Pentagonal Icositetrahedron


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The pentagonal icositetrahedron is the 24-faced dual polyhedron of the snub cube. It is illustrated above together with a wireframe version and a net that can be used for its construction.

The mineral cuprite (Cu_2O) forms in pentagonal icositetrahedral crystals (Steinhaus 1999, pp. 207 and 209).

It is Wenninger dual W_(17).


Because it is the dual of the chiral snub cube, the pentagonal icositetrahedron also comes in two enantiomorphous forms, known as laevo (left) and dextro (right). An attractive dual of the two enantiomers superposed on one another is illustrated above.

Solids inscribed in a pentagonal icositetrahedron

A cube, octahedron, and stella octangula can all be inscribed on the vertices of the pentagonal icositetrahedron (E. Weisstein, Dec. 25, 2009).

Surprisingly, the tribonacci constant t is intimately related to the metric properties of the pentagonal icositetrahedron cube.

Its irregular pentagonal faces have vertex angles of

theta_1=cos^(-1)[(4x^3-4 x^2+1)_1]
=114.812... degrees

(four times) and

=80.7517... degrees

(once), where (P(x))_n is a polynomial root and t is the tribonacci constant.

The dual formed from a snub cube with unit edge length has side lengths


The circumradius R is given by


The surface area S given by


and volume V given by


See also

Archimedean Dual, Archimedean Solid, Icositetrahedron, Pentagonal Icositetrahedral Graph, Snub Cube, Snub Cube-Pentagonal Icositetrahedron Compound, Tribonacci Constant

Explore with Wolfram|Alpha


Holden, A. Shapes, Space, and Symmetry. New York: Columbia University Press, p. 55, 1971.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999.Wenninger, M. J. Dual Models. Cambridge, England: Cambridge University Press, p. 28, 1983.

Cite this as:

Weisstein, Eric W. "Pentagonal Icositetrahedron." From MathWorld--A Wolfram Web Resource.

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