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ORTHOGONAL COLORINGS OF THE SPHERE
Published online by Cambridge University Press: 04 February 2016
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 
${\it\pi}/2$ 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.
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- Copyright © University College London 2016
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