Great complex icosidodecahedron

Great complex icosidodecahedron
Great complex icosidodecahedron.png
Type Uniform star polyhedron
Elements F = 32, E = 60 (30x2)
V = 12 (χ = -16)
Faces by sides 20{3}+12{5/2}
Wythoff symbol 5 | 3 5/3
Symmetry group Ih, [5,3], *532
Index references U-, C-, W-
Dual polyhedron Great complex icosidodecacron
Vertex figure Great complex icosidodecahedron verf.png
Bowers acronym Gacid

In geometry, the great complex icosidodecahedron is a degenerate uniform star polyhedron. It has 12 vertices, and 60 (doubled) edges, and 32 faces, 12 pentagrams and 20 triangles. All edges are doubled (making it degenerate), sharing 4 faces, but are considered as two overlapping edges as topological polyhedron.

It can be constructed from a number of different vertex figures.

As a compoundEdit

The great complex icosidodecahedron can be considered a compound of the small stellated dodecahedron, {5/2,5}, and great icosahedron, {3,5/2}, sharing the same vertices and edges, while the second is hidden, being completely contained inside the first.

Its two-dimensional analogue would be the compound of a regular pentagon, {5}, and regular pentagram, {5/2}. These shapes would share vertices, similarly to how its 3D equivalent shares edges.

Compound polyhedron
Small stellated dodecahedron Great icosahedron Compound
Compound polygon
Pentagon Pentagram Compound

See alsoEdit


  • Coxeter, Harold Scott MacDonald; Longuet-Higgins, M. S.; Miller, J. C. P. (1954), "Uniform polyhedra", Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences, 246: 401–450, doi:10.1098/rsta.1954.0003, ISSN 0080-4614, JSTOR 91532, MR 0062446 (Table 6, degenerate cases)
  • Weisstein, Eric W. "Great complex icosidodecahedron". MathWorld.
  • Klitzing, Richard. "3D uniform polyhedra o5/3x3o5*a and o3/2x5/2o5*a - gacid".