If I ask you to place N equally space points on a circle,
you would have no particular trouble doing so. The same
is not true for equally spaced points on a sphere, as is
evidenced by the irregular-looking pattern of dimples
on a golf ball.
But what does equally spaced points on a sphere mean?
I have come up with a working definition which is probably
not original to me: The points are equally spaced if and
only if the minimum distance from a point to any other
point is maximized. That is, find the closest neighbor
to every point, then find the smallest such distance,
and make sure that it is as large as possible. So, for
a 2-point example, the points would be antipodal -- that's
as far away from each other than they can get. For three
points, they will all lie on a great circle. For four
points, they form the vertices of a regular tetrahedron
(a tetrahedron of which all edges have the same length).
Suppose one places 8 equally-spaces points (according to
the definition above) on a unit sphere (a sphere of radius
1). What would be the shortest euclidean distance between
any pair of points? Euclidean distance is the straight-line
distance (not the distance over the surface of the sphere)
between the points. How would you describe the figure
whose vertices are those points?