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Triakis Icosahedron


TriakisIcosahedronSolidWireframeNet

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The (semiregular) triakis icosahedron is the dual polyhedron of the truncated dodecahedron (Holden 1971, p. 55). It has 60 faces and is illustrated above together with a wireframe version and a net that can be used for its construction.

Note that Wenninger (1989, p. 46) uses the term "triakis icosahedron" and Coxeter et al. (1999) the terms "a 'triakisicosahedron"' (Coxeter et al. 1999, p. 13) and "triakisicosahedron" (Coxeter et al. 1999, p. 64) to refer to the small triambic icosahedron.

The triakis icosahedron is Wenninger dual W_(10).

Solids inscribed in a triakis icosahedron

A tetrahedron 10-compound, cube 5-compound, icosahedron, and dodecahedron can be inscribed on the vertices of the triakis icosahedron (E. Weisstein, Dec. 25-27, 2009).

Taking the dual of a truncated dodecahedron with unit edge lengths gives a triakis icosahedron with edge lengths

s_1=5/(22)(7+sqrt(5))
(1)
s_2=1/2(5+sqrt(5)).
(2)

The surface area and volume are

S=(75)/(11)sqrt(1/2(313+117sqrt(5)))
(3)
V=(125)/(44)(19+9sqrt(5)).
(4)

See also

Archimedean Dual, Archimedean Solid, Hexecontahedron, Small Triambic Icosahedron

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References

Coxeter, H. S. M.; Du Val, P.; Flather, H. T.; and Petrie, J. F. The Fifty-Nine Icosahedra. Stradbroke, England: Tarquin Publications, 1999.Holden, A. Shapes, Space, and Symmetry. New York: Columbia University Press, p. 55, 1971.Wenninger, M. J. Dual Models. Cambridge, England: Cambridge University Press, pp. 19-20, 1983.Wenninger, M. J. Polyhedron Models. New York: Cambridge University Press, p. 46, 1989.

Cite this as:

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

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