Penrose tilings, infinite friezes, and the A_∞-singularity

2025/11/10 by Esentepe, Özgür, Faber, Eleonore
#Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2511.07530

Abstract

We study Penrose tilings of the plane ℝ² 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 C₂=CM_ℤ(ℂ[x,y]/(x²)), the singularity category of the curve singularity of type A_∞. We relate Penrose tilings to certain triangulations of the completed infinity-gon, and thus to the corresponding subcategories of C₂. 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.

Citations

• Cluster characters for 2-Calabi-Yau Frobenius extriangulated categories
• Frieze patterns
• Frieze patterns of integers
• Categories for Grassmannian cluster algebras of infinite rank
• Transfinite mutations in the completed infinity-gon
• Infinite friezes
• On the Enlargement by Prüfer Objects of the Cluster Category of type A_∞
• Cluster categories coming from cyclic posets
• On cluster algebras with coefficients and 2-Calabi-Yau categories
• From triangulated categories to cluster algebras
• Cluster algebras as Hall algebras of quiver representations
• Pentaplexity A Class of Non-Periodic Tilings of the Plane

Related

• Infinite friezes
• Infinite friezes and triangulations of the strip
• Cluster structures for the A_∞ singularity
• Frieze patterns and combinatorics of curve singularities
• Arithmetic infinite friezes from punctured discs