Truncated Icosahedron

Make Your Own Truncated Icosahedron

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The truncated icosahedron is the 32-faced Archimedean solid with 60 vertices corresponding to the facial arrangement . It is also the uniform polyhedron with Maeder index 25 (Maeder 1997), Wenninger index 9 (Wenninger 1989), Coxeter index 27 (Coxeter et al. 1954), and Har'El index 30 (Har'El 1993). It has Schläfli symbol t and Wythoff symbol . It is illustrated above together with a wireframe version and a net that can be used for its construction.

Leonardo da Vinci drew the skeletal truncated icosahedron illustrated above for Luca Pacioli's Divina proportione (Pacioli and da Vinci 1509, Viana 2024).

Several symmetrical projections of the truncated icosahedron are illustrated above.

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

The lenses used for focusing the explosive shock waves of the detonators in the Fat Man atomic bomb were constructed in the configuration of a truncated icosahedron (Rhodes 1996, p. 195). It did not however become a familiar household shape until the 1970 introduction of the Adidas Telstar soccer ball, whose white hexagons surrounding black pentagons forming a truncated icosahedron are now iconically associated with the sport of soccer. The truncated icosahedron is also known to chemists as the structure of pure carbon known as a buckyball (a.k.a. fullerenes).

The dual polyhedron of the truncated icosahedron is the pentakis dodecahedron, both of which are illustrated above together with their common midsphere. The inradius of the dual, midradius of the solid and dual, and circumradius of the solid, for unit edge length (), 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)

The unit truncated icosahedron has Dehn invariant

			(8)
			(9)
			(10)

(OEIS A377787), where the first expression uses the basis of Conway et al. (1999, p. 331).

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 , a doubly periodic function with a parallelogram as a fundamental domain for its periods in the complex plane.