Tessellation (tiling of the plane)

Types and characteristics

Common distinctions include regular, semi-regular and aperiodic tessellations. Only three regular tessellations exist in the Euclidean plane, made from equilateral triangles, squares and regular hexagons. Semi-regular tilings combine a few regular polygons in repeating arrangements, while aperiodic sets produce nonrepeating patterns.

Key concepts

• Vertex configuration: the sequence of polygons meeting at a point.
• Symmetry groups: plane patterns fall into one of 17 wallpaper groups based on their symmetries.
• Prototiles and aperiodicity: certain small sets of tiles force nonperiodic order.

History and development

Tessellated surfaces appear in ancient mosaics and traditional geometric ornament, notably in Islamic art. Mathematicians have studied tilings for centuries; early scientific interest grew into a modern theory that links symmetry, combinatorics and geometry. Artists such as M.C. Escher popularized imaginative interlocking figures inspired by mathematical ideas.

Uses and examples

Tessellation plays a role in architecture and floor design, in computer graphics (texture mapping and mesh generation), and in materials science where tiling models help describe crystal and quasicrystal structures. Natural examples include honeycomb cells and certain biological patterns. Engineers also exploit tessellations in efficient packing and manufacturing.

Notable facts and extensions

Tessellation theory extends to non-Euclidean geometries (for example, hyperbolic tilings) and to higher dimensions as space-filling polyhedra. Research continues on classification problems, aperiodic sets and algorithmic questions about when a given set of tiles can fill the plane. For illustrations of basic ideas and examples of tile shapes, see shapes.