Small Triambic Icosahedron


The small triambic icosahedron is the dual polyhedron of the small ditrigonal icosidodecahedron U_(30), and is Wenninger model W_(70). It can be constructed by augmentation of a unit edge-length icosahedron by a pyramid with height sqrt(15)/15, giving a solid with edge lengths sqrt(2/5) and 1.

Wenninger (1989, p. 49) calls this solid the triakis octahedron (which is a term more commonly used for the dual of one of the Archimedean solids).

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

The convex hull of the small ditrigonal icosidodecahedron is a regular dodecahedron whose dual is the icosahedron, so the dual of the small ditrigonal icosidodecahedron (i.e., the small triambic icosahedron) is one of the icosahedron stellations (Wenninger 1983, p. 42). In fact, it is the second icosahedron stellation in the enumeration of Maeder (1994).

Small triambic icosahedron vertex groups

The small triambic icosahedron has 32 vertices, 90 edges, and 70 faces. Its vertices are arranged in two concentric groups of 12 (indicated in red in the above illustration) and 20 (blue).

Polyhedra determined by vertex groups

For a small triambic icosahedron with edge lengths


the vertices determine a regular icosahedron and regular dodecahedron with circumradii



The solid in the right figure is the dodecahedron-small triambic icosahedron compound.

The surface area and volume of the small triambic icosahedron are


The solid small triambic icosahedron has moment of inertia tensor

 I=[1/(300)(53+19sqrt(5))Ma^2 0 0; 0 1/(300)(53+19sqrt(5))Ma^2 0; 0 0 1/(300)(53+19sqrt(5))Ma^2]

for uniform density solid of mass M.

See also

Dodecahedron-Small Triambic Icosahedron Compound, Dual Polyhedron, Echidnahedron, Great Triakis Octahedron, Icosahedron Stellations, Small Ditrigonal Icosidodecahedron, Small Triakis Octahedron, Triakis Icosahedron,

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Maeder, R. E. "The Stellated Icosahedra." Mathematica in Education 3, 5-11, 1994., M. J. Dual Models. Cambridge, England: Cambridge University Press, pp. 42 and 46-47 1983.Wenninger, M. J. Polyhedron Models. New York: Cambridge University Press, p. 46, 1989.

Cite this as:

Weisstein, Eric W. "Small Triambic Icosahedron." From MathWorld--A Wolfram Web Resource.

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