Discovered by


A substitution for the aperiodic hat monotile found in 2022 and published in SMCG2023, see also BGS2023. The “metatiles” in the first paper have some free parameter. Most of these parameters yield a substitution that is not selfsimilar. Here we show the selfsimilar version of the substitution rule. This tiling was used to derive the cut-and-project description of the hat tiling in BGS2023.

Substitution Rule

Rule CAP


Patch CAP download vectorformat CAP


Smith, D. and Myers, J. S. and Caplan, C. S. and Goodman-Strauss, C.
An Aperiodic Monotile
arXiv 2023,

Baake, M. and Gähler, F. and Sadun, L.
Dynamics and topology of the Hat family of tilings