I wanted an excuse to use Harvey Mudd’s large format printer, so I made a movie-sized (27″×40″) poster for my office based on the second term of OEIS sequence A368138(n): \(A368138(2) = 154\). The idea here is that you have a a collection of tiles like 
, which you can rotate and mirror; you then choose a \(n \times n\) grid of these tiles, and repeat that pattern infinitely over the plane. I recently learned that this particular example (for \(n = 2\)) was first enumerated in a 1996 paper by Dan Davis, On a Tiling Scheme from M. C. Escher, in The Electronic Journal of Combinatorics.
In my paper, “Counting Tilings of the \(n \times m\) Grid, Cylinder, and Torus,” we give a method for counting these kinds of problems in full generality. While revising the paper, I learned from Doris Schattschneider’s 1990 book Visions of Symmetry (pp 44-48) that the artist M.C. Escher was perhaps the first person to attempt counting this. (In particular, Escher successfully enumerated A368145(2) = 23.)
Notice how the whitespace and the black space are \(180^\circ\) rotations of each other. Ami Radunskaya pointed out to me that this looks like “op art.”
If you had to choose one of these patterns to tile your bathroom floor, which would you pick?
Leave a Reply