  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.
