Matching Rules as Cocycle Conditions: Discrete Potentials on Penrose and Canonical Projection Tilings

Abstract

Aperiodic tilings support two classically studied but hitherto separately presented structures: matching rules, which enforce global order via local constraints, and height functions, which encode global geometry through integer-valued potentials. Their precise relationship has remained implicit in the literature. This paper bridges them via a cochain-first framework, establishing a four-way equivalence -- between matching rules, Ammann bar continuity, cycle closure of the associated $1$-cochains, and height-function existence -- proved for candidate tilings without presupposing any of the four conditions. The proof proceeds via a half-edge/gluing construction: for each Ammann bar family, we assign to every directed edge a signed bar-crossing count, yielding an antisymmetric $1$-cochain. A tile-side crossing function and a global cochain are built in two stages; the global cochain exists precisely when adjacent tiles agree on shared edges. Gluing implies cycle closure; the discrete Poincaré lemma then produces a scalar potential coinciding with the classical Ammann height function. The framework extends uniformly to canonical projection tilings (CPTs) from $\mathbb{Z}^N$: lattice-coordinate cochains reconstruct vertex positions via $v = \sum_{k=1}^N x_k(v)\,\mathbf{e}_k^*$, and (for CPTs with generic window) form a $\mathbb{Z}$-basis for $\check{H}^1 \cong \mathbb{Z}^N$ (Forrest--Hunton--Kellendonk), yielding a conservation-forced structure with recognition gap $\mathcal{R}(\mathcal{T}) \cong \mathbb{Z}^N$. The framework is verified for the Fibonacci chain, Penrose P2, Ammann--Beenker, and the icosahedral Ammann tiling; whether conservation forcing characterises exactly the Pisot substitution CPTs is left as an open conjecture.

Figures (4)

• Figure 1: The five Ammann bar families for Penrose P2 can be indexed by unit normals $\mathbf{n}_k$ related by $72^\circ$ rotation. A "bar family direction" means the direction of the bar lines; its normal $\mathbf{n}_k$ is used to assign signed crossing numbers to directed edges.
• Figure 2: Sign convention for Definition \ref{['def:half-edge-crossing']}: a crossing with $\mathbf{n}\!\cdot\!(v-u)>0$ contributes $+1$; the reversed directed edge contributes $-1$.
• Figure 3: Schematic of the gluing condition on a shared edge. Tiles $A$ and $B$ induce tile-side values $\Delta^{(A)}_{\mathbf{n}}$ and $\Delta^{(B)}_{\mathbf{n}}$ on the same directed edge $u\to v$. The Penrose matching rule on $\{u,v\}$ (for the unique relevant family) is equivalent to equality of these induced values (Lemma \ref{['lem:gluing']}).
• Figure 4: Schematic version of Examples \ref{['ex:dart-patch']}--\ref{['ex:violation']}. The blue segment represents one local Ammann bar segment from the chosen family. In the matching case the local segments glue to a single straight bar; in the mismatched case they fail to glue, so no global cochain (and hence no global potential) exists for that family.

Theorems & Definitions (70)

• Definition 2.1: Candidate decorated Penrose tiling
• Definition 2.2: Local and global Ammann bars
• Definition 2.3: Tiling adjacency graph
• Remark 2.4: Basic graph properties
• Lemma 2.5: Local bar segments are decoration-determined
• proof
• Remark 2.6: Transversality and uniqueness properties
• Definition 2.7: Canonical projection tiling
• Definition 2.8: Lattice-coordinate map
• Remark 2.9: Gradient structure