Introduction

In the figure below, a regular hexagon with side length 12 is tiled with “lozenges,” sort of like triangular dominoes, each consisting of a pair of unit-length equilateral triangles joined at a common side.

Each lozenge is in one of three possible orientations. Although the orientations appear random, if we count carefully, we find that there are exactly the same number of lozenges in each of the three orientations. In fact, no matter how we tile the hexagon, it is always the case that the number of lozenges (144 in this case) of each “type” is fixed. Can you prove this?

Proof without words

Although the (proof of the) Pythagorean theorem seems to be the most common example of a proof without words— usually a picture or diagram that doesn’t require any words to explain– this problem, with its “proof” below, is probably my favorite.

One reason I like this problem, particularly as motivation for student discussion, is that it is somewhat controversial. Is the above “proof without words” really a satisfactory proof?

Theorem without words

Moving away from random tilings for the moment, the original motivation for this post wasn’t actually a proof without words, but a theorem without words. I recently learned an interesting result I had not seen before, involving no more than high school-level physics, that I thought could be presented as an animation without any accompanying explanation:

Unfortunately, I’m not sure this quite succeeds as a “theorem without words.” As with the tiling proof, there is more going on here than the above animation arguably conveys. In particular, one of the most interesting things about this problem– that the animation doesn’t really show– is that the shape (that is, the eccentricity) of the ellipse of apogees is invariant: it does not depend on the speed of the projectile, or gravity, or any relationship between the two.

More on random lozenge tilings

Finally, some source code: although there are plenty of papers and web sites with images of random tilings like those above, I wanted to make my own images, and working code is harder to find. See here for my Python implementation for generating a random lozenge tiling of a hexagon of a given size. For example, the following figure shows a random tiling of a hexagon with side length 64:

My implementation is quick and dirty in the sense that it simply iterates the Markov chain of single-step up-or-down moves in the corresponding family of non-intersecting lattice paths, essentially shuffling long enough for the resulting random tiling to be approximately uniform. (Note that the tilings in the above images are more random than they might appear, despite the “frozen” regions near the corners.) It would be interesting to extend this implementation to use coupling from the past, to generate an exactly uniform random tiling.

References:

David, G. and Tomei, C., The Problem of the Calissons, American Mathematical Monthly, 96(5) May 1989, p. 429-431 [JSTOR] Wilson, D., Mixing Times of Lozenge Tiling and Card Shuffling Markov Chains, Annals of Applied Probability, 14(1) 2004, p. 274–325 [PDF]