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Penrose tilings, infinite friezes, and the $A_\infty$-singularity

Özgür Esentepe, Eleonore Faber

TL;DR

The work builds a bridge between Penrose tilings, infinite frieze patterns, and the $A_ inf$-singularity by interpreting Penrose tilings as special cluster-tilting phenomena in the Frobenius category $\\mathcal{C}_2=\\mathrm{CM}_{\\mathbb{Z}}(\\mathbb{C}[x,y]/(x^2))$ via triangulations of the completed $\\infty$-gon. It extends the Paquette–Yıldırım cluster character to the Frobenius and triangulated settings, producing infinite friezes with coefficients that encode the Penrose tilings and their combinatorics, including mutation relations and a between-class equivalence for fountain triangulations. A central result is a bijection between special right fountain triangulations modulo a finite-mutation equivalence and Penrose tilings modulo isometry, tying together nonperiodic tilings, frieze theory, and representation theory of $A_ inf$-singularities. The paper also provides concrete examples (fountain and leapfrog triangulations) to illustrate the construction of infinite friezes and their relation to Penrose tilings, highlighting the interplay between combinatorics, cluster theory, and noncommutative geometry.

Abstract

We study Penrose tilings of the plane $\mathbb{R}^2$ and nonperiodic infinite frieze patterns from the point of view of Cohen--Macaulay representation theory: Triangulations of the completed infinity-gon correspond to subcategories of the Frobenius category $\mathcal{C}_2=\mathrm{CM}_{\mathbb{Z}}(\mathbb{C}[x,y]/(x^2))$, the singularity category of the curve singularity of type $A_\infty$. We relate Penrose tilings to certain triangulations of the completed infinity-gon, and thus to the corresponding subcategories of $\mathcal{C}_2$. We then extend the cluster character of Paquette and Yıldırım for a triangulated category modelling said triangulations to our setting. This allows us to define nonperiodic infinite friezes patterns coming from triangulations of the completed infinity-gon and in particular from Penrose tilings.

Penrose tilings, infinite friezes, and the $A_\infty$-singularity

TL;DR

The work builds a bridge between Penrose tilings, infinite frieze patterns, and the -singularity by interpreting Penrose tilings as special cluster-tilting phenomena in the Frobenius category via triangulations of the completed -gon. It extends the Paquette–Yıldırım cluster character to the Frobenius and triangulated settings, producing infinite friezes with coefficients that encode the Penrose tilings and their combinatorics, including mutation relations and a between-class equivalence for fountain triangulations. A central result is a bijection between special right fountain triangulations modulo a finite-mutation equivalence and Penrose tilings modulo isometry, tying together nonperiodic tilings, frieze theory, and representation theory of -singularities. The paper also provides concrete examples (fountain and leapfrog triangulations) to illustrate the construction of infinite friezes and their relation to Penrose tilings, highlighting the interplay between combinatorics, cluster theory, and noncommutative geometry.

Abstract

We study Penrose tilings of the plane and nonperiodic infinite frieze patterns from the point of view of Cohen--Macaulay representation theory: Triangulations of the completed infinity-gon correspond to subcategories of the Frobenius category , the singularity category of the curve singularity of type . We relate Penrose tilings to certain triangulations of the completed infinity-gon, and thus to the corresponding subcategories of . We then extend the cluster character of Paquette and Yıldırım for a triangulated category modelling said triangulations to our setting. This allows us to define nonperiodic infinite friezes patterns coming from triangulations of the completed infinity-gon and in particular from Penrose tilings.

Paper Structure

This paper contains 18 sections, 19 theorems, 44 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Maximal Cohen--Macaulay modules over the $A_\infty$-singularity
  4. The combinatorics
  5. Penrose tilings
  6. Friezes
  7. Penrose tilings and friezes from triangulations of the $\infty$-gon
  8. Penrose tilings from right fountains
  9. Proof of Theorem \ref{['intro-theorem-penrose']}
  10. Friezes and Penrose Tilings
  11. From cluster characters to friezes
  12. Extensions and the Auslander--Reiten quiver of $\mathcal{C}_2$ and $\mathcal{C}$
  13. Paquette--Yıldırım cluster character for $\mathcal{C}$
  14. Cluster character for $\mathcal{C}_2$
  15. The frieze relations
  16. ...and 3 more sections

Key Result

Theorem A

Let $2^\mathbb{N}$ denote the set of all sequences on $\{0,1\}$ and $\mathcal{P}$ denote the space of Penrose tilings.

Figures (2)

  • Figure 1: A fountain triangulation.
  • Figure 2: The five different types of triangulations of the $\infty$-gon.

Theorems & Definitions (52)

  • Theorem A
  • Theorem B: see Theorem \ref{['Thm:half-friezes-coeffs']} and Definition \ref{['Def:inf-frieze-coeff']}
  • Remark 2.1
  • Example 2.2
  • Theorem 2.3: Conway--Coxeter CoCo1CoCo2
  • Definition 3.1
  • Remark 3.2
  • Example 3.3
  • Example 3.4
  • Lemma 3.5
  • ...and 42 more