Octagonal tilings with three prototiles

TL;DR

The paper introduces a new family of octagonal tilings built from three prototiles, parameterized by non-negative integers $m$ and $n$, with inflation factor $\delta_{(m,n)}=m+n(1+\sqrt{2})$. It develops primitive substitution rules $\sigma_{(m,n)}$ by edge dissections, analyzes their statistical structure via a substitution matrix $M_{(m,n)}$, and derives area fractions and vertex configurations, including projection-window insights in perpendicular space. Variants of the substitution, odd-$m+n$ cases, and connections to existing tilings (Ammann-Beenker, AB; Watanabe-Ito-Soma tiling) are explored, along with a replacement-rule path to rhombi-and-squares tilings and a comparison to Fayen et al.'s non-random octagonal tilings. The work suggests applications to modeling quasicrystal-like phases and outlines future directions for odd-$m+n$ cases and broader window analyses. The combination of a constructive tiling framework, exact area-frequency analysis, and connections to established octagonal tilings provides a versatile toolkit for exploring ordered and near-ordered octagonal structures in mathematics and materials science.

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 $δ_{(m,n)}=m+n (1+\sqrt{2})$. 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.

Figures (19)

• Figure 1: Substitution rules for the Ammann-Beenker tiling.
• Figure 2: The 2-order supertile of the first prototile of $\omega$.
• Figure 3: The patch $\mathcal{V} \subset \omega^2 (T_1)$ yields the patch $\mathcal{V}'$ that is a seed for the AB tiling.
• Figure 4: (a) The prototiles; (b) edge labeling.
• Figure 5: Edge substitution rule for the case $m+n$ is even; the substitution for $c$ is the same as $a$.