Hat Monotile Coronas

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Consider a grid of squares. The central square is completely surrounded by eight squares. By shifting squares along the top and bottom, the central square may be completely surrounded by six or seven squares. Touching at the corners is allowed. All of these produce examples of a square surrounded with a 1-corona of squares. Surrounding the 1-corona to make a 2-corona requires anywhere between 19 and 25 squares.


The Hat monotile is a new family of polygons that tiles the plane aperiodically [1]. This Demonstration uses only the unmodified hat monotile. If all possible 3-coronas are evaluated, there are exactly 188 ways to make a 2-corona with hats. This Demonstration shows all 188 2-coronas of the hat.


Contributed by: Ed Pegg Jr (June 13)
Open content licensed under CC BY-NC-SA



[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.


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