Truncated Icosahedron

The truncated icosahedron is the 32-faced Archimedean solid corresponding to the facial arrangement . It is the shape used in the construction of soccer balls, and it was also the configuration of the lenses used for focusing the explosive shock waves of the detonators in the Fat Man atomic bomb (Rhodes 1996, p. 195). The truncated icosahedron has 60 vertices, and is also the structure of pure carbon known as buckyballs (a.k.a. fullerenes).

The truncated icosahedron is uniform polyhedron and Wenninger model . It has Schläfli symbol t and Wythoff symbol .

It is implemented in the Wolfram Language as PolyhedronData["TruncatedIcosahedron"].

Several symmetrical projections of the truncated icosahedron are illustrated above.

The dual polyhedron of the truncated icosahedron is the pentakis dodecahedron. The inradius of the dual, midradius of the solid and dual, and circumradius of the solid for are

			(1)
			(2)
			(3)

The distances from the center of the solid to the centroids of the pentagonal and hexagonal faces are given by

			(4)
			(5)

The surface area and volume are

			(6)
			(7)

M. Trott illustrates how a torus can be continuously deformed into two concentric soccer balls of identical size and orientation with no tearing of the surface in this transition. In particular, the animation (a few frames of which are illustrated above) shows a smooth homotopy between the identity map and a particular map involving the Weierstrass elliptic function , which is a doubly-periodic function whose natural domain is a periodic parallelogram in the complex -plane.

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