Filling the Plane Without Gaps
A tessellation covers an infinite plane with shapes that fit together perfectly — no gaps, no overlaps. The simplest examples use regular polygons: equilateral triangles, squares, and hexagons are the only regular polygons that can tile the plane alone. Their interior angles divide evenly into 360°, allowing them to meet flush at every vertex.
Beyond Regular Tilings
Semi-regular (Archimedean) tilings use two or more types of regular polygons with the same vertex configuration everywhere. There are exactly 8 of these. The Cairo tiling uses irregular pentagons. Going further, any triangle or quadrilateral can tile the plane — a surprising result that extends to certain pentagons and hexagons with specific angle constraints.
Aperiodic Wonders
In 1974, Roger Penrose discovered tilings that cover the plane but never repeat. Using just two tile shapes (kites and darts), Penrose tilings have five-fold rotational symmetry — something impossible for periodic tilings. In 1984, Dan Shechtman discovered quasicrystals with the same forbidden symmetry, earning the 2011 Nobel Prize in Chemistry.
From Islamic Art to Modern Materials
Tessellations have been explored by artists for millennia. Islamic geometric art achieved extraordinary complexity using compass-and-straightedge constructions. M.C. Escher transformed simple tilings into interlocking birds, fish, and lizards. Today, tessellation theory underpins metamaterial design, architectural facades, and procedural generation in video games.