# Great snub dodecicosidodecahedron

Great snub dodecicosidodecahedron Rank3
TypeUniform
SpaceSpherical
Notation
Bowers style acronymGisdid
Coxeter diagrams5/3s5/2s3*a (   )
Elements
Faces20+60 triangles, 24 pentagrams
Edges60+60+60
Vertices60
Vertex figureIrregular hexagon, edge lengths 1, 1, 1, (5–1)/2, 1, (5–1)/2 Measures (edge length 1)
Circumradius$\frac{\sqrt2}{2} ≈ 0.70711$ Volume$\frac{5\sqrt2}{3} ≈ 2.35702$ Dihedral angles5/2–3 #1: $\arccos\left(-\sqrt{\frac{5+2\sqrt5-4\sqrt{5\sqrt5-10}}{15}}\right) ≈ 125.77490°$ 3–3: $\arccos\left(-\frac13\right) ≈ 109.47122°$ 5/2–3 #2: $\arccos\left(\sqrt{\frac{5+2\sqrt5+4\sqrt{5\sqrt5-10}}{15}}\right) ≈ 16.30368°$ Central density10
Number of pieces660
Level of complexity42
Related polytopes
ArmySemi-uniform srid
RegimentGisdid
DualGreat hexagonal hexecontahedron
ConjugateGreat snub dodecicosidodecahedron
Abstract properties
Euler characteristic-16
Topological properties
OrientableYes
Properties
SymmetryH3+, order 60
ConvexNo
NatureTame

The great snub dodecicosidodecahedron, or gisdid, is a uniform polyhedron. It consists of 60 snub triangles, 20 more triangles, and 24 pentagrams that fall in coplanar pairs of one prograde, one retrograde. Four triangles and two pentagrams meet at each vertex.

It is the only chiral uniform polyhedron with an achiral convex hull. As such, it cannot be made into a compound with its reflection. If the pentagrams are removed, however, the disnub icosahedron is formed.

This polyhedron's edges are a subset of those of the great dirhombicosidodecahedron, and it shares the same vertices.

## Vertex coordinates

A great snub dodecicosidodecahedron of edge length 1 has vertex coordinates given by all even permutations of:

• $\left(±\sqrt{\frac{\sqrt5-1-2\sqrt{\sqrt5-2}}{2}},\,±\sqrt{\frac{3-\sqrt5-\sqrt{10\sqrt5-22}}{8}},\,±\sqrt{\frac{2+\sqrt{2\sqrt5-2}}{8}}\right),$ • $\left(0,\,±\frac{\sqrt{3-\sqrt5}}{2},\,±\frac{\sqrt{\sqrt5-1}}{2}\right),$ • $\left(±\sqrt{\frac{3-\sqrt5+\sqrt{10\sqrt5-22}}{8}},\,±\sqrt{\frac{2-\sqrt{2\sqrt5-2}}{8}},\,±\sqrt{\frac{\sqrt5-1+2\sqrt{\sqrt5-2}}{8}}\right).$ ## Related polyhedra

o5/3o5/2o3*a truncations
Name OBSA CD diagram Picture
Great complex icosidodecahedron (degenerate, sissid+gike) gacid x5/3o5/2o3*a (   )
Great dodecicosidodecahedron gaddid x5/3x5/2o3*a (   )
(degenerate, double cover of gissid) o5/3x5/2o3*a (   )
(degenerate, ditdid+gidtid) o5/3x5/2x3*a (   )
Great complex icosidodecahedron (degenerate, sissid+gike) gacid o5/3o5/2x3*a (   )
(degenerate, double cover of sidhei) x5/3o5/2x3*a (   )
(degenerate, giddy+12(10/2)) x5/3x5/2x3*a (   )
Great snub dodecicosidodecahedron gisdid s5/3s5/2s2*a (   )