Orthogonal colorings of the sphere

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.

References in corpus (1)

• Plank theorems via successive inradii