Octants are Cover Decomposable

TL;DR

This paper proves that octants are cover-decomposable, meaning any 12-fold covering with their translates can be split into two, with implications for triangles and the plane, but also presents an indecomposable 3-fold covering.

Contribution

It establishes the cover-decomposability of octants for 12-fold coverings and extends results to triangles and the entire plane, introducing new decomposition techniques.

Findings

Octants are cover-decomposable for 12-fold coverings.

Any 12-fold covering of the plane with triangles can be decomposed into two.

Existence of an indecomposable 3-fold covering.

Abstract

We prove that octants are cover-decomposable, i.e., any 12-fold covering of any subset of the space with a finite number of translates of a given octant can be decomposed into two coverings. As a corollary, we obtain that any 12-fold covering of any subset of the plane with a finite number of homothetic copies of a given triangle can be decomposed into two coverings. We also show that any 12-fold covering of the whole plane with open triangles can be decomposed into two coverings. However, we exhibit an indecomposable 3-fold covering.