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The hat monotile is a new family of polygons that tiles the plane aperiodically [1]. 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.

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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 [1] is still under review, so this sketch is conjecture.

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Contributed by: Ed Pegg Jr (May 8)
Open content licensed under CC BY-NC-SA

## Details

References

[1] D. Smith, J. S. Myers, C. S. Kaplan and C. Goodman-Strauss, "An Aperiodic Monotile." arxiv.org/abs/2303.10798.

[2] E. Pegg. "Einstein Problem Solved (Aperiodic Monotile Discovery)" from Wolfram Community—A Wolfram Web Resource. community.wolfram.com/groups/-/m/t/2861234.

[3] B. Klee. "Hat Tilings via HTPF Equivalence." from Wolfram Community—A Wolfram Web Resource. community.wolfram.com/groups/-/m/t/2858759.

## Permanent Citation

Ed Pegg Jr ""
http://demonstrations.wolfram.com/HatMonotileFamily/
Wolfram Demonstrations Project
Published: May 2023 8

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