How to convert celestial sphere coordinates into a truncated icosahedron?

Question

Taking the figure of a truncated icosahedron where a pentagonal face is established as the north pole and another pentagonal face on the opposite side as the south pole, the celestial coordinates of the center of its faces would be RA 0h, DEC + 90 ° and RA 0h, DEC -90 °. How to calculate the celestial coordinates for each vertex and center of each of the remaining faces of the polyhedron ?. I am trying to build a planetarium projector and I need to transfer the positions of the stars on the faces of the truncated icosahedron.