Spectre: a chiral aperiodic 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 May 2023 (see: https://arxiv.org/abs/2305.17743). It is an example of an "einstein"—a single shape that tiles the plane aperiodically. Unlike their earlier tiles—the "hat" and the "turtle"—this tile, which the authors call a "spectre" does not require any tiles to be flipped upside down in order to tile the plane periodically. Moreover, if we are not allowed to flip the tile, then it can only tile aperiodically. (If we allow reflections, it is possible to tile the plane periodically with the spectre tile.) The second tile was obtained from the first by adding or removing semicircles from each unit-length edge. This tile can only tile aperiodically—even if one flips the tile over.