Weak local rules for planar octagonal tilings

Abstract

We provide an effective characterization of the planar octagonal tilings which admit weak local rules. As a corollary, we show that they are all based on quadratic irrationalities, as conjectured by Thang Le in the 90s.

Figures (6)

• Figure 1: A celebrated octagonal tiling: the Ammann-Beenker tiling.
• Figure 2: The window of an Ammann-Beenker tiling, divided into some regions (each of which corresponds to a pattern), with a circled coincidence (left). The slope is slightly changed so that the circled coincidence breaks and a new region appears (right). This corresponds to a new pattern which does not appear in an Ammann-Beenker tiling (compare with Fig. \ref{['fig:ammann_beenker_tiling']}).
• Figure 3: When an Ammann-Beenker tiling is shifted, it creates lines of flips whose directions are determined by its four subperiods (left). A smaller shift yields a similar picture, but the lines become sparser (right).
• Figure 4: The irrational slope $(1,\sqrt{2},\sqrt{3},2\sqrt{2},3\sqrt{3},\sqrt{6})$ has one subperiod of type $3$ and one of type $4$. A generic shift thus creates two sets of sparse lines and two sets of sparse flips (left). However, a shift along $\vec{e}_4$ "neutralizes" the lines of flips directed by the subperiod of type $4$ (right).
• Figure 5: The points in $E_1$ are depicted by squares, while the points in $E_2$ and $E_3$ are respectively depicted by triangles and hexagons. The two lines are at distance at least $3r$ from each other, as well as any two triangles and any two hexagons. The step edge goes between the two lines and stay at distance at least $r/2$ from any point in $E(\vec{s})$, that is, outside the shaded region.

Theorems & Definitions (21)

• Theorem 1
• Corollary 1
• Corollary 2
• Definition 1: subperiod
• Definition 2: weak local rules
• Proposition 1
• Definition 3: coincidence
• Proposition 2
• Proposition 3
• proof