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

TL;DR

This paper proves that any tiling of a convex set in the plane with mutually incongruent equilateral triangles must include arbitrarily small tiles, revealing a fundamental property of such tilings.

Contribution

It establishes that all such tilings necessarily contain arbitrarily small tiles, using an elementary proof linked to a novel connection with random walks on directed graphs.

Findings

Any tiling of a convex set with mutually incongruent equilateral triangles contains arbitrarily small tiles.

The proof involves a surprising connection to a random walk on a directed graph.

The approach is elementary except for a specific family of tilings of the plane.

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.