Pentagonal Antiprism

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The pentagonal antiprism is the antiprism having a regular pentagon for the top and bottom bases. Since the bases are rotated with respect to each other, it has a ribbon of 10 triangular faces connecting those pentagons, giving a total of 12 faces, making it a (non-regular) dodecahedron.

The equilateral pentagonal antiprism is also the uniform polyhedron with Maeder index 77 (Maeder 1997) and Har'El index 2 (Har'El 1993) as well as a canonical polyhedron.

👁 PentagonalAntiprismConvexHulls

The pentagonal antiprism is the convex hull of the pentagrammic crossed antiprism.

The unit pentagonal antiprism has volume

👁 V=1/6(5+2sqrt(5))

(1)

and Dehn invariant

👁 D	👁 =	👁 30<3>_5+5<5>_1	(2)
👁 Image	👁 =	👁 5[3cot^(-1)(2/(sqrt(5)))+tan^(-1)2],	(3)

where the first expression uses the basis of Conway et al. (1999). It can be dissected into the metabidiminished icosahedron (E. Weisstein, Sep. 6, 2023).

👁 PentagonalAntiprismAndDual

Its dual polyhedron is the pentagonal trapezohedron, with which it is illustrated above.

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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.Har'El, Z. "Uniform Solution for Uniform Polyhedra." Geometriae Dedicata 47, 57-110, 1993.Johnson, N. W. "Convex Polyhedra with Regular Faces." Canad. J. Math. 18, 169-200, 1966.Maeder, R. E. "77: Pentagonal Antiprism." 1997. https://www.mathconsult.ch/static/unipoly/7.html.

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Pentagonal Antiprism

Cite this as:

Weisstein, Eric W. "Pentagonal Antiprism." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PentagonalAntiprism.html

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