Tessellation Patterns

Every of the flat repeating patterns that we find beautiful can be reduced to 17 symetrical patterns. Our quilts, tilings, mosaics can only have so many symmetrical combinations, and mathematicians can prove it. We see patterns like these in tiling, quilts, brickwork and Islamic art. The artist M. C. Escher was inspired by Moorish artwork in Spain, though he worked with figures instead of shapes to create his art. On this page exolore the 17 tessellation patterns, and links to other tessellation tools, such as our tesselation art page where you can experiments with patterns.

The 17 patterns

Why only 17? We can prove this because patterns that repeat by translation in two directions, and tile a surface with no overlaps or gaps, can only have a few kinds of rotational symmetry. The ones that work are 2-fold (180°), 3-fold (120°), 4-fold (90°) and 6-fold (60°), plus patterns with no rotational symmetry at all. Combine those with the presence or absence of mirrors and glide reflections, where the slide runs parallel to the mirror line, and exactly 17 distinct tessellation patterns are possible.

In 1891, the Russian crystallographer Evgraf Fedorov proved that every pattern repeating in two directions on a flat surface belongs to one of exactly 17 symmetry groups, each built from some combination of translation, rotation, reflection and glide reflection. The same year he worked out the 230 groups that describe crystals in three dimensions. Snowflakes follow a related rule: water molecules freeze into a lattice of hexagonal rings, so ice crystals grow with six-fold symmetry, even though most real snowflakes come out irregular due to microscopic fluxuations of temperature and humidity.

The Hungarian mathematician George Pólya reached the 17 groups on his own in 1924 and drew a sample tiling for each one. In 1937, Escher’s half-brother Berend, a geology professor at Leiden, sent him a list of Polya’s papers, and Escher copied the work by hand, illustrations and all. From it he built his own system for classifying the tilings he designed, working as an artist rather than a mathematician, with birds, fish and lizards in place of geometric shapes. Mathematicians later found examples of all 17 groups in his work.