R_primary × (ρ_primary / ρ_satellite)^(1/3). 4 parameters.
- [Tidal Heating & Io-Style Dissipation](/lab/planetary-science/tidal-heating): Simulate tidal heating in moons like Io — explore how orbital eccentricity, distance, and interior rigidity drive volcanic activity through gravitational flexing Formula: P = (21/2) × (k₂/Q) × (G M²_p r⁵ n e²) / a⁶. 4 parameters.

### Plasma Physics & Fusion (5 simulations)
- [Debye Shielding & Charge Screening](/lab/plasma-physics/debye-shielding): Visualize how a plasma shields electric charges — watch the Debye cloud form around a test charge and see the exponential potential decay Formula: λ_D = √(ε₀ × k_B × T / (n_e × e²)) — Debye length. 4 parameters.
- [Langmuir Plasma Oscillation Simulator](/lab/plasma-physics/langmuir-waves): Watch electrons oscillate collectively in a plasma — visualize Langmuir waves, the plasma frequency, and the Bohm-Gross dispersion relation Formula: ω_pe = √(n_e × e² / (ε₀ × m_e)) — electron plasma frequency. 4 parameters.
- [Magnetic Plasma Confinement Simulator](/lab/plasma-physics/plasma-confinement): Simulate charged particles spiraling along magnetic field lines — explore how magnetic mirrors and toroidal fields confine hot plasma for fusion Formula: r_L = m × v_perp / (q × B) — Larmor radius. 4 parameters.
- [Plasma Frequency Simulator](/lab/plasma-physics/plasma-frequency): Visualize how electromagnetic waves interact with plasma — watch waves reflect below the plasma frequency and propagate above it Formula: f_pe = (1/2*pi)*sqrt(n_e*e^2 / (m_e*epsilon_0)) — electron plasma frequency. 4 parameters.
- [Tokamak Fusion Reactor Cross-Section](/lab/plasma-physics/tokamak-fusion): Explore the anatomy of a tokamak fusion reactor — visualize the magnetic field geometry, plasma cross-section, and the Lawson criterion for ignition Formula: q(a) = 2π × a² × B_T / (μ₀ × R₀ × I_p) — cylindrical safety factor. 4 parameters.

### Quantum Field Theory & Particle Physics (5 simulations)
- [Casimir Effect & Vacuum Force](/lab/quantum-field-theory/casimir-effect): Simulate the Casimir effect — explore how plate separation, area, and temperature determine the attractive force between conducting plates from quantum vacuum fluctuations Formula: F = -π²ℏcA / (240d⁴) (Casimir force between parallel plates). 4 parameters.
- [Feynman Diagram & Scattering Amplitude](/lab/quantum-field-theory/feynman-diagram): Simulate Feynman diagram scattering — explore how coupling strength, particle masses, and momentum transfer determine cross sections in quantum electrodynamics Formula: dσ/dΩ = α² / (4s) × |M(s,t)|² (differential cross section). 4 parameters.
- [Lamb Shift & Radiative Corrections](/lab/quantum-field-theory/lamb-shift): Simulate the Lamb shift — explore how quantum vacuum fluctuations and radiative corrections lift the degeneracy between hydrogen energy levels Formula: ΔE_Lamb(nS) = (4α⁵m_ec²)/(3πn³) × [ln(1/(α²)) + C] (leading order). 4 parameters.
- [Running Coupling Constants & Asymptotic Freedom](/lab/quantum-field-theory/running-coupling): Simulate running coupling constants — explore how electromagnetic, weak, and strong coupling strengths change with energy scale through renormalization group flow Formula: αs(μ) = αs(μ₀) / [1 + (b₀αs(μ₀)/2π)ln(μ/μ₀)] (1-loop running). 4 parameters.
- [Vacuum Fluctuations & Zero-Point Energy](/lab/quantum-field-theory/vacuum-fluctuation): Simulate quantum vacuum fluctuations — explore how field mode frequency, spatial extent, and observation time affect virtual particle pair creation and zero-point energy Formula: E₀ = ½ℏω (zero-point energy per mode). 4 parameters.

### Quantum Mechanics (5 simulations)
- [Double-Slit Experiment Simulator](/lab/quantum-mechanics/double-slit): Watch quantum interference emerge photon by photon — adjust wavelength, slit geometry, and photon count to see the famous interference pattern build up from individual particle detections Formula: Interference pattern: I(theta) = I_0 cos^2(pi d sin(theta)/lambda) sinc^2(pi a sin(theta)/lambda). 4 parameters.
- [Particle in a Box Simulator](/lab/quantum-mechanics/particle-in-a-box): Explore quantized energy levels and wave functions of a particle confined in an infinite square well — visualize standing waves, probability densities, and quantum superposition states Formula: Wave function: psi_n(x) = sqrt(2/L) sin(n pi x / L). 4 parameters.
- [Quantum Entanglement & Bell's Inequality Simulator](/lab/quantum-mechanics/quantum-entanglement): Test Bell's inequality with entangled particle pairs — set detector angles for Alice and Bob and watch the CHSH parameter S reveal whether correlations exceed classical limits Formula: CHSH inequality: |S| = |E(a,b) - E(a,b') + E(a',b) + E(a',b')| <= 2 (classical). 5 parameters.
- [Quantum Tunneling Simulator](/lab/quantum-mechanics/quantum-tunneling): Visualize how particles tunnel through potential energy barriers — adjust energy, barrier height, width, and particle mass to see transmission and reflection coefficients change in real time Formula: Transmission coefficient: T = 1/[1 + (kappa^2 + k1^2)^2 sinh^2(kappa L)/(4 k1^2 kappa^2)]. 4 parameters.
- [Qubit Bloch Sphere Simulator](/lab/quantum-mechanics/qubit-bloch-sphere): Visualize qubit states on the Bloch sphere and apply quantum gates — see how Hadamard, Pauli-X, Pauli-Z, and T-gate transform the quantum state vector in 3D Formula: Qubit state: |psi> = cos(theta/2)|0> + e^(i phi) sin(theta/2)|1>. 3 parameters.

### Quantum Optics & Photonics (5 simulations)
- [Cavity QED & Jaynes-Cummings Model](/lab/quantum-optics/cavity-qed): Simulate cavity quantum electrodynamics — explore vacuum Rabi oscillations, atom-photon coupling strengths, and the Jaynes-Cummings energy ladder in optical and microwave cavities Formula: Ω_n = 2g√(n+1) (Rabi frequency for n photons). 5 parameters.
- [Entanglement Witness & Bell Inequality](/lab/quantum-optics/entanglement-witness): Simulate entanglement detection — explore Bell's CHSH inequality, witness operators, and the boundary between classical correlations and quantum entanglement Formula: E(θ_A, θ_B) = -V cos(2(θ_A - θ_B)). 4 parameters.
- [Hanbury Brown-Twiss Experiment](/lab/quantum-optics/hbt-experiment): Simulate the HBT experiment — explore photon bunching and antibunching by measuring intensity correlations g⁽²⁾(τ) for thermal, coherent, and single-photon sources Formula: g⁽²⁾(τ) = 1 + exp(-2|τ|/τ_c) (thermal). 4 parameters.
- [Photon Counting Statistics](/lab/quantum-optics/photon-statistics): Simulate photon number distributions — compare Poissonian (coherent), super-Poissonian (thermal), and sub-Poissonian (quantum) light sources in real time Formula: P(n) = ⟨n⟩ⁿ e^{-⟨n⟩} / n! (Poisson). 4 parameters.
- [Squeezed Light & Quantum Noise](/lab/quantum-optics/squeezed-light): Simulate squeezed states of light — visualize how squeezing reduces quantum noise in one quadrature below the vacuum level at the expense of the conjugate quadrature Formula: r = r_dB / (20 log₁₀ e) ≈ r_dB / 8.686. 4 parameters.

### Radio Astronomy & Interferometry (5 simulations)
- [Aperture Synthesis Imaging](/lab/radio-astronomy/aperture-synthesis): Simulate aperture synthesis — explore how array configuration, observation duration, and declination affect UV coverage and synthesized beam quality Formula: N_baselines = N(N-1)/2. 4 parameters.
- [Interferometer Baseline & Resolution](/lab/radio-astronomy/interferometer-baseline): Simulate a two-element radio interferometer — explore how baseline length, wavelength, and source angle produce fringe patterns and angular resolution Formula: θ_res ≈ λ / B_max (angular resolution in radians). 4 parameters.
- [Pulsar Timing & Period Analysis](/lab/radio-astronomy/pulsar-timing): Simulate pulsar radio emission — explore how spin period, period derivative, dispersion measure, and pulse width shape the observed timing profile Formula: τ_char = P / (2 Ṗ) (characteristic age). 4 parameters.
- [Radio Spectrum & Source Classification](/lab/radio-astronomy/radio-spectrum): Simulate radio source spectra — explore how spectral index, turnover frequency, and flux density characterize synchrotron, thermal, and self-absorbed emission Formula: S(ν) = S₀ (ν/ν₀)^α (power-law spectrum). 4 parameters.
- [Radio Telescope Beam Pattern](/lab/radio-astronomy/radio-telescope): Simulate a parabolic radio telescope's beam pattern — explore how dish diameter, wavelength, and feed offset shape the primary beam and sidelobes Formula: θ_HPBW ≈ 1.22 λ / D (half-power beamwidth in radians). 4 parameters.

### Rheology & Flow Behavior (5 simulations)
- [Extensional Flow & Elongational Viscosity](/lab/rheology/extensional-flow): Simulate extensional flow of polymer solutions — explore how strain rate, relaxation time, and extensibility govern filament thinning and elongational viscosity Formula: Wi = λ_r × ε̇ (Weissenberg number). 4 parameters.
- [Rheometry & Oscillatory Testing](/lab/rheology/rheometry): Simulate oscillatory rheometry — explore how frequency, strain amplitude, and material viscoelasticity determine storage modulus G' and loss modulus G'' Formula: G'(ω) = G₀ × (ωλ)² / (1 + (ωλ)²). 4 parameters.
- [Thixotropy & Time-Dependent Flow](/lab/rheology/thixotropy): Simulate thixotropic behavior — explore how viscosity evolves over time during shearing and recovery as microstructure breaks down and rebuilds Formula: dλ/dt = k_r × (1 − λ) − k_b × γ̇ × λ. 4 parameters.
- [Viscosity Models & Shear Response](/lab/rheology/viscosity-models): Simulate non-Newtonian viscosity models — explore how shear rate, consistency index, and power-law exponent govern shear-thinning and shear-thickening behavior Formula: τ = τ₀ + K × γ̇ⁿ (Herschel-Bulkley). 4 parameters.
- [Yield Stress & Plastic Flow](/lab/rheology/yield-stress): Simulate yield stress phenomena — explore how Bingham plastics and soft solids transition from solid-like to flowing behavior under applied stress Formula: τ = τ₀ + μₚ × γ̇ (Bingham model). 4 parameters.

### Semiconductor Physics (5 simulations)
- [LED Emission Spectrum & Efficiency Simulator](/lab/semiconductor/led-emission): Simulate how band gap energy, temperature, and current density determine an LED's emission wavelength, spectral width, and luminous efficiency Formula: λ_peak = 1240 / E_g (nm). 4 parameters.
- [MOSFET Operation & Switching Simulator](/lab/semiconductor/mosfet-operation): Simulate MOSFET output characteristics, transfer curves, and switching behavior with adjustable gate voltage, threshold, and channel parameters Formula: I_D = k × (V_GS − V_th)² / 2 (saturation). 4 parameters.
- [PN Junction Diode Characteristics Simulator](/lab/semiconductor/pn-junction): Simulate the IV curve, depletion region, and built-in potential of a PN junction diode with adjustable doping concentrations, temperature, and applied voltage Formula: I = I_s × (exp(V / (n × V_T)) − 1). 4 parameters.
- [Semiconductor Band Gap & Doping Simulator](/lab/semiconductor/band-gap): Explore how temperature, doping concentration, and material type affect the band gap, Fermi level, and carrier concentrations in semiconductors Formula: E_g(T) = E_g(0) − α × T² / (T + β). 4 parameters.
- [Solar Cell IV Curve & Efficiency Simulator](/lab/semiconductor/solar-cell): Simulate the current-voltage characteristics, maximum power point, and efficiency of a solar cell with adjustable irradiance, temperature, and cell parameters Formula: I = I_ph − I_0 × (exp(V / V_T) − 1). 4 parameters.

### Sound & Music (5 simulations)
- [Chladni Patterns (Vibrating Plates)](/lab/sound-music/chladni-patterns): Visualize the beautiful geometric patterns that form on vibrating plates — sand settles along nodal lines revealing the mathematics of standing waves in 2D. Formula: Square plate: w(x,y) = sin(m*pi*x/L) * sin(n*pi*y/L). 4 parameters.
- [Harmonic Series & Timbre](/lab/sound-music/harmonic-series): Visualize how harmonics combine to create timbre — watch individual overtones stack into complex waveforms that define every instrument's unique sound. Formula: f_n = n * f₀ (nth harmonic frequency). 4 parameters.
- [Musical Intervals & Consonance](/lab/sound-music/musical-intervals): Explore why some note combinations sound harmonious and others clash — visualize the waveform interference that creates consonance and dissonance. Formula: cents = 1200 * log₂(f₂ / f₁). 4 parameters.
- [Resonance & Natural Frequency](/lab/sound-music/resonance): Drive an oscillator at different frequencies and watch amplitude explode at resonance — explore how damping controls the sharpness of the resonance peak. Formula: A(w) = F₀ / sqrt((w₀² - w²)² + (2bw)²). 4 parameters.
- [Waveform Synthesis](/lab/sound-music/waveform-synthesis): Generate and visualize classic waveforms — sine, square, sawtooth, triangle — with amplitude modulation and real-time spectral analysis. Formula: Sine: x(t) = A * sin(2*pi*f*t). 5 parameters.

### Special & General Relativity (4 simulations)
- [Length Contraction Visualizer](/lab/relativity/length-contraction): See how objects physically shorten along the direction of motion at relativistic speeds Formula: L' = L₀/γ = L₀·√(1 − v²/c²). 2 parameters.
- [Mass-Energy Equivalence Calculator](/lab/relativity/mass-energy): Explore E=mc² and relativistic energy with real-world comparisons from grams to megatons Formula: E₀ = mc² (rest energy). 3 parameters.
- [Time Dilation Calculator](/lab/relativity/time-dilation): Visualize how time slows down for objects moving at relativistic speeds with the Lorentz factor Formula: γ = 1/√(1 − v²/c²). 2 parameters.
- [Twin Paradox Simulator](/lab/relativity/twin-paradox): Simulate the twin paradox with a Minkowski spacetime diagram showing how the traveling twin ages less Formula: Earth time (constant v): T_earth = 2D/v. 3 parameters.

### Spectroscopy & Spectral Analysis (5 simulations)
- [Absorption Line Profiles & Beer-Lambert Law](/lab/spectroscopy/absorption-lines): Simulate absorption spectroscopy — explore concentration, path length, molar absorptivity, and line width to visualize Beer-Lambert transmission and absorption line shapes Formula: A = ε × c × l (Beer-Lambert law). 4 parameters.
- [Atomic Emission Spectrum Generator](/lab/spectroscopy/emission-spectrum): Simulate atomic emission spectra — explore atomic number, excitation energy, temperature, and line broadening to visualize characteristic spectral lines from quantum transitions Formula: λ = hc / ΔE (photon wavelength from energy transition). 4 parameters.
- [Mass Spectrometry & Fragmentation](/lab/spectroscopy/mass-spectrometry): Simulate mass spectrometry — explore molecular mass, ionization energy, acceleration voltage, and magnetic field to visualize ion trajectories and mass spectra Formula: r = (1/B)√(2mV/q) (magnetic deflection radius). 4 parameters.
- [Raman Scattering & Vibrational Modes](/lab/spectroscopy/raman-scattering): Simulate Raman spectroscopy — explore laser wavelength, molecular bond strength, reduced mass, and temperature to visualize Stokes and anti-Stokes shifts Formula: ν = (1/2π)√(k/μ) (harmonic oscillator frequency). 4 parameters.
- [X-Ray Diffraction & Crystal Structure](/lab/spectroscopy/xray-diffraction): Simulate X-ray diffraction from crystal lattices — explore lattice spacing, X-ray wavelength, crystal type, and beam angle to visualize Bragg peaks and diffraction patterns Formula: nλ = 2d sin θ (Bragg's law). 4 parameters.

### Stellar Evolution & Astrophysics (5 simulations)
- [Chandrasekhar Mass Limit](/lab/stellar-evolution/chandrasekhar-limit): Explore the Chandrasekhar limit — simulate how electron degeneracy pressure supports white dwarfs up to 1.44 M☉ and what happens when mass exceeds this critical threshold Formula: M_Ch = (ℏc/G)^(3/2) × 5.83 / (μ_e m_H)². 4 parameters.
- [Hertzsprung-Russell Diagram Explorer](/lab/stellar-evolution/hr-diagram): Plot stars on the HR diagram — explore how mass determines luminosity, temperature, and evolutionary stage across the main sequence, giant branch, and white dwarf cooling track Formula: L = L☉ × (M/M☉)^3.5. 4 parameters.
- [Main-Sequence Hydrogen Burning](/lab/stellar-evolution/main-sequence): Model hydrogen fusion on the main sequence — explore how stellar mass controls core temperature, fusion rate, luminosity, and lifetime through the pp-chain and CNO cycle Formula: L = L☉ × (M/M☉)^3.5. 3 parameters.
- [Stellar Nucleosynthesis Pathways](/lab/stellar-evolution/stellar-nucleosynthesis): Trace element production in stars — explore how mass and core temperature control fusion stages from hydrogen through silicon, building the periodic table layer by layer Formula: 4¹H → ⁴He + 2e⁺ + 2ν_e + 26.7 MeV. 4 parameters.
- [White Dwarf Cooling Curves](/lab/stellar-evolution/white-dwarf-cooling): Simulate white dwarf thermal evolution — explore how mass, composition, and crystallization control cooling rates over billions of years Formula: L = (4πR²)σT⁴ (Stefan-Boltzmann luminosity). 4 parameters.

### String Theory & Extra Dimensions (5 simulations)
- [AdS/CFT Correspondence & Holographic Duality](/lab/string-theory/ads-cft): Experience the holographic principle — watch how a gravitational theory in anti-de Sitter space maps to a conformal field theory on its boundary Formula: S_BH = A / (4 × G_N) (Bekenstein-Hawking entropy = area / 4G). 4 parameters.
- [Brane Worlds & Bulk Space](/lab/string-theory/brane-worlds): Navigate through higher-dimensional bulk space and observe how our universe may exist as a 3-brane floating in a larger dimensional reality Formula: G_4D = G_(4+n)D / V_n (effective 4D gravitational constant, V_n = volume of compact dimensions). 4 parameters.
- [Extra Dimensions & Calabi-Yau Manifolds](/lab/string-theory/extra-dimensions): Visualize the hidden dimensions of string theory — watch Calabi-Yau manifolds unfold and explore how extra dimensions shape the physics we observe Formula: D_critical = 10 (superstrings) or 11 (M-theory). 4 parameters.
- [String Vibration Modes & Particle Spectrum](/lab/string-theory/string-vibrations): Watch fundamental strings vibrate in different modes and see how each vibration pattern corresponds to a different elementary particle in the Standard Model Formula: M² = (n - a) × 2T / ℏc (mass-shell condition, n = mode, a = intercept, T = tension). 4 parameters.
- [Supersymmetry & Partner Particles](/lab/string-theory/supersymmetry): Explore supersymmetry by visualizing the boson-fermion partner spectrum — every known particle has a superpartner with different spin statistics Formula: δm²_H(boson) + δm²_H(fermion) = 0 (exact SUSY: quantum corrections cancel). 4 parameters.

### Superconductivity & Zero Resistance (5 simulations)
- [BCS Theory: Cooper Pair Formation & Energy Gap](/lab/superconductivity/bcs-theory): Simulate Cooper pair condensation and the superconducting energy gap as a function of temperature and coupling strength Formula: Δ(T) = Δ₀ × tanh(1.74 × √(Tc/T - 1)). 4 parameters.
- [Critical Field: Hc vs Temperature Phase Diagram](/lab/superconductivity/critical-field): Explore the superconducting phase diagram — critical magnetic field as a function of temperature for type-I superconductors Formula: Hc(T) = Hc₀ × (1 - (T/Tc)²). 4 parameters.
- [Josephson Junction & SQUID](/lab/superconductivity/josephson-junction): Simulate the DC and AC Josephson effects — supercurrent tunneling across a weak link and SQUID flux sensitivity Formula: Is = Ic × sin(φ) (DC Josephson relation). 4 parameters.
- [Meissner Effect: Magnetic Flux Expulsion](/lab/superconductivity/meissner-effect): Visualize how a superconductor expels magnetic flux below Tc — the Meissner effect with London penetration depth Formula: B(x) = B₀ × exp(-x/λ). 4 parameters.
- [Type-II Vortex: Abrikosov Vortex Lattice](/lab/superconductivity/type-ii-vortex): Visualize the Abrikosov vortex lattice in type-II superconductors — flux quantization, vortex interactions, and lattice melting Formula: a₀ = 1.075 × √(Φ₀/B) (triangular lattice spacing). 4 parameters.

### Thermodynamics & Statistical Mechanics (4 simulations)
- [Boltzmann Brain Timescale Calculator](/lab/thermodynamics/boltzmann-brain): Explore the mind-bending timescales of thermal fluctuations — how long until random chance assembles a conscious brain from the void Formula: T ~ exp(ΔS / k_B). 3 parameters.
- [Carnot Cycle Efficiency Calculator](/lab/thermodynamics/carnot-efficiency): Visualize the Carnot cycle on a PV diagram and calculate the theoretical maximum efficiency of any heat engine operating between two temperatures Formula: η = 1 - T_cold / T_hot. 4 parameters.
- [Entropy & Particle Diffusion Simulator](/lab/thermodynamics/entropy-simulator): Watch particles diffuse from order to disorder and observe entropy increase in real time — a visual demonstration of the Second Law of Thermodynamics Formula: S = k_B · ln(W). 4 parameters.
- [Maxwell–Boltzmann Speed Distribution](/lab/thermodynamics/maxwell-boltzmann): Explore how temperature and molecular mass shape the distribution of particle speeds in a gas — and why some molecules escape Earth's gravity Formula: f(v) = 4π·n·(m/(2πk_BT))^(3/2)·v²·exp(-mv²/(2k_BT)). 3 parameters.

### Vacuum Science & Technology (5 simulations)
- [Helium Leak Detection & Sensitivity](/lab/vacuum-science/leak-detection): Simulate helium leak rate measurements, minimum detectable leak size, and response time for mass-spectrometer leak detectors Formula: P_He = Q_leak / S_pump. 4 parameters.
- [Mean Free Path & Knudsen Number](/lab/vacuum-science/mean-free-path): Visualize molecular collisions, mean free path length, and Knudsen number across vacuum pressure regimes Formula: λ = kT / (√2 × π × d² × P). 4 parameters.
- [Outgassing & Bakeout Kinetics](/lab/vacuum-science/outgassing): Model surface desorption rates, outgassing decay curves, and bakeout temperature optimization for UHV systems Formula: q(t) = q₁ × t⁻¹ (Pa·m³/s/m²). 4 parameters.
- [Pumping Speed & Throughput](/lab/vacuum-science/pump-speed): Calculate effective pumping speed, throughput, and pumpdown time for vacuum chambers with conductance-limited connections Formula: Seff = Sp × C / (Sp + C). 4 parameters.
- [Sputtering Rate & Thin Film Growth](/lab/vacuum-science/thin-film-deposition): Model magnetron sputtering deposition rate, film thickness, and mean free path of sputtered atoms as a function of power, pressure, and target material Formula: R = (Y × J × M) / (ρ × NA × A) — deposition rate. 4 parameters.

### Waves & Optics (6 simulations)
- [Doppler Effect Simulator](/lab/waves-optics/doppler-effect): Watch wavefronts compress and stretch as a source moves, demonstrating the Doppler shift that changes the pitch of sirens and the color of stars. Formula: f_obs = f₀ × (v_w ± v_obs) / (v_w ∓ v_s). 4 parameters.
- [Electromagnetic Spectrum Explorer](/lab/waves-optics/electromagnetic-spectrum): Navigate the full electromagnetic spectrum from radio waves to gamma rays, exploring wavelength, frequency, energy, and real-world applications of each band. Formula: c = λf. 4 parameters.
- [Single-Slit Diffraction](/lab/waves-optics/diffraction-single-slit): Watch light spread and form diffraction patterns through a single slit, revealing the wave nature of light and the counter-intuitive relationship between slit width and pattern width. Formula: I(θ) = I₀ × (sin(β)/β)²  where β = πa sin(θ)/λ. 4 parameters.
- [Snell's Law & Refraction](/lab/waves-optics/snells-law): Watch light bend as it passes between materials of different refractive indices, and discover the critical angle for total internal reflection. Formula: n₁ sin(θ₁) = n₂ sin(θ₂). 4 parameters.
- [Standing Waves & Harmonics](/lab/waves-optics/standing-waves): Watch a vibrating string form standing wave patterns with nodes and antinodes, exploring how tension, density, and harmonic number determine pitch. Formula: f_n = n/(2L) × √(T/μ). 4 parameters.
- [Wave Interference & Superposition](/lab/waves-optics/wave-interference): Visualize how two waves combine through superposition, producing constructive and destructive interference patterns including beats. Formula: y = A₁sin(2πf₁t) + A₂sin(2πf₂t + φ). 5 parameters.

## psychology (2 topics, 9 simulations)

### Cognitive Biases & Decision Science (4 simulations)
- [Anchoring Effect Simulator](/lab/cognitive-biases/anchoring-effect): Simulate how an initial anchor biases a population of estimates through insufficient adjustment, visualized as histogram and scatter plot Formula: estimate = anchor + α · (true_value - anchor) + ε. 5 parameters.
- [Bayes' Theorem Visualizer](/lab/cognitive-biases/bayes-theorem): Explore how prior beliefs update with evidence using Bayes' theorem, with probability trees and natural frequency displays Formula: P(H|E) = P(E|H) · P(H) / [P(E|H) · P(H) + P(E|¬H) · P(¬H)]. 3 parameters.
- [Prospect Theory Value Function](/lab/cognitive-biases/prospect-theory): Visualize Kahneman-Tversky's S-shaped value function and probability weighting — the foundation of behavioral economics Formula: v(x) = x^α for x ≥ 0 (gains). 5 parameters.
- [Survivorship Bias Simulator](/lab/cognitive-biases/survivorship-bias): Simulate selection bias in company returns — see how looking only at survivors inflates perceived performance Formula: P(survive all years) = survival_rate^years. 5 parameters.

### Psychoacoustics & Hearing Science (5 simulations)
- [Auditory Masking & Critical Bands](/lab/psychoacoustics/masking-effect): Simulate auditory masking — explore how a louder masker tone renders nearby frequencies inaudible and how critical bandwidth shapes perceptual audio coding Formula: CB(f) = 25 + 75 × [1 + 1.4·(f/1000)²]^0.69 (critical bandwidth). 4 parameters.
- [Binaural Sound Localization](/lab/psychoacoustics/binaural-localization): Simulate binaural sound localization — explore how interaural time difference (ITD) and interaural level difference (ILD) enable the brain to determine sound source direction Formula: ITD = (r/c) × (sin θ + θ) (Woodworth model). 4 parameters.
- [Equal-Loudness Contours (Fletcher-Munson)](/lab/psychoacoustics/equal-loudness): Simulate equal-loudness contours — explore how the human ear perceives loudness differently across frequency and how phon curves shape audio engineering Formula: S = 2^((L_N − 40)/10) (sone from phon). 4 parameters.
- [Hearing Threshold & Audiogram](/lab/psychoacoustics/hearing-threshold): Simulate absolute hearing thresholds — explore how the ear's sensitivity varies with frequency and how age, noise exposure, and hearing conditions affect the audiogram Formula: Threshold(f,age) = MAP(f) + 0.5·(age − 20)·k(f) dB. 4 parameters.
- [Pitch Perception & Basilar Membrane](/lab/psychoacoustics/pitch-perception): Simulate pitch perception — explore how frequency maps to perceived pitch through the basilar membrane's tonotopic organization, mel scale, and place-rate theory Formula: m = 2595 × log₁₀(1 + f/700) (mel scale). 4 parameters.

## science (6 topics, 30 simulations)

### Archaeology & Dating Methods (5 simulations)
- [Archaeological Site Survey Grid Simulator](/lab/archaeology/site-survey): Plan and execute a systematic archaeological survey using grid squares — visualize artifact density heatmaps and optimize sampling strategies for site discovery Formula: Detection probability ≈ 1 - (1 - site_area/total_area)^(n_samples). 4 parameters.
- [Artifact Typology & Seriation Simulator](/lab/archaeology/artifact-classification): Classify artifacts into typological groups and arrange assemblages chronologically using frequency seriation — the statistical backbone of relative dating Formula: Brainerd-Robinson coefficient = 200 - Σ|pᵢ - qᵢ| for assemblage similarity. 4 parameters.
- [Radiocarbon C-14 Decay Dating Simulator](/lab/archaeology/radiocarbon-dating): Watch Carbon-14 atoms decay exponentially and see how archaeologists convert remaining C-14 ratios into calendar ages for organic samples up to 50,000 years old Formula: N(t) = N₀ × (1/2)^(t / t½) where t½ = 5,730 years. 4 parameters.
- [Stratigraphic Layer Analysis Simulator](/lab/archaeology/stratigraphy): Build and analyze sediment layers to understand the law of superposition — how archaeologists read time from the vertical sequence of deposits at an excavation site Formula: Deposition rate = layer thickness / time interval. 4 parameters.
- [Tree Ring Dating & Climate Proxy Simulator](/lab/archaeology/dendrochronology): Analyze tree ring patterns to date wooden artifacts and reconstruct past climate — cross-matching ring width sequences that extend calendars back thousands of years Formula: Ring width index = actual_width / expected_width from age-detrending curve. 4 parameters.

### Astrophysics & Cosmology (5 simulations)
- [Cosmic Inflation & Early Universe](/lab/astrophysics/cosmic-inflation): Simulate the first fraction of a second after the Big Bang — watch the universe inflate exponentially and seed the structure of everything Formula: Scale factor during inflation: a(t) = a₀ × exp(H × t). 4 parameters.
- [Dark Energy & Accelerating Expansion](/lab/astrophysics/dark-energy): Simulate the accelerating expansion of the universe — explore how dark energy density, matter content, and the cosmological constant shape cosmic fate Formula: Friedmann equation: H² = H₀²[Ω_m × a⁻³ + Ω_Λ × a^(-3(1+w))]. 4 parameters.
- [Gravitational Wave Detection & Mergers](/lab/astrophysics/gravitational-waves): Simulate gravitational waves from merging compact objects — visualize spacetime ripples and the chirp signal detected by LIGO Formula: Chirp mass: M_c = (m1 × m2)^(3/5) / (m1 + m2)^(1/5). 4 parameters.
- [Neutron Star Density & Structure](/lab/astrophysics/neutron-star): Explore the extreme physics of neutron stars — compress solar masses into city-sized spheres and observe how density, gravity, and rotation behave Formula: Average density: ρ = 3M / (4πR³). 4 parameters.
- [Supernova Explosion Lifecycle](/lab/astrophysics/supernova-lifecycle): Watch a massive star collapse and explode as a supernova — simulate the shockwave, nucleosynthesis, and remnant formation in real time Formula: Chandrasekhar limit: M_Ch ≈ 1.4 M☉ (maximum mass for white dwarf / iron core). 4 parameters.

### Cartography & Navigation (5 simulations)
- [Geographic Coordinate System Transforms](/lab/cartography/coordinate-systems): Convert between geographic coordinate systems — visualize how latitude/longitude, UTM, and MGRS grids map to positions on the globe Formula: UTM Zone = floor((lon + 180) / 6) + 1. 4 parameters.
- [GPS Trilateration Positioning](/lab/cartography/gps-trilateration): Visualize how GPS receivers calculate position from satellite signals — see trilateration circles intersect and understand dilution of precision Formula: d_i = c × (t_received - t_transmitted) = distance to satellite i. 4 parameters.
- [Great Circle Distance & Navigation Routes](/lab/cartography/great-circle): Calculate and visualize great circle routes between any two points on Earth — see why airplanes fly curved paths on flat maps and compare to rhumb lines Formula: d = R × arccos(sin(φ₁)×sin(φ₂) + cos(φ₁)×cos(φ₂)×cos(Δλ)). 4 parameters.
- [Map Projections & Distortion Comparison](/lab/cartography/map-projections): Compare how Mercator, Mollweide, and other map projections distort area, shape, and distance — see Tissot indicatrices reveal the hidden warping Formula: Mercator: x = λ, y = ln(tan(π/4 + φ/2)). 4 parameters.
- [Topographic Contour Line Generation](/lab/cartography/contour-mapping): Generate topographic contour maps from elevation data — watch contour lines trace terrain features like peaks, valleys, saddles, and ridgelines Formula: Slope = arctan(Δelevation / Δhorizontal_distance). 4 parameters.

### Food Science & Chemistry (5 simulations)
- [Emulsion Stability Simulator](/lab/food-science/emulsification): Simulate oil-in-water emulsion stability — adjust droplet size, emulsifier concentration, and viscosity to see how quickly an emulsion separates Formula: v = 2r²(ρ_oil - ρ_water)g / 9η — Stokes' law for creaming velocity. 4 parameters.
- [Fermentation Kinetics Simulator](/lab/food-science/fermentation): Model ethanol fermentation kinetics — adjust temperature, sugar concentration, and yeast density to watch how ethanol yield evolves over time Formula: C₆H₁₂O₆ → 2 C₂H₅OH + 2 CO₂ (Gay-Lussac equation). 4 parameters.
- [Maillard Reaction Simulator](/lab/food-science/maillard-reaction): Explore how temperature, time, pH, and moisture affect Maillard browning intensity — the chemistry behind golden crusts, roasted coffee, and seared steaks Formula: k = A × exp(-Ea / RT) — Arrhenius rate equation for browning kinetics. 4 parameters.
- [Pasteurization Time-Temperature Simulator](/lab/food-science/pasteurization): Visualize pasteurization thermal kill curves — adjust temperature, hold time, and target pathogen to calculate log-reduction and safety margins Formula: D_T = D_ref × Math.pow(10, (T_ref - T) / z) — D-value at temperature T. 4 parameters.
- [Sugar Crystallization Simulator](/lab/food-science/crystallization): Model sugar crystallization from supersaturated solutions — control temperature, supersaturation, and agitation to grow or prevent crystals Formula: S = C / C_sat(T) — supersaturation ratio. 4 parameters.

### Forensic Analysis & Crime Science (5 simulations)
- [Blood Spatter Pattern Analysis](/lab/forensic-analysis/blood-spatter): Analyze blood spatter patterns — compute impact angles, origin points, and velocity categories from stain geometry Formula: sin(α) = width / length. 4 parameters.
- [DNA Profiling & STR Match Probability](/lab/forensic-analysis/dna-profiling): Simulate DNA profiling — explore how STR loci count, allele frequencies, and population genetics determine match probability Formula: RMP = Π(2·p_i·p_j) for heterozygous loci, Π(p_i²) for homozygous. 4 parameters.
- [Fingerprint Minutiae Matching Simulator](/lab/forensic-analysis/fingerprint-matching): Simulate fingerprint matching — explore how minutiae count, false acceptance rate, and match threshold affect identification reliability Formula: S = Nm² / (N₁ × N₂)  (Minutiae match score). 4 parameters.
- [Forensic Ballistics Trajectory Reconstructor](/lab/forensic-analysis/ballistics-trajectory): Reconstruct bullet trajectories — compute muzzle velocity, drop, drift, and impact angle for forensic ballistics analysis Formula: Δy = ½g(d/v₀)² (simplified drop, flat fire). 4 parameters.
- [Forensic Toxicology Screening Model](/lab/forensic-analysis/toxicology-screening): Model toxicology pharmacokinetics — simulate drug absorption, distribution, and elimination to estimate blood concentration over time Formula: C(t) = (F·D/Vd) × exp(-0.693·t/t½). 4 parameters.

### Sports Science & Athletics (5 simulations)
- [Athletic Reaction Time Distribution](/lab/sports-science/reaction-time): Model the statistical distribution of reaction times in competitive sports — from sprint starts to goalkeeper saves and false start thresholds Formula: P(RT < t) = Φ((t - μ) / σ) — cumulative normal distribution. 4 parameters.
- [Cycling Drafting & Aerodynamic Advantage](/lab/sports-science/drafting-aerodynamics): Simulate the aerodynamic drag reduction of drafting in cycling — see how gap distance, speed, and group size affect power savings Formula: F_drag = 0.5 × ρ × CdA × v². 4 parameters.
- [Elo Rating System & Tournament Simulation](/lab/sports-science/elo-rating): Simulate round-robin tournaments with the Elo rating system — watch ratings converge to true skill levels through matchplay dynamics Formula: E_A = 1 / (1 + Math.pow(10, (R_B - R_A) / 400)). 4 parameters.
- [Sprint Biomechanics & Force Curves](/lab/sports-science/sprint-biomechanics): Visualize ground reaction forces during a 100m sprint — see how stance time, peak force, and stride frequency evolve from start to maximum velocity Formula: Peak GRF = m × g × (1 + v / (g × t_contact)). 4 parameters.
- [Training Load Periodization Model](/lab/sports-science/training-periodization): Model the fitness-fatigue relationship in athletic training — visualize how training load, recovery, and tapering affect performance readiness over time Formula: Performance(t) = p₀ + k₁ × Σ w(i) × Math.exp(-(t-i)/τ₁) - k₂ × Σ w(i) × Math.exp(-(t-i)/τ₂). 4 parameters.

## social-science (3 topics, 15 simulations)

### Economics & Finance (5 simulations)
- [Compound Interest & Wealth Growth Simulator](/lab/economics/compound-interest): Visualize how compound interest transforms savings over time, comparing contributions vs interest earned with the Rule of 72 and milestone tracking Formula: Compound interest: A = P·(1 + r/n)^(nt). 5 parameters.
- [Gini Coefficient & Income Inequality Simulator](/lab/economics/gini-inequality): Explore income inequality through the Lorenz curve and Gini coefficient, with Pareto-distributed incomes, redistribution effects, and real-world country comparisons Formula: Gini = 1 - 2·∫₀¹ L(x)dx. 4 parameters.
- [Inflation & Money Supply Simulator](/lab/economics/inflation-money-supply): Model the Quantity Theory of Money (MV=PY) to visualize how money supply growth, velocity changes, and real GDP growth determine inflation and purchasing power erosion Formula: Quantity equation: M · V = P · Y. 5 parameters.
- [Market Bubble Dynamics Simulator](/lab/economics/market-bubbles): Model speculative market bubbles with positive feedback loops, herd behavior, and sudden crashes — visualizing the divergence between fundamental value and market price Formula: Price dynamics: P(t+1) = P(t) · (1 + fundamental_return + speculation · sentiment(t)). 5 parameters.
- [Supply & Demand Equilibrium Simulator](/lab/economics/supply-demand): Visualize how supply and demand curves determine market equilibrium price and quantity, with consumer/producer surplus and deadweight loss from taxation Formula: Demand: Q_d = a - b·P. 5 parameters.

### Sociology & Social Dynamics (5 simulations)
- [Condorcet Voting Paradox](/lab/sociology/voting-paradox): Discover why there is no perfect voting system — compare plurality, runoff, Borda, and Condorcet methods and watch them produce different winners from identical ballots Formula: Borda score for candidate i = Σ_voters (rank position points). 4 parameters.
- [Information Cascade in Networks](/lab/sociology/social-network-cascade): Watch ideas spread through social networks — seed nodes trigger adoption cascades whose success depends on network structure, connection density, and individual thresholds Formula: Adoption rule: node i adopts if (adopted neighbors / total neighbors) ≥ threshold. 4 parameters.
- [Schelling Segregation Model](/lab/sociology/schelling-segregation): Watch neighborhoods self-segregate from mild individual preferences — Schelling's Nobel Prize-winning model reveals how tolerance at the micro level produces segregation at the macro level Formula: Happiness condition: (same-type neighbors / total neighbors) ≥ threshold. 4 parameters.
- [Social Opinion Dynamics](/lab/sociology/opinion-dynamics): Watch opinions cluster, polarize, and fragment — a bounded confidence model showing how social influence, extremism, and noise shape public opinion landscapes Formula: Opinion update: xᵢ(t+1) = (1/|Nᵢ|) × Σⱼ∈Nᵢ xⱼ(t) where Nᵢ = {j : |xᵢ−xⱼ| < ε}. 4 parameters.
- [Wealth Distribution Dynamics](/lab/sociology/wealth-distribution): Watch inequality emerge from fair rules — random wealth exchanges between equal agents naturally produce exponential distributions, Pareto principles, and Gini coefficients matching real economies Formula: Gini = (2 × Σᵢ i × wᵢ) / (N × Σᵢ wᵢ) − (N+1)/N. 4 parameters.

### Voting & Democracy (5 simulations)
- [Arrow's Impossibility Theorem](/lab/voting-democracy/arrows-impossibility): See how the same voters produce different winners under plurality, Borda, Condorcet, ranked-choice, and approval voting — proving Arrow's impossibility theorem visually Formula: Borda score for candidate i: sum over all voters of (num_candidates - rank_of_i). 4 parameters.
- [Electoral College vs Popular Vote](/lab/voting-democracy/electoral-college): Simulate how a candidate can win the presidency while losing the popular vote — explore the mathematics of the Electoral College Formula: Electoral votes per state: floor(population_i / avg_pop * base_seats) + 2. 4 parameters.
- [Gerrymandering Geometry](/lab/voting-democracy/gerrymandering): Visualize how drawing district boundaries on the same electorate can manufacture wildly different election outcomes — the dark geometry of gerrymandering Formula: Efficiency Gap = |wasted_blue - wasted_red| / total_votes. 4 parameters.
- [Political Polarization Model](/lab/voting-democracy/political-polarization): Watch a moderate population split into extremes as echo chambers, media bias, and radicalization drive opinions apart over generations Formula: Opinion update: x_i(t+1) = x_i(t) + echo * (x_neighbor - x_i) + media_bias * sign(x_i - 0.5). 5 parameters.
- [Proportional vs Winner-Take-All](/lab/voting-democracy/proportional-representation): Compare D'Hondt, Sainte-Lague, and largest-remainder seat allocation methods — see how electoral thresholds and rounding rules reshape legislatures Formula: D'Hondt quotient: votes_i / (seats_allocated_i + 1). 4 parameters.

## social-sciences (2 topics, 10 simulations)

### Demography & Population Studies (5 simulations)
- [Demographic Transition Model](/lab/demography/demographic-transition): Simulate the demographic transition — explore how development stage, healthcare access, industrialization, and urbanization drive the shift from high to low birth and death rates Formula: RNI = CBR - CDR — rate of natural increase. 4 parameters.
- [Fertility Rate & Replacement Level](/lab/demography/fertility-rate): Simulate total fertility rate dynamics — explore how desired family size, contraception prevalence, female education, and child mortality affect TFR and replacement thresholds Formula: TFR = Σ ASFR(x) — sum of age-specific fertility rates. 4 parameters.
- [Life Table & Survival Analysis](/lab/demography/life-table): Simulate actuarial life tables — explore how infant mortality, adult mortality hazard, maximum lifespan, and initial cohort size shape survival curves and life expectancy Formula: μ(x) = α × exp(βx) — Gompertz mortality hazard. 4 parameters.
- [Migration Flow & Push-Pull Model](/lab/demography/migration-model): Simulate migration flows between regions — explore how wage differential, distance, origin population, and border friction govern migration rates using the gravity model Formula: M_ij = k × P_i × Δw / (d^α × f) — gravity migration model. 4 parameters.
- [Population Pyramid & Age-Sex Distribution](/lab/demography/population-pyramid): Simulate population pyramids — explore how birth rate, death rate, growth rate, and migration shape the age-sex structure of a population over time Formula: t₂ = ln(2) / r ≈ 70 / r% — population doubling time. 4 parameters.

### Urban Planning & City Design (5 simulations)
- [Green Space Distribution & Urban Ecology](/lab/urban-planning/green-space): Model how parks, urban forests, and green corridors affect biodiversity, air quality, heat island reduction, and resident well-being across a city grid. Formula: Species-Area: S = cA^z (z ≈ 0.25 for urban patches). 4 parameters.
- [Population Density & Infrastructure Costs](/lab/urban-planning/population-density): Explore how population density affects infrastructure costs per capita, service delivery efficiency, and the economic viability of public transit — the core trade-off in urban form. Formula: Infrastructure cost/capita ∝ 1/density^0.3 (economies of scale). 4 parameters.
- [Traffic Network & Congestion Flow](/lab/urban-planning/traffic-network): Simulate vehicular traffic flow through a city road network — observe congestion patterns, Braess's paradox, and how signal timing affects throughput. Formula: Congestion Index = Actual Travel Time / Free-Flow Travel Time. 4 parameters.
- [Walkability Score & Pedestrian Access](/lab/urban-planning/walkability): Calculate neighborhood walkability scores based on intersection density, land-use mix, block length, and amenity proximity — the metrics that predict whether people walk or drive. Formula: Walk Score = 0.3×IntersectionScore + 0.25×MixScore + 0.25×AmenityScore + 0.2×BlockScore. 4 parameters.
- [Zoning & Land Use Model](/lab/urban-planning/zoning-model): Interactive zoning simulator that models how residential, commercial, and industrial zone allocations shape city structure, property values, and livability. Formula: FAR = Total Floor Area / Lot Area. 4 parameters.