tessellationin

Tessellation, often called tiling, is the arrangement of shapes, or tiles, that cover a surface without gaps or overlaps. The term tessellationin is not standard and may reflect a typographical variant or a context-specific use of tessellation. This article reviews tessellation in mathematics and art.

In two dimensions, plane tilings use polygons to cover the plane. The simplest are regular tilings, made from one type of regular polygon: equilateral triangles, squares, or regular hexagons. Semi-regular, or Archimedean, tilings use more than one polygon type but have a uniform vertex configuration around every tile. There are eleven such tilings in the plane. Edge-to-edge tilings require tile edges to coincide fully with neighbor edges, although many tilings allow partial edge contacts at vertices.

Aperiodic tilings do not repeat periodically. The best-known example is the Penrose tiling, which uses two shapes (kites and darts or rhombs) to produce non-repeating patterns with apparent fivefold symmetry. Substitution tilings grow from a recursive rule to generate larger tilings from a small seed.

Tilings can also be considered on curved surfaces. On the sphere, tilings correspond to polyhedra with a finite number of faces. On the hyperbolic plane, tiled polygons can meet with angular sums that exceed Euclidean limits, producing rich, nonintuitive patterns.

Applications of tessellation include architectural ornament, graphic design, crystallography and materials science, computational geometry, and mathematical education. Ancient mosaics and Islamic art traditions illustrate early and sophisticated uses of tessellation, while modern mathematics formalizes the classification and construction of tilings.