The small triambic icosahedron is the
dual polyhedron of the small ditrigonal icosidodecahedron
with Maeder index 30 (Maeder 1997), Weinninger index 70 (Wenninger 1971, p. 106-107),
Coxeter index 39 (Coxeter et al. 1954), and Har'El index 35 (Har'El 1993).
Note that while Wenninger (1989, p. 49) calls this solid the triakis octahedron,
this term more commonly used for the dual of one of the Archimedean solids.
The small triambic icosahedron is implemented in the
Language as [ PolyhedronData "SmallTriambicIcosahedron"].
It is illustrated above together with its
The small triambic icosahedron consists of 20 equilateral irregular hexagons with alternating angles
where the first expression uses the basis of Conway
et al. (1999).
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).
skeleton is the rhombic
The hull of the small triambic icosahedron can be constructed by
augmentation of a unit edge-length icosahedron by a pyramid with
giving a solid with edge lengths and 1.
The small triambic icosahedron hull 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).
For a small triambic icosahedron hull with edge lengths
the vertices determine a
and regular dodecahedron with circumradii
The solid in the right figure is the
triambic icosahedron compound.
surface area and volume
of the small triambic icosahedron hull are
The solid small triambic icosahedron has moment of inertia tensor
for uniform density solid of mass
See also Dodecahedron-Small Triambic Icosahedron Compound
Small Ditrigonal Icosidodecahedron
Small Triakis Octahedron
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References 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. Har'El, Z. "Uniform
Solution for Uniform Polyhedra." Geom. Dedicata 47, 57-110, 1993.
http://www.math.technion.ac.il/~rl/docs/uniform.pdf. Maeder, R. E. "The Stellated Icosahedra." Mathematica
in Education 3, 5-11, 1994. http://library.wolfram.com/infocenter/Articles/2519/. Maeder,
R. E. "30: Small Ditrigonal Icosidodecahedron." 1997. https://www.mathconsult.ch/static/unipoly/30.html. Wenninger,
M. J. Cambridge, England: Cambridge University Press, pp. 42 and 46-47
Models. Wenninger, M. J.
New York: Cambridge University Press, p. 46, 1989. Polyhedron
Models. Cite this as:
Weisstein, Eric W. "Small Triambic Icosahedron."
From --A Wolfram Web Resource. MathWorld https://mathworld.wolfram.com/SmallTriambicIcosahedron.html