Dimensions
If the edge length of a truncated icosahedron is a, the radius of a circumscribed sphere (one that touches the truncated icosahedron at all vertices) is:
where φ is the golden ratio.
This result is easy to get by using one of the three orthogonal golden rectangles drawn into the original icosahedron (before cut off) as the starting point for our considerations. The angle between the segments joining the center and the vertices connected by shared edge (calculated on the basis of this construction) is approx. 23.281446°.
Read more about this topic: Truncated Icosahedron
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—Thomas Jefferson (1743–1826)
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—Ralph Waldo Emerson (1803–1882)