Tilings of regular pentagons with similar right triangles

Abstract

We say that a triangle $\mathrm{\Delta }$ tiles the polygon $\mathcal{𝒫}$ if $\mathcal{𝒫}$ can be decomposed into finitely many nonoverlapping triangles similar to $\mathrm{\Delta }$. Let $\mathcal{ℛ}$ be a regular pentagon. In this paper we prove that a right triangle tiles $\mathcal{ℛ}$ if and only if this triangle has an acute angle $\frac{\pi }{5}$.