Tessellations

A tessellation, or tiling, of the plane is a covering of the plane by shapes without gaps or overlaps. Tiles may be polygons or curved shapes, and copies of a tile may be rotated, reflected, or translated. Tessellations study how shapes fit together and what symmetries the pattern exhibits.

Regular tessellations use congruent regular polygons in a uniform way around every vertex. In the plane, only three exist: triangles, squares, and regular hexagons. They occur because the interior angles of these polygons allow exactly six, four, or three polygons to meet around a point to fill space (6×60°, 4×90°, 3×120°).

Semiregular or Archimedean tessellations use more than one type of regular polygon but have the same sequence of polygons around every vertex. There are 11 distinct Archimedean tilings in the plane (including the three regular ones). They commonly feature mixtures such as triangles with squares or hexagons around each vertex.

Aperiodic tilings cover the plane without gaps or overlaps but do not repeat periodically. Penrose tilings are a famous example, using a small set of shapes that enforce nonperiodic order and have connections to quasicrystals in materials science.

Tessellations appear in mathematics, art, and design. They are used to study symmetry and tiling theory and are found in architecture, floor patterns, and computer graphics. The field also extends to tilings of curved surfaces and non-Euclidean geometries.