Sorry if this video was short I had to speed it up by 10x otherwise it would be like a slideshow.
This is not an actual sphere, it actually has 8x8x8x8x8 segments.
The formula for each point is:
x = cos(x angle)*sin(y angle)*sin(w angle)*sin(v angle)
y = sin(x angle)*sin(y angle)*sin(w angle)*sin(v angle)
z = cos(z angle)*sin(w angle)*sin(v angle)
w = cos(w angle)*sin(v angle)
v = cos(v angle)
Add a point for every permutation of the 5 different angles values.
Also any polytope of any dimension can be calculated using this. Just multiply each dimension by another layer of sin(n angle) and make the last dimension cos(n angle).