any specific platonic solid in mind or do you to be able to calculate the points for all of them?

the wiki page has a lot of useful formulas and lengths on them.

the octahedron is the simplest, it's 3d points (assuming sphere radius r and origin at (0, 0, 0) and no rotation) are: (0, r, 0), (-r, 0, 0), (0, 0, -r), (r, 0, 0), (0, 0, r), (0, -r, 0).

using the numbers at wiki, calculating the 3d points for the tetrahedron:

top point obviously at (0, r, 0), bottom y-value at -(1 - 1/sqrt(2))r.

the bottom 3 points's x- and z-values: assume it's rotated so that one of them is at x = r, z = 0.

then the other 2's coordinates are: (-(1 - 1/sqrt(2))r, -(1 - 1/sqrt(2))r, r) and (-(1 - 1/sqrt(2))r, -(1 - 1/sqrt(2))r, -r).

so the 4 points are: (0, r, 0), (r, -(1 - 1/sqrt(2))r, 0), (-(1 - 1/sqrt(2))r, -(1 - 1/sqrt(2))r, r) and (-(1 - 1/sqrt(2))r, -(1 - 1/sqrt(2))r, -r).

not sure if i could make this into an easy formula... to calculate the points for the other platonic solids, try using the numbers in wiki.

sorry it's so messy but i haven't figured out how to make nice latex formulas yet