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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 
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The unit trigyrate rhombicosidodecahedron has volume
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(1)
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and Dehn invariant
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(2)
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(3)
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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.
