Truncated Icosahedron

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The truncated icosahedron is the 32-faced Archimedean solid with 60 vertices corresponding to the facial arrangement 👁 20{6}+12{5}
. 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👁 {3,5}
and Wythoff symbol 👁 25|3
. It is illustrated above together with a wireframe version and a net that can be used for its construction.

Several symmetrical projections of the truncated icosahedron are illustrated above.

It is implemented in the Wolfram Language as [].

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 became 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 👁 C_(60)
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 👁 r
of the dual, midradius 👁 rho
of the solid and dual, and circumradius 👁 R
of the solid for 👁 a=1
are

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

The surface area and volume are

The unit truncated icosahedron has Dehn invariant

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

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 👁 P(z;g_2,g_3)
, which is a doubly-periodic function whose natural domain is a periodic parallelogram in the complex 👁 z
-plane.

See also

Archimedean Solid, Conext 21 Polyhedron, Equilateral Zonohedron, Hexecontahedron, Jabulani Polyhedron, Soccer Ball, Truncation

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References

Aldersey-Williams, H. The Most Beautiful Molecule. New York: Wiley, 1997.Chung, F. and Sternberg, S. "Mathematics and the Buckyball." Amer. Sci. 81, 56-71, 1993.Conway, J. H.; Radin, C.; and Sadun, L. "On Angles Whose Squared Trigonometric Functions Are Rational." Discr. Computat. Geom. 22, 321-332, 1999.Coxeter, H. S. M.; Longuet-Higgins, M. S.; and Miller, J. C. P. "Uniform Polyhedra." Phil. Trans. Roy. Soc. London Ser. A 246, 401-450, 1954.Cundy, H. and Rollett, A. "Truncated Icosahedron. 👁 5.6^2
." §3.7.10 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 110, 1989.Geometry Technologies. "Truncated Icosahedron." http://www.scienceu.com/geometry/facts/solids/tr_icosa.html.Har'El, Z. "Uniform Solution for Uniform Polyhedra." Geometriae Dedicata 47, 57-110, 1993.Kabai, S. Mathematical Graphics I: Lessons in Computer Graphics Using Mathematica. Püspökladány, Hungary: Uniconstant, p. 131, 2002.Kasahara, K. "Three More Semiregular Polyhedrons Become Possible." Origami Omnibus: Paper-Folding for Everyone. Tokyo: Japan Publications, p. 225, 1988.Harris, J. W. and Stocker, H. Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 101, 1998.Maeder, R. E. "25: Truncated Icosahedron." 1997. https://www.mathconsult.ch/static/unipoly/25.html.Rhodes, R. Dark Sun: The Making of the Hydrogen Bomb. Touchstone Books, 1996.Sloane, N. J. A. Sequence A377787 in "The On-Line Encyclopedia of Integer Sequences."👁 Image
Trott, M. "Constructing a Buckyball with Mathematica: A Combination of Geometry and Algebra from Classical and Modern Mathematics." http://library.wolfram.com/infocenter/Demos/106/.Trott, M. "Bending a Soccer Ball." http://www.mathematicaguidebooks.org/soccer/.Wenninger, M. J. "The Truncated Icosahedron." Model 9 in Polyhedron Models. Cambridge, England: Cambridge University Press, p. 23, 1989.

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Truncated Icosahedron

Cite this as:

Weisstein, Eric W. "Truncated Icosahedron." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TruncatedIcosahedron.html

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