Other Tessellation Worlds

Beyond our 35 symmetries are other interesting tessellation worlds to explore. (All links below are to other sites. Images shown with permission; click for full version.)

Many artworks use more than one tile shape — like these birds and fish by M.C. Escher.

Often such an artwork uses one of our 35 symmetries, with each tile divided into two or more distinct tiles. Alain Nicolas gives a nice presentation, with examples such as this leaf and rabbit.

And sometimes several figures in a single-tile tessellation share a differently-shaped tile, as in soccer4 by Makoto Nakamura (at left).

There are also 58 other isohedral symmetries, distinct from the 35 shown on this site. Some have unshapeable mirrored edges (like cats21 by Makoto Nakamura), some have inner rotational symmetry (like Half Card-trick by Alain Nicolas, discussed just above here), and some have multiple inner mirrors (one such is included with TesselManiac).

Some artists have explored tiling three-dimensional surfaces. Dominique Ribault’s site shows his tessellations on a sphere, cone, donut, and knot, as well as two types of spiral. (Scroll down to “SURFACES 3D”). And Makiya Torigoe's site includes tessellations aligned on four platonic solids, such as his Ray on a cube. Note in each artwork how the pattern is different at the corners. (Section “Ju+Ten kaizu”).

And there are fascinating worlds of non-periodic tessellations, where tiles fill space without translational symmetry; Alain Nicolas has many fine examples such as Boules Player shown here. See also Penrose tiles, as well as John A. L. Osborn's Bats & Lizards and Beetles.