Beyond Euclid's Fifth Postulate
For two thousand years, mathematicians tried to prove Euclid's parallel postulate from the other four axioms. In the 19th century, Bolyai, Lobachevsky, and Gauss independently discovered that denying it produces a perfectly consistent geometry — hyperbolic geometry — where space curves away from itself at every point. In this strange world, triangles have angles summing to less than 180°, circles grow exponentially with radius, and infinitely many parallel lines pass through any point.
The Poincaré Disk Model
Henri Poincaré devised an elegant way to visualize the infinite hyperbolic plane inside a finite disk. Straight lines become circular arcs that meet the disk boundary at right angles. Distances are distorted: objects near the edge appear tiny but are hyperbolically just as large as objects in the center. This conformal model preserves angles, making it ideal for studying tilings.
Regular Hyperbolic Tilings
In Euclidean geometry, only three regular tilings exist: squares, triangles, and hexagons. Hyperbolic geometry is far richer — infinitely many regular tilings {p,q} are possible whenever (p-2)(q-2) > 4. The simulation above renders these tilings recursively, showing how exponentially many tiles crowd toward the boundary disk as the recursion depth increases.
Escher's Circle Limit
M.C. Escher created his famous Circle Limit woodcuts after learning about hyperbolic tilings from geometer H.S.M. Coxeter. The repeating fish, angels, and devils in these prints follow hyperbolic symmetry groups, with each figure the same hyperbolic size despite appearing to shrink near the boundary. This simulation lets you create your own tilings in the same mathematical framework Escher used.