Orthogonal colorings of the sphere

TL;DR

This paper investigates orthogonal colorings of the sphere, proving that if each color class has a non-empty interior, then the coloring must be octahedral, thus characterizing a class of such colorings.

Contribution

It establishes that orthogonal 4-colorings of the sphere with non-empty interior classes are necessarily octahedral, addressing a key open question.

Findings

Orthogonal 4-colorings with non-empty interior classes are octahedral.

Minimum four parts are needed for orthogonal colorings of the sphere.

Related results on sphere colorings are also presented.

Abstract

An orthogonal coloring of the two-dimensional unit sphere $\mathbb{S}^2$, is a partition of $\mathbb{S}^2$ into parts such that no part contains a pair of orthogonal points, that is, a pair of points at spherical distance $\pi/2$ apart. It is a well-known result that an orthogonal coloring of $\mathbb{S}^2$ requires at least four parts, and orthogonal colorings with exactly four parts can easily be constructed from a regular octahedron centered at the origin. An intriguing question is whether or not every orthogonal 4-coloring of $\mathbb{S}^2$ is such an octahedral coloring. In this paper we address this question and show that if every color class has a non-empty interior, then the coloring is octahedral. Some related results are also given.