Tilings

Another topic I have spent a lot of time exploring are parallelogram tilings. This exploration began when I was working on my thesis, and stumbled on a fascinating construction to relate sandpile elements and bases of oriented arithmetic matroids. Since then, I have gotten away from the original motivation and plan to continue studying the tiling until the proof no longer seems like a miracle. 

The original construction can be phrased to depend on two integer matrices, one in some sense dual to the other. Nevertheless, in joint work with Joseph Doolittle, we showed that a similar construction is possible with any pair of integer matrices of the correct dimensions. The catch is that this construction gives a signed tiling: some parallelepipeds are positive, and some are negative. While there may be some overlap between tiles, we proved that every point in n-space is covered by exactly one more positive tile than negative tile. 

Below are the positive tiles that make up a running example from our paper, as well as an animation which shows how they overlap in the tiling.