Octagonal tilings with three prototiles

April Lynne D. Say-awen; Sam Coates

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arXiv · 2502.04133

Octagonal tilings with three prototiles

April Lynne D. Say-awen·Sam Coates

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Abstract

Motivated by theoretically and experimentally observed structural phases with octagonal symmetry, we introduce a family of octagonal tilings which are composed of three prototiles. We define our tilings with respect to two non-negative integers, $m$ and $n$, so that the inflation factor of a given tiling is $\delta_{(m,n)}=m+n (1+\sqrt{2})$. As such, we show that our family consists of an infinite series of tilings which can be delineated into separate `cases' which are determined by the relationship between $m$ and $n$. Similarly, we present the primitive substitution rules or decomposition of our prototiles, along with the statistical properties of each case, demonstrating their dependence on these integers.

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April Lynne D. Say-awen, Sam Coates. 2025-02-06. Octagonal tilings with three prototiles. https://arxiv.org/abs/2502.04133

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