An effective construction for cut-and-project rhombus tilings with global n-fold rotational symmetry

TL;DR

This work provides an explicit, constructive approach to build rhombus cut-and-project tilings with global $n$-fold rotational symmetry by dualizing regular $n$-fold multigrids. The core technical advance is a rigorous regularity result: regular multigrids yield dual tilings that are edge-to-edge rhombus tilings and inherit quasiperiodicity; the regularity is established by translating potential singular intersections into trig Diophantine equations and applying Conway–Jones-type classifications of vanishing sums of roots of unity. The construction uses specific offsets, notably $G_n(\tfrac{1}{2})$ and $G_n(\tfrac{1}{n})$, producing dual tilings $P_n(\tfrac{1}{2})$ and $P_n(\tfrac{1}{n})$ with global symmetry, applicable for all $n\ge 3$, and providing nonperiodic, uniformly recurrent tilings. The paper also offers a practical sufficient condition for even $n$ that guarantees regularity, broadening the toolkit for generating symmetry-rich aperiodic tilings and informing substitution-based constructions in related literature.

Abstract

We give an explicit and effective construction for rhombus cut-and-project tilings with global n-fold rotational symmetry for any n. This construction is based on the dualization of regular n-fold multigrids. The main point is to prove the regularity of these multigrids, for this we use a result on trigonometric diophantine equations.

Figures (6)

• Figure 1: Examples of multigrids
• Figure 2: Example of a regular grid and its dual tiling, some elements of the multigrid and their dual in the tiling have been colored.
• Figure 3: Possible intersection points in $G_{5}(\gamma)$ and their dual tiles
• Figure 4: Central patch of the multigrid dual tiling with exactly $n$-fold rotational symmetry for $n\in\{7,8,9,10,11,12\}$
• Figure 5: Intersection of three lines

Theorems & Definitions (9)

• Theorem 1
• Theorem 2
• Proposition 1: Regularity of multigrids and trigonometric equations
• proof
• Theorem 3: ConwayJones, '76
• Corollary 1
• proof
• Lemma 1: Sine and Cosine
• proof