# Angle between vertex and center of face in Platonic solids?

I'm trying to find the angle between a vertex and the center of one of the nearest faces in a dodecahedron. This would be nice to know the formula and/or number for all the Platonic solids though. I'm using these to model some 3D shapes in Blender and managed to work around the regular icosahedron modeling by using the dihedral angle then parenting relative faces and rotating them around eachother's axis's, but the same trick won't work (or at least I can't see how to make it work) to put the dodecahedron's faces into the icosahedron's vertices when starting from a point where each shape contains a face sharing a common Z axis (pictured because I know I don't have the proper terminology.)

The short of this is that I'm trying to rotate a dodecahedron such that the face-centered objects are aligned to the vertexes of a regular icosahedron starting from a position where they share a common axis between a pair of faces (each cylinder in the above image is face-centered.)

• Wikipedia has coordinates for the vertices, which is enough to get you what you need. en.wikipedia.org/wiki/Platonic_solid – Ethan Bolker Feb 6 '18 at 13:19
• @EthanBolker Thanks, but I'm a perfectionist, it was N[180-(ArcCos[-Sqrt[(1/15) (5 + 2 Sqrt[5])]]*(180/Pi)),79] – CoryG Feb 6 '18 at 15:58
• If you're a perfectionist you shouldn't be calling N either – Rahul Feb 8 '18 at 15:00
• @Rahul Perfection has it's limits (and N[, 79] is well beyond the double point precision allowed by a 3D modeling program - while the 2-digit precision in the Wiki page would fall apart as soon as you apply more than a single rotation or scaling effect.) – CoryG Feb 8 '18 at 20:06

## 1 Answer

I found the answer on Wolfram Alpha via the following formula:

N[180-(ArcCos[-Sqrt[(1/15) (5 + 2 Sqrt[5])]]*(180/Pi)),79]