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Gyrate Rhombicosidodecahedron

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J72
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The gyrate rhombicosidodecahedron is a convex equilateral solid obtained by rotating one of the pentagonal cupolas of a small rhombicosidodecahedron by 1/10 of a turn. It is Johnson solid J_(72).

The unit gyrate rhombicosidodecahedron has volume

V=20+(29)/3sqrt(5)
(1)

and Dehn invariant

D=60<3>_5-30<5>_1
(2)
=30sin^(-1)((5-4sqrt(5))/(15)),
(3)

where the first expression uses the basis of Conway et al. (1999). It can be dissected into the metabigyrate rhombicosidodecahedron, parabigyrate rhombicosidodecahedron, small rhombicosidodecahedron, and trigyrate rhombicosidodecahedron (E. Weisstein, Aug. 17, 2023).

See also

Johnson Solid, Metabigyrate Rhombicosidodecahedron, Parabigyrate Rhombicosidodecahedron, Small Rhombicosidodecahedron, Trigyrate Rhombicosidodecahedron

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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. "Gyrate Rhombicosidodecahedron." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GyrateRhombicosidodecahedron.html

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