I'm working on a project that requires an arbitrary number of points approximately evenly spaced around a sphere of arbitrary dimensions. (In practice, probably d < 20.) Ideally, this would be more uniform than you'd get from uniformly or randomly choosing points along each axis and projecting them to the sphere.

In 3D the easiest way to do this is to take an octahedron and subdivide each face into smaller triangles, projecting each point onto the surface of the sphere. I'm struggling to figure out how to generalize this to higher dimensions.

Do I just need to subdivide each 2d face (equilateral triangle) the way I would subdivide the faces of an octahedron? Or do I need to subdivide the (d-1)-face? If so, how do I do this, since it isn't possible to subdivide a regular tetrahedron into smaller regular tetrahedra?