Ammann Bars for Octagonal Tilings

Abstract

Ammann bars are formed by segments (decorations) on the tiles of a tiling such that forming straight lines with them while tiling forces non-periodicity. Only a few cases are known, starting with Robert Ammann's observations on Penrose tiles, but there is no general explanation or construction. In this article we propose a general method for cut and project tilings based on the notion of subperiods and we illustrate it with an aperiodic set of 36 decorated prototiles related to what we called Cyrenaic tilings.

Figures (12)

• Figure 1: Left: Penrose tiles with Ammann segments (in orange). On each rhombus the dashed line is an axis of symmetry and the sides have length $\varphi=(1+\sqrt{5})/2$. Right: Ammann bars on a valid pattern of Penrose tiles, where each segment is correctly prolonged on adjacent tiles. The red vectors are "integer versions" of one subperiod.
• Figure 2: Set $\mathcal{C}$ of 36 decorated prototiles obtained from Cyrenaic tilings. Any tiling by these tiles where segments extend to lines is non-periodic.
• Figure 3: Examples. Left: Rauzy tiling from which you can visualize the lift in $\mathbb{R}^3$. Center: Ammann-Beenker tiling. Right: Penrose tiling.
• Figure 4: Golden octagonal tiling with the usual valid projection (left) and a non-valid projection on the same slope (right). Colors of the tiles are the same with respect to the $\pi(e_i)$'s, with an opacity of 50% in both images.
• Figure 5: Shadows of Ammann-Beenker and Penrose tilings.

Theorems & Definitions (7)

• Theorem 1
• Definition 1
• Proposition 1
• proof
• Lemma 1
• proof
• Corollary 1