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

TL;DR

This paper establishes a unified cochain-based framework linking matching rules, height functions, and geometric structures in aperiodic tilings, extending to canonical projection tilings and verified on several classical examples.

Contribution

It introduces a cochain-first approach that unifies matching rules and height functions, providing new insights into aperiodic tilings and extending to CPTs with a verification on key examples.

Findings

Established equivalence between matching rules and height functions.

Extended the framework to canonical projection tilings from bZ^N.

Verified the approach on Fibonacci, Penrose, Ammann--Beenker, and icosahedral 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 Poincar\'{e} 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.