Tiling Font

by Erik Demaine, Martin Demaine, Scott Kim, and Yushi Uno, 2021

Hover/click on a letter to reveal a tiling.
 Hue: Saturation: Lightness:

Tiling the plane. Every letter and digit in this type face tiles the plane, meaning that infinitely many copies of that one shape can fill two dimensions without leaving any gaps between the tiles. There are three fonts: an uppercase font, a lowercase font, and a mixture of the two (according to how you type the characters). A fun puzzle is to figure out how each character tiles the plane. The letter J, in each font, has a particularly interesting tiling. • If you hover your mouse or tap your finger on any rendered character, then the corresponding tiling fills the screen. Click or tap again to redraw with a new random set of colors. • You can control the random colors assigned to tiles by setting the range of allowed hue, saturation, and lightness using the sliders. For example, if you force lightness to 100, you get a black-and-white drawing.

History. In 1986, Scott Kim designed a lower-case tessellating alphabet, which is identical to the lowercase font presented here (modernized to no longer require Flash). This alphabet was part of Scott's video game Letterforms & Illusion. Answering a challenge posed by Scott in 2000, the four of us collaboratively designed the uppercase font, mostly via email.

Mathematics. A famous open problem in the mathematics of tilings is whether there is an efficient algorithm to decide whether a single polygon tiles the plane (or whether the problem is NP-hard). The problem is unsolved even for polyominoes — polygons made by gluing together equal-size squres along edges, as in this font. The tilings for the letters in this typeface are all periodic and furthermore isohedral, meaning that every transformation mapping one tile to another is a symmetry on the whole tiling. In this special case, there are efficient algorithms to determine whether a polyomino tiles isohedrally. The unsolved case is thus nonisohedral tilings; for example, it is unsolved whether a single tile ever tiles the plane aperiodically (but two tiles do, as in the famous Penrose tiling). If we allow tiny gaps between tiles, no algorithm can decide whether a polyomino “nearly” tiles the plane.

Check out other mathematical and puzzle fonts. • Feedback or not working? Email Erik.