Great retrosnub icosidodecahedron

Great retrosnub icosidodecahedron
Great retrosnub icosidodecahedron.png
Type Uniform star polyhedron
Elements F = 92, E = 150
V = 60 (χ = 2)
Faces by sides (20+60){3}+12{5/2}
Wythoff symbol | 2 3/2 5/3
Symmetry group I, [5,3]+, 532
Index references U74, C90, W117
Dual polyhedron Great pentagrammic hexecontahedron
Vertex figure Great retrosnub icosidodecahedron vertfig.png
Bowers acronym Girsid

In geometry, the great retrosnub icosidodecahedron or great inverted retrosnub icosidodecahedron is a nonconvex uniform polyhedron, indexed as U74. It has 92 faces (80 triangles and 12 pentagrams), 150 edges, and 60 vertices.[1] It is given a Schläfli symbol sr{3/2,5/3}.

3D model of a great retrosnub icosidodecahedron

Cartesian coordinatesEdit

Cartesian coordinates for the vertices of a great retrosnub icosidodecahedron are all the even permutations of

(±2α, ±2, ±2β),
(±(α−βτ−1/τ), ±(α/τ+β−τ), ±(−ατ−β/τ−1)),
(±(ατ−β/τ+1), ±(−α−βτ+1/τ), ±(−α/τ+β+τ)),
(±(ατ−β/τ−1), ±(α+βτ+1/τ), ±(−α/τ+β−τ)) and
(±(α−βτ+1/τ), ±(−α/τ−β−τ), ±(−ατ−β/τ+1)),

with an even number of plus signs, where

α = ξ−1/ξ


β = −ξ/τ+1/τ2−1/(ξτ),

where τ = (1+5)/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 (U29), great snub icosidodecahedron (U57), great inverted snub icosidodecahedron (U69), and great retrosnub icosidodecahedron (U74).

See alsoEdit


  1. ^ Maeder, Roman. "74: great retrosnub icosidodecahedron". MathConsult.

External linksEdit