What are the center coördinates of the planes of a regular dodecahedron?

Question

I'm programming a regular dodecahedron using this Wiki page. But I'm wondering, what are the center coördinates of the planes of a dodecahedron?

Answer 1

I assume you mean the faces instead of the planes. The faces are regular pentagons, so you can just take the average of the coordinates of the vertices.

E.g. in the picture below, the face which is turned towards us, and has one pink, two orange and two green vertices, has vertex coordinates $(\phi,0,1/\phi), (1,1,-1), (1,-1,-1), (0,1/\phi,-\phi), (0,-1/\phi,-\phi)$. The average is $(\frac{2+\phi}5, 0, \frac{1/\phi-2-2\phi}5)$ = $(\frac{5+\sqrt{5}}{10},0,\frac{-7-\sqrt5}{10})$. You can find similar coordinates for the other faces.

_{(source: Wikiwand)}