Tilings of Convex Sets by Mutually Incongruent Equilateral Triangles Contain Arbitrarily Small Tiles

摘要

We show that every tiling of a convex set in the Euclidean plane$mathbb{R}^2$ by equilateral triangles of mutually different sizes containsarbitrarily small tiles. The proof is purely elementary up to the discussion ofone family of tilings of the full plane $mathbb{R}^2$, which is based on asurprising connection to a random walk on a directed graph.