Tessellatio

Tessellatio, or tessellation, is a pattern formed by repeating shapes that cover a plane without gaps or overlaps. The shapes are called tiles, and the arrangement is a tessellation. Tessellations can use a single tile shape or multiple shapes, and they may exhibit translational symmetry or be non-periodic.

In the plane, a necessary condition for tiling by congruent copies of a polygon is that the angles around each vertex sum to 360 degrees. For regular polygons, this constraint yields only a few possibilities and produces three regular tilings: with equilateral triangles, squares, or regular hexagons. In these tilings, around every vertex the numbers of polygon corners are 6, 4, or 3 respectively.

Semi-regular (Archimedean) tilings use more than one type of regular polygon but maintain the same vertex configuration throughout the pattern. Examples include tilings with one square and two octagons (4.8.8) or one triangle and two dodecagons (3.12.12). There are 11 uniform Archimedean tilings in the Euclidean plane.

Non-periodic tilings, or aperiodic tilings, do not repeat translationally. The best-known example is the Penrose tiling, which uses a small set of shapes with matching rules to create an aperiodic, space-filling pattern.

Tessellations appear in art, architecture, and science. They are central in Islamic geometric art and in the work of M. C. Escher, and they also play a role in crystallography and computer graphics. Tilings can be extended to spherical or hyperbolic geometries, producing tilings of curved surfaces.