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Pentagonal Gyrocupolarotunda

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The pentagonal gyrocupolarotunda is a convex equilateral solid that is Johnson solid J_(33).

The unit pentagonal gyrocupolarotunda has volume

V=5/(12)(11+5sqrt(5))
(1)

and Dehn invariant

D=-15<3>_5+(25)/2<5>_1
(2)
=-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 pentagonal orthocupolarotunda, from which it differs only by the relative rotation of the top and bottom cupola and rotunda.

See also

Augmentation, Truncated Dodecahedron

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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.Johnson, N. W. "Convex Polyhedra with Regular Faces." Canad. J. Math. 18, 169-200, 1966.

Cite this as:

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

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