Tessellation with regular hexagons

A tessellation is semi-regular if it consists of regular polygons, not all congruent. It can be shown that there are just eight of these, one of which has two forms that are mirror-images of each other. They use triangles, squares, hexagons, octagons, and dodecagons. For example, one consisting of octagons and squares and another consisting of hexagons and triangles are shown here:

Tessellation with octagons and squares

Tessellation with triangles and hexagons

Regular pentagons cannot tessellate the Euclidean plane (see Euclidean space). There are currently 15 known ways to tessellate the plane with congruent pentagons, the latest being found in 2015. Tesselations of the sphere and hyperbolic plane are also studied, where regular pentagonal tessellations do exist.

See also Penrose tilings.