tilingud

Tilings, also known as tessellations, are coverings of a plane using one or more shapes without gaps or overlaps. The shapes, called tiles, are arranged so copies may be placed next to each other under isometries such as translations, rotations, and reflections. A tiling is periodic if it repeats itself through a lattice of translations; it is aperiodic if it has no translational symmetry.

Regular tilings of the plane use a single tile type: equilateral triangles, squares, or regular hexagons. More generally, tilings may use a set of prototiles that fit together in various ways. The study of tilings intersects geometry, combinatorics, and materials science. Common examples include square tiling and hexagonal tiling, visible in architecture and design. Triangular tilings and semi-regular tilings combine different shapes around vertices to achieve uniform coverage. In contrast, Penrose tilings use a small, non-periodic set of tiles that enforce aperiodicity and exhibit fivefold local symmetry.

History and applications: tilings have been studied since ancient times, appearing in Islamic art and architectural decorations. In the 20th century, tiling theory advanced with the discovery of aperiodic tile sets and the broader development of tessellation concepts. Applications span computer graphics, architectural design, and the study of quasicrystals in materials science.

See also: tessellation, quasicrystal, Penrose tiling, aperiodicity.