Octagonal tilings with three prototiles

TL;DR

This paper introduces a family of octagonal tilings with three prototiles, characterized by two parameters, and explores their inflation properties, substitution rules, and statistical features, motivated by observed octagonal symmetry in structures.

Contribution

The paper defines an infinite series of octagonal tilings based on two parameters, providing their inflation factors, substitution rules, and statistical properties, expanding understanding of octagonal symmetry.

Findings

Infinite series of tilings parameterized by m and n

Explicit inflation factors for each tiling case

Derived substitution rules and statistical properties

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.