Tessellation Patterns

Wallpaper and Tessellation Patterns

Every repeating pattern that we find beautiful can be reduced to 17 patterns, no more, no less. Our quilts, tilings, mosaics can only have so many patterns, and mathematicians can prove it. We see patterns like these in tiling, quilts, brick work, and Islamic art. The famous artist, M.C Escher himself, was inspired by Arabic artwork in Spain though he worked with figures instead of shapes to create his art.

How do we describe the way objects can move? Mathematicians who study geometry classify movement as translation, rotation, reflection and glide reflection. Translation is when things slide from one point to another, rotation is spinning, reflection is flipping like a mirror and glide reflection is a single operation where a shape is reflected and translated at the same time.

Why only 17? We can prove this because only a few rotational symmetries are compatible with tiling a surface with no overlap or space. 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, and exactly 17 distinct tessellation patterns combinations are possible.

In 1891, mathematician Evgraf Fedorov proved there are exactly 17 ways to tile a flat surface using translation, rotation, reflection and glide. His work with crystal structures also explains why snowflakes always show six-fold symmetry, they are constrained by their atomic structure to only fold certain ways. Escher encountered that mathematical theorem through his brother, a geologist who shared crystallography papers with him. Escher worked independently to establish the same precept held true artistically.

The interactive tool below is made with the letter F, which has no symmetry of its own, so any symmetry you see in the pattern must come from the tiling rule, not the shape itself.