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 .

The unit trigyrate rhombicosidodecahedron has volume

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.