Summary

This is a 3d-printable tile based on the ones discovered by David Smith, Joseph Samuel Meyers, Craig S. Kaplan, and Chaim Goodman-Strauss in March 2023 (see: https://arxiv.org/pdf/2303.10798.pdf). It is the first example of an "einstein"—a single shape that tiles the plane aperiodically. Unlike the tile highlighted in the paper (the "hat"), this is another one in the family, which people have called the "turtle."

The aperiodic tiling requires some of the tiles to be reflected (flipped over). This turtle tile can be flipped over, of course. But if you want to print most of the tiles in one color and the reflected tiles in another, then it is good to know that they are in a ratio of (phi)^4 (which is approximately 6.85) to 1. So, print roughly seven times more of one tile than the other.