Indecomposable coverings with homothetic polygons

TL;DR

This paper demonstrates that for certain convex and concave polygons, there exist multiple coverings of the plane with homothetic copies that cannot be split into simpler coverings, highlighting limitations in decomposability.

Contribution

It establishes the existence of indecomposable multiple coverings of the plane using homothetic polygons for a broad class of polygons.

Findings

Existence of indecomposable coverings for convex polygons with at least four sides.

Existence of indecomposable coverings for certain concave polygons with no parallel sides.

These coverings cannot be decomposed into two simpler coverings.

Abstract

We prove that for any convex polygon $S$ with at least four sides, or a concave one with no parallel sides, and any $m>0$, there is an $m$-fold covering of the plane with homothetic copies of $S$ that cannot be decomposed into two coverings.