Great retrosnub icosidodecahedron


In geometry, the great retrosnub icosidodecahedron or great inverted retrosnub icosidodecahedron is a nonconvex uniform polyhedron, indexed as U74. It has 92 faces, 150 edges, and 60 vertices. It is given a Schläfli symbol sr.

Cartesian coordinates

for the vertices of a great retrosnub icosidodecahedron are all the even permutations of
with an even number of plus signs, where
and
where τ = /2 is the golden mean and
ξ is the smaller positive real root of ξ3−2ξ=−1/τ, namely
or approximately 0.3264046.
Taking the odd permutations of the above coordinates with an odd number of plus signs gives another form, the enantiomorph of the other one. Taking the odd permutations with an even number of plus signs or vice versa results in the same two figures rotated by 90 degrees.
The circumradius for unit edge length is
where is the appropriate root of. The four positive real roots of the sextic in
are the circumradii of the snub dodecahedron, great snub icosidodecahedron, great inverted snub icosidodecahedron, and great retrosnub icosidodecahedron.