tiling

Tilings or tessellations are arrangements of shapes that cover a plane without gaps or overlaps. The tiles can be geometric shapes such as polygons, and can be congruent or of varying sizes. Tilings are studied in mathematics as a way to understand symmetry, repetition, and coverage.

In mathematics, tilings are categorized as periodic if the pattern repeats regularly in two directions, and aperiodic if it never repeats exactly yet still covers the plane. The study includes regular tilings (made with one type of regular polygon) and semi-regular or Archimedean tilings (patterns where vertex configurations are uniform). The concept connects to wallpaper groups, which classify possible symmetries of plane tilings, and to tiling problems that seek efficient coverings with minimum waste.

Regular tilings of the plane are possible with equilateral triangles, squares, and regular hexagons. Semi-regular tilings combine different regular polygons in repeating vertex configurations. Beyond periodic tilings, aperiodic tilings such as the Penrose tiling use a finite set of tiles that enforce nonperiodic order, revealing complex, quasi-crystalline structures.

In practical use, tiling refers to laying tiles for floors, walls, and surfaces. Common materials include ceramic and porcelain tile, natural stone, glass, and metal. Installation involves layout planning, substrate preparation, cutting, setting with adhesive, and grouting. Design choices include color, texture, orientation, and grout lines.

The concept has ancient roots in mosaics and decorative tilings across cultures. In mathematics, tiling theory intersects geometry, group theory, and combinatorics, with applications in crystallography and computer graphics.