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

Abstract

We show that every tiling of a convex set in the Euclidean plane ${\mathbb{R}}^2$ 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 ${\mathbb{R}}^2$, which is based on a surprising connection to a random walk on a directed graph.

11 pages, 8 figures