Octants are Cover-Decomposable into Many Coverings

TL;DR

This paper proves that octants can be decomposed into multiple coverings, and as a consequence, similar decompositions are possible for triangles in the plane, advancing understanding of cover-decomposability in geometric coverings.

Contribution

It establishes the first general proof that octants are cover-decomposable into multiple coverings, extending to triangles in the plane.

Findings

Octants are cover-decomposable into multiple coverings.

Any finite covering of the plane with homothetic triangles can be decomposed into multiple coverings.

Provides bounds for the number of coverings needed for decomposition.

Abstract

We prove that octants are cover-decomposable into multiple coverings, i.e., for any k there is an m(k) such that any m(k)-fold covering of any subset of the space with a finite number of translates of a given octant can be decomposed into k coverings. As a corollary, we obtain that any m(k)-fold covering of any subset of the plane with a finite number of homothetic copies of a given triangle can be decomposed into k coverings. Previously only some weaker bounds were known for related problems.