Quadrilateral reptiles

TL;DR

This paper characterizes cyclic quadrilaterals that are reptiles, proving they must be trapezoids, and concludes that all convex reptiles are either triangles or trapezoids, advancing understanding of polygon decomposition.

Contribution

It proves that cyclic quadrilaterals that are reptiles are trapezoids and classifies all convex reptiles as triangles or trapezoids, extending prior results.

Findings

Cyclic reptile quadrilaterals are trapezoids.

All convex reptiles are triangles or trapezoids.

Provides a classification of convex reptile polygons.

Abstract

A polygon $P$ is called a reptile, if it can be decomposed into $k\ge 2$ nonoverlapping and congruent polygons similar to $P$. We prove that if a cyclic quadrilateral is a reptile, then it is a trapezoid. Comparing with results of U. Betke and I. Osburg we find that every convex reptile is a triangle or a trapezoid.