Tilings of convex sets by mutually incongruent equilateral triangles\n contain arbitrarily small tiles

2017/11/24 by Christian Richter, Richter, Christian, Melchior Wirth +1
Computer Science · Mathematics · #05C81 #51M20 #52C20 (primary) #60J10 (secondary) #FOS: Mathematics #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #Point processes and geometric inequalities #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1711.08903

openalex publication_date 2017/11/24 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28

Abstract

We show that every tiling of a convex set in the Euclidean plane\n\ℝ² by equilateral triangles of mutually different sizes contains\narbitrarily small tiles. The proof is purely elementary up to the discussion of\none family of tilings of the full plane \ℝ², which is based on a\nsurprising connection to a random walk on a directed graph.\n

Related

• Tilings of convex polygons by equilateral triangles of many different\n sizes
• Old problem revisited: Which equilateral convex polygons tile the plane?
• Families of convex tilings
• Incongruent equipartitions of the plane
• Tiling the plane with equilateral triangles