Aperiodic "hat" (or "t-shirt" or "einstein") monotile

thingiverse

This 3d-printable tile is based on the one 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. There are three printable objects here based on two different designs. The aperiodic tiling requires some of the tiles to be reflected (flipped over). The blank tile with no decorations can serve that purpose—it can be placed with either side up. The second and third objects are based on a design by Daniel Piker. The tiles are placed so the curves on the top meet up and make a continuous path when the tiles are placed edge-to-edge. In this case, you would have to print tiles of both types. 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.