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What we should know about dodecahedron and other platonic solids.
http://en.wikipedia.org/wiki/Dodecahedron
http://en.wikipedia.org/wiki/Platonic_solid
http://en.wikipedia.org/wiki/Schl%C3%A4fli_symbol
Equations in model:
{p,q} is a Schläfli symbol, for dodecahedron {5,3}
p=5 //the number of parties at the brink
q=3 //the number of edges adjacent to the vertices
a=51 //lenght of rib
sintheta=(cos (180/q)/sin (180/p)) //separate variable angle to simplify the formula of theta
theta=2*asin (sintheta) //dihedral angle
epsilon=2*sin (36) //constant need to equal radius of inscribed sphere
phi=2*cos (36) // phi is "golden ratio" also is (1+sqrt(5))/2. 36=pi/5=180/5,
rad=a*0.5*phi*phi/epsilon // radius of inscribed sphere which is tangent to each face at the center of the face.
That's what I did.
Answered with a tutorial: https://grabcad.com/tutorials/tutorial-how-to-make-dodecahedron-pot