Tessellate

Tessellate is a verb meaning to cover a plane or surface with shapes, called tiles, so that there are no gaps or overlaps. The resulting pattern is a tessellation, or tiling. Tessellation can involve polygons or other shapes and may be periodic, repeating a pattern, or aperiodic, lacking translational symmetry.

In two dimensions, tessellations can be classified by the nature of the tiles and their arrangement. Regular tessellations use congruent regular polygons; only three exist for the plane: equilateral triangles, squares, and regular hexagons. Semi-regular (or Archimedean) tilings use more than one type of regular polygon but maintain a uniform vertex configuration. Non-periodic tilings, such as Penrose tilings, do not repeat exactly in a regular grid yet still cover the plane without gaps.

Tessellation also applies to three-dimensional space. Space-filling tessellations, or honeycombs, fill space with polyhedra so that there are no gaps or overlaps. Examples include the cubic honeycomb (filled with cubes) and the truncated octahedral honeycomb.

History and applications: tessellations have a long artistic and mathematical history. They appear in ancient mosaics and Islamic geometric patterns, illustrating tiling principles. In modern contexts, tessellations are used in architecture, decorative arts, and computer graphics, including texture mapping and procedural tiling in digital models.

Etymology: tessellate derives from French tesselation, from Latin tessella, a small tile.