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 R-2 by equilateral triangles of mutually different sizes contains arbitrarily small tiles. The proof is purely elementary up to the discussion of one family of tilings of the full plane R-2, which is based on a surprising connection to a random walk on a directed graph.