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Trigyrate Rhombicosidodecahedron


J75
J75Net

The trigyrate rhombicosidodecahedron is a convex equilateral solid obtained by rotating three of the pentagonal cupolas of a small rhombicosidodecahedron by 1/10 of a turn each. It is Johnson solid J_(75).

The unit trigyrate 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 gryrate rhombicosidodecahedron, parabigyrate rhombicosidodecahedron, and small rhombicosidodecahedron, from which it differs only by the relative positions of the three cupolas.


See also

Gyrate Rhombicosidodecahedron, Johnson Solid, Parabigyrate Rhombicosidodecahedron, Small 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. "Trigyrate Rhombicosidodecahedron." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TrigyrateRhombicosidodecahedron.html

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