I have successfully calculated the icosahedron inside the dodecahedron using an a simple formula and some ugly brute force.
Here is how I did this:
Starting with the vertices of a unit dodecahedron, I find the midpoint of each pentagon in the dodecahedron which defines the icosahedron. Given the vertices of each pentagon, you can easy find the midpoint by averaging the points in the pentagon. EASY!
What’s not as easy is grouping the vertices of the dodecahedron into pentagon faces. I used some ugly brute force to find this:
I tested every combination of 5 vertices to find the ones farthest from the center. These vertices define the related icosahedron.
Here we see the tori defined by the icosahedron nested inside the vertices of the dodecahedron:
Now we can just solve for the midpoint of each edge of the dodecahedron. Here are the edges: