Hat Monotile Family
The hat monotile is a new family of polygons that tiles the plane aperiodically . This Demonstration defines a function that draws the polygon with eight segments of length and six segments of length . The main shape explored has been , since it is four-thirds of a 3-hexagon triangle.[more]
A polyiamond is a shape formed by a finite number of equilateral triangles.
Three members of the family admit a periodic tiling:
: the tetriamond (four equilateral triangles)
: the octiamond (eight equilateral triangles)
: a new shape ([1, Figure 6.2])
Six kites are formed by connecting the center of a regular hexagon to the midpoints of its edges. A shape made from multiple connected kites is known as a polykite. The shapes and are both polykites.
Both the aperiodic tiling and the proof of aperiodicity are complicated [2, 3].
Remarkably, the family may provide a simpler proof sketched here as:
1. Suppose the large tiling is periodic. Then there is a minimal periodic patch with translation vector .
2. The polyiamond shapes and would each have translation vectors.
3. Since the area of is 4 and the area of is 8, a term is needed in one of the translation vectors.
4. A translation vector of does not work on the triangular grid.
The paper  is still under review, so this sketch is conjecture.[less]
 D. Smith, J. S. Myers, C. S. Kaplan and C. Goodman-Strauss, "An Aperiodic Monotile." arxiv.org/abs/2303.10798.
 E. Pegg. "Einstein Problem Solved (Aperiodic Monotile Discovery)" from Wolfram Community—A Wolfram Web Resource. community.wolfram.com/groups/-/m/t/2861234.
 B. Klee. "Hat Tilings via HTPF Equivalence." from Wolfram Community—A Wolfram Web Resource. community.wolfram.com/groups/-/m/t/2858759.