The Icosahedron and Dodecahedron are 'geometric duals' of each other. Their faces and vetices share the same symmetry axes, which is why they fit together so neatly.
The five symmetry forms are entirely golden ratio proportions. Compared to the length of one edge, any opther measurement between vertexes, face centres, and even the mid points between them ... everything will be some simple relationship to phi (the golden ratio - 1.618... )
The five symmetry is the essence of life as shown by xray cross sections of the DNA double helix. The pentagonal forms do not fit neatly into the grid of 3 dimensional matter. Any temporary fit is incomplete and unstable, but life requires this instability for its process of constant change. However, in higher dimensions this problem is transcended.