  We present a method to assign, for any radius $r$ greater than about 12.44,
one of seven colors to each point in $\mathbb{R}^3$ lying at distance $r$ from
the origin, such that no two points at unit distance from each other are
assigned the same color. The existence of such a construction contrasts with
the recent demonstration that, for any positive value $\varepsilon$, if no two
points assigned the same color lie at any distance in $[1,1+\varepsilon]$ (and
with certain other restrictions that are also satisfied with our coloring),
then eight colors are needed for any finite $r\ge18$, even though seven colors
suffice in the plane when $\varepsilon \leq\frac{\sqrt{7}}{2} - 1$.
