Hat

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A pretty simple aperiodic monotile found in 2022 and published in SMCG2023. The tile is a tridecagon (13-gon) built from 16 basic triangles with edge lengths $1,2, \sqrt{3}$. The shape alone ensures that every tiling of the plane you can build from this tile will be aperiodic. In this sense it has the simplest matching rules imaginable.

This one is not really a substitution tiling. However, its properties (e.g., being an aperiodic monotile) are deduced from related substitution tilings, see for instance the CAP tiling. The exact relations are rather tricky for this particular one.

In the patch below all tiles are of the same shape. However, the colours indicate some structure: for instance, dark blue tiles are reflected copies of the monotile, the others are not. Each reflected tile is surrounded in the same way by three others (light blue).

Patch

download vectorformat Hat

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References

[SMCG2023]
Smith, D. and Myers, J. S. and Kaplan, C. S. and Goodman-Strauss, C.
An Aperiodic Monotile
arXiv 2023, https://arxiv.org/abs/2303.10798

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