Tiling of regular polygon with similar right triangles

TL;DR

This paper characterizes the possible angles of similar right triangles used to tile regular polygons with five or more sides, excluding the case of 28 sides, extending previous related results.

Contribution

It provides a new necessary condition on the angles of similar right triangles that can tile regular polygons with n ≥ 5 sides, except for n=28.

Findings

If a regular n-gon (n ≥ 5, n ≠ 28) is tiled with similar right triangles, one angle must be in a specific set.

The result generalizes and refines earlier findings by Laczkovich and Szegedy.

The paper establishes constraints on tiling configurations for regular polygons.

Abstract

A tiling is a decomposition of a polygon into finitely many non-overlapping triangles. We prove that if a regular n-gon, $n \geq 5$, $n \neq 28$, can be tiled with similar right triangles, then one of the angles of these triangles is in $\left\{\frac{\pi}{n},\frac{2\pi}{n}, \frac{\pi} {6}+\frac{2\pi}{3n}\right\}$. Some related results were previously obtained by M.Laczkovich and B. Szegedy.