I found this:
https://cage.ugent.be/~hs/polyhedra/dodeicos.html
So it appears that the edge of the icosahedron is the greatest of the parts in which the edge of the cube is divided by the golden section,The regular icosahedron
Although Euclid (book XIII proposition 16) constructs a regular icosahedron starting with a given sphere, there's a nice method to construct a regular icosahedron starting from a given cube
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in cube 6 this would mean that the edge of the icosahedron is 6 x 0.61803398875 = 3.7082039325
The edge of the icosahedron that touches with six vertices the six faces of the cube 6 must be √8 / cos36 = 2.82842712475 / 0.80901699437 = 3.49546735735
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