[Webmath] Platonic Solids & Metranon's Cube

[Jonathan Pike]

Hi;
The Platonic solids nest perfectly one within another. This configuration, starting with the octahedron, is known as Metranon's Cube. I would like to build a mathematically perfect model of the same. Andrew Robbins was kind enough to point me to this page:
http://mathworld.wolfram.com/PlatonicSolid.html
which gives me a lot of pertinent data; however, the data for exactly nesting one solid in another seems lacking by half. It gives the formulas for determining the spherical radii of nested solids vis-a-vis their "dual" solids; however, those duals are matched icosahedron-dodecahedron, octahedron-cube and tetrahedron-tetrahedron (for a star tetrahedron). In the Metranon's Cube as described above, the tetrahedron is between the octahedron and the cube, so how does one figure out the "duals" or the radii of nested solids? This problem compounds itself with the matter of bridging the cube to the icosahedron. How do I resolve these matters?
TIA,
Jonathan
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