Tessellating Alphabet 1986
SYMMETRY. Each letter fits together with copies of itself to tile the plane. Click a letter to see how it tiles. Some letters tile in more than one way.
INSPIRATION. Created as part my computer game Letterforms & Illusion.
STORY. I have long admired M. C. Escher’s ingenious tiling patterns with animal and human shapes. One day it occured to me that I might be able to do something similar with letters. So I set to work designing an entire alphabet of letters, each of which can tile the plane. I drew all letters on a square grid, so I wouldn’t have to worry about fitting curved edges together, and to give the font some overall visual unity.
The letter that gave me the most trouble at first was O. How can a letter with a hole tile the plane without leaving any gaps? My solution was to allow letters to overlap; the corners of the O cross through each other in the O tiling. The letter V has a similar corner.
With my artist hat on, I tried to make the letters look like they belonged to one font, with consistent x-height, descender length and ascender length. Some of the letters are exactly the same: BDPQ, MW, NU and SZ. With my puzzle hat on, I tried to create a wide range of types of tessellations, with different symmetries and different degrees of difficulty. Some tilings, like K, have all letters in the same orientation. One of the tilings of R has the letter in eight different orientations. The most difficult tessellation to figure out, given the letter shape, is J. Can you figure out how it tiles before looking at the answer?
I originally drew this alphabet in a painting program. Painting programs make it easy and fun to multiply a single shape into a tiling: copy the shape and flip it around, then copy the copies. The design will keep doubling, and before you know it the pattern will fill the screen. (If you want to explore tilings patterns on computer, I recommend the program Kaleidomania, published by Key Curriculum Press. It’s sold as educational software, but don’t let that stop you from enjoying it as a creative tool.)
To see how the letters tile, click on them. Most letters have only one possible tessellation, but a few, like E, have more than one. The letters R and V have particularly interesting alternate tessellations. I’m not sure I found all the tessellations for all the letters; I suspect V may have another answer.
By the way, all these patterns are what mathematicians call “regular” tessellations, meaning that every tile is the same shape, and every tile is related to its neighbors in exactly the same way. It’s also possible to create irregular tessellations for some of these letters, but I don’t find them as aesthetically pleasing.
In 1999 I revisited the problem of designing tessellating letters in a puzzle I created for Discover magazine. I wanted to create puzzles based on the work of mathematician Gödel, artist Escher, and musician Bach, in honor of the book Gödel, Escher, Bach. For the Escher puzzle I created new versions of the letters MCESCHER, and challenged readers to figure out the tessellations.
Notice that my tessellating alphabet has only lowercase letters. Can you design tessellating uppercase letters to go with the lower case?
Speaking of tessellating uppercase letters, here is Erik Demaine’s solution to that problem. Check out his wonderful origami and other mathematical alphabets.
And here are some other tessellating letters by other artists.