Only three regular polygons tile the plane forever
A tessellation covers a surface with tiles that neither overlap nor leave gaps. On the Euclidean plane, only equilateral triangles, squares, and regular hexagons can form a fully regular tiling—identical regular polygons meeting identically at every corner. Everything else trades that purity for richer rules.
Periodic tilings repeat; their patterns fall into seventeen wallpaper groups, a classification Yevgraf Fyodorov proved in 1891 for every periodic plane tiling. Semi-regular, or Archimedean, tilings mix more than one regular polygon while keeping the same vertex arrangement—eight such types, or nine if mirror pairs count separately. Aperiodic sets of prototiles refuse any repeating pattern; Penrose tilings, built from two quadrilateral shapes, are the best-known examples. Space-filling analogues are honeycombs in higher dimensions.
History runs deep. Sumerians around 4000 BC decorated walls with patterned clay tiles. Classical mosaics used small squared tesserae—Latin tessella means a little square, from Greek four. Johannes Kepler studied regular and semiregular tessellations in Harmonices Mundi in 1619 and pondered honeycomb and snowflake hexagons. Islamic art at sites such as the Alhambra pushed geometric tiling sophistication; some claim all seventeen wallpaper groups appear there, though that claim is disputed. M. C. Escher later bent both Euclidean and hyperbolic tilings into animal-shaped art.
Formal language helps. An edge joins two tiles; a vertex is where three or more meet. Regular hexagon tilings carry Schläfli symbol {6,3} or vertex configuration 6.6.6. Edge-to-edge tilings share full sides; brick walls fail that test because each long brick edge meets two neighbours. Monohedral tilings use one congruent prototile; Heinz Voderberg’s 1936 spiral used a nonconvex nine-sided tile, and the 1985 Hirschhorn tiling deploys irregular pentagons—regular pentagons cannot tile the plane because their interior angle is not a divisor of a full turn.
Ludwig Schläfli generalised the game to polytopes in higher dimensions. No universal rule decides whether an arbitrary shape tiles, so open problems abound. Nature still votes with hexagons: honeycomb cells are a living tessellation.
Source: Tessellation
