Patch frequencies in Penrose rhombic tilings

TL;DR

This work addresses the problem of exactly determining patch frequencies in Penrose rhombic tilings by leveraging a dualisation framework based on the root lattice $A_4$. By projecting dual Voronoi–Delone structures onto parallel and perpendicular subspaces and employing a cutting-plane construction, the authors obtain an algebraic, inflation-free method to compute frequencies for arbitrary finite patches, and they extend the approach to Ammann–Beenker tilings. The main contributions include a concrete algorithm that reduces frequency computation to area ratios of projected windows, the identification of a Penrose frequency module $\mathcal{M}_{\mathscr{T}_{Pen}}=\frac{1}{10}\mathbb{Z}[\tau]$, and explicit frequencies for several large patches, along with a parallel treatment of the AB tiling where $\mathcal{M}_{\mathscr{T}_{AB}}=\frac{1}{2}\mathbb{Z}[\lambda]$ with $\lambda=1+\sqrt{2}$. The results enhance understanding of IDS gaps and provide a generalizable framework for patch frequencies in a broad class of dualisation tilings.

Abstract

This short exposition presents an algorithm for an exact calculation of patch frequencies for the rhombic Penrose tiling. We recall a construction of Penrose tilings via dualisation, and by extending the known method for obtaining vertex configurations, we obtain the desired algorithm. It is then used to determine the frequencies of several particular large patches which appear in the literature. The analogous approach is also explained for the Ammann-Beenker tiling.

Figures (26)

• Figure 1: The Dynkin diagram $A_4$. Every node represents a basis vector, and their geometry is encoded via the lines. If two vertices are connected, their scalar product is -1. Otherwise, they are orthogonal.
• Figure 2: Projections of the standard basis $\boldsymbol{e}_1, \dots, \boldsymbol{e}_5$ into the two subspaces $\mathcal{S}^{\parallel}$ and $\mathcal{S}^{\perp}$, respectively.
• Figure 3: Images of the 2-boundary $P(-+\bigcirc \bigcirc+)$ and its dual $P^{*}(-+\bigcirc \bigcirc+)$ under the projections $\pi_{{}_\parallel}$ and $\pi_{{}_\perp}$, respectively. The solid blue rhombuses correspond to projections of the 2-boundary, whereas the dashed line indicates the projection of its dual. The gray points are the $20^{\mathrm{th}}$ roots of unity scaled by $\sqrt{\tfrac{2}{5}}$.
• Figure 4: Projections of the different (modulo translation) 2-boundaries $P$ in the $\mathcal{S}^{\parallel}$ which result in a thick rhombus. The solid rhombi correspond to the label, whereas the dashed rhombi are their space inversion. The red point attached to a given rhombus indicates the shallow hole. The gray points are the $20^{\mathrm{th}}$ roots of unity scaled by the factor $\sqrt{\tfrac{2}{5}}$.
• Figure 5: Projections $\pi_{{}_\perp}\bigl( V^{*}(\boldsymbol{v}^{*}_{i})\bigr)\subset \mathcal{S}^{\perp}$ corresponding to the windows. The blue pentagons carry the $\pi_{{}_\perp}$-projections of shallow holes, whereas the black ones comprise the projections of deep holes. Note that for every window, there exists its own lattice. Thus even though there is a non-trivial intersection of windows, the resulting points must differ, as one expects. The gray points are the $20^{\mathrm{th}}$ roots of unity scaled by the factor $\sqrt{\tfrac{2}{5}}$.

Theorems & Definitions (2)

• Remark 2.1
• Proposition 4.1