Canonical projection tilings defined by patterns

TL;DR

The paper establishes a necessary and sufficient equivalence between characterizing slopes by finite forbidden patterns (local rules) and by coincidences in canonical projection tilings, reducing the problem to solving polynomial equations in Grassmann coordinates. This yields an effective, algebraic criterion: a slope $E\in G'(n,d)$ is pattern-characterizable if and only if it is coincidence-characterizable, implying such slopes are algebraic. Through explicit examples (Ammann-Beenker and Penrose tilings) it demonstrates both the power and limitations of the approach, showing that some slopes are not pattern-characterizable despite having well-defined coincidences, while Penrose tilings can be captured by forbidden patterns even when non-generic. Overall, the work links the algebraic structure of $d$-planes in $\mathbb{R}^n$ with the combinatorics of their digitizations, enabling finite local-rule characterizations in a broad class of tilings and clarifying the role of algebraicity in these characterizations.

Abstract

We give a necessary and sufficient condition on a $d$-dimensional affine subspace of $\mathbb{R}^n$ to be characterized by a finite set of patterns which are forbidden to appear in its digitization. This can also be stated in terms of local rules for canonical projection tilings, or subshift of finite type. This provides a link between algebraic properties of affine subspaces and combinatorics of their digitizations. The condition relies on the notion of {\em coincidence} and can be effectively checked. As a corollary, we get that only algebraic subspaces can be characterized by patterns.

Figures (12)

• Figure 1: Top: projection on a slope $E$ of the basis vectors of $\mathbb{R}^4$, a pattern formed by a unique edge and a more complicated pattern (from left to right). Bottom: the same objects projected in the (octagonal) window, where the darkest polygons depict the regions associated with each pattern.
• Figure 2: Concentric $r$-patterns with $r$ ranging from $0$ to $4$ (left) and partition of the window by $0$-patterns for a $4\to 2$ planar tiling (namely an Ammann-Beenker tiling, see Sec. \ref{['sec:ammann_beenker']}), with each $0$-pattern being depicted inside its region. A point on the boundary between two regions belong to the region whose associated pattern has the more edges for inclusion (there is always such an inclusion).
• Figure 3: How to permute the edges of a path with endpoints in the window so that it lies completly in the window.
• Figure 4: Lemma \ref{['lem:point_in_window']}: $r$-patterns can force a point $x$ to project inside the window.
• Figure 5: A $4\to 2$ canonical planar tiling with a typical generic algebraic slope.

Theorems & Definitions (13)

• Theorem 1
• Proposition 1
• Definition 1
• Proposition 2
• Proposition 3
• Proposition 4
• Corollary 1
• Proposition 5
• Lemma 1
• Lemma 2