A two-vertex theorem for normal tilings

2021/10/05 by G. Domokos, Ákos G. Horváth, Domokos, Gábor +3 · 1 citation
Computer Science · Materials Science · Mathematics · #52C20 #Differential Geometry (math.DG) #FOS: Mathematics #Limits and Structures in Graph Theory #Quasicrystal Structures and Properties #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2110.02323

openalex publication_date 2021/10/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We regard a smooth, d=2-dimensional manifold M and its normal tiling M, the cells of which may have non-smooth or smooth vertices (at the latter, two edges meet at 180 degrees.) We denote the average number (per cell) of non-smooth vertices by v^⋆ and we prove that if M is periodic then v^⋆ ≥ 2 and we show the same result for the monohedral case by an entirely different argument. Our theory also makes a closely related prediction for non-periodic tilings. In 3 dimensions we show a monohedral construction with v^⋆=0.

Cited by

• Prismatic Soft Cubes

Related

• Monohedral Tilings of a Convex Disc with a Smooth Boundary
• On k-normality and Regularity of Normal Toric Varieties
• Periodicity and decidability of translational tilings by rational polygonal sets
• A new simple family of non-periodic tilings with square tiles
• Corona limits of tilings : Periodic case