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A tessellation, also known as a tiling, is a collection of plane figures that completely cover a plane with no overlaps or gaps. These figures are called tiles, and they can be polygons such as squares, triangles, or hexagons. The study of tessellations falls under the domain of geometry.

There are several types of tessellations. A regular tessellation is one made up entirely of congruent regular polygons, with one type of polygon used throughout. Only three regular polygons can tessellate the plane: equilateral triangles, squares, and regular hexagons. This is because the interior angles of these polygons divide 360 degrees evenly. An equilateral triangle has an interior angle of 60 degrees, a square has an angle of 90 degrees, and a regular hexagon has an angle of 120 degrees.

A semi-regular tessellation, also known as an Archimedean tessellation, is made up of two or more different types of regular polygons. In a semi-regular tessellation, the vertices are all identical. This means that the same combination of polygons meets at each vertex, and they are arranged in the same order around each vertex. There are eight such semi-regular tessellations.

A k-uniform tessellation is a tessellation where the tiles are all similar, and the symmetry group of the tessellation acts transitively on the orbits of tiles. This means that all tiles are equivalent under the symmetries of the tiling.

Tessellations have numerous applications in art, architecture, and design. For instance, they can be seen in the intricate patterns of Islamic art, the mosaics of ancient Rome, and the stained glass windows of Gothic cathedrals. In nature, patterns resembling tessellations can be observed in honeycomb structures, the scales of some animals, and the arrangement of cells in tissues. The concept also extends to three-dimensional space, where polyhedra can fill space, known as space-filling polyhedra or honeycombs.