# Great snub dodecicosidodecahedron

Great snub dodecicosidodecahedron
Rank3
TypeUniform
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${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Volume${\displaystyle {\frac {5{\sqrt {2}}}{3}}\approx 2.35702}$
Dihedral angles5/2–3 #1: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}-4{\sqrt {5{\sqrt {5}}-10}}}{15}}}\right)\approx 125.77490^{\circ }}$
3–3: ${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
5/2–3 #2: ${\displaystyle \arccos \left({\sqrt {\frac {5+2{\sqrt {5}}+4{\sqrt {5{\sqrt {5}}-10}}}{15}}}\right)\approx 16.30368^{\circ }}$
Central density10
Number of external pieces660
Level of complexity42
Related polytopes
ArmySemi-uniform srid, edge lengths ${\displaystyle {\sqrt {\frac {3-{\sqrt {5}}-{\sqrt {10{\sqrt {5}}-22}}}{2}}}}$ (pentagons), ${\displaystyle {\sqrt {\frac {{\sqrt {5}}-1-2{\sqrt {{\sqrt {5}}-2}}}{2}}}}$ (triangles)
RegimentGisdid
DualGreat hexagonal hexecontahedron
ConjugateGreat snub dodecicosidodecahedron
Abstract & topological properties
Flag count720
Euler characteristic-16
OrientableYes
Genus9
Properties
SymmetryH3+, order 60
ChiralYes
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:

• ${\displaystyle \left(\pm {\sqrt {\frac {{\sqrt {5}}-1-2{\sqrt {{\sqrt {5}}-2}}}{8}}},\,\pm {\sqrt {\frac {3-{\sqrt {5}}-{\sqrt {10{\sqrt {5}}-22}}}{8}}},\,\pm {\sqrt {\frac {2+{\sqrt {2{\sqrt {5}}-2}}}{8}}}\right),}$
• ${\displaystyle \left(0,\,\pm {\frac {\sqrt {3-{\sqrt {5}}}}{2}},\,\pm {\frac {\sqrt {{\sqrt {5}}-1}}{2}}\right),}$
• ${\displaystyle \left(\pm {\sqrt {\frac {3-{\sqrt {5}}+{\sqrt {10{\sqrt {5}}-22}}}{8}}},\,\pm {\sqrt {\frac {2-{\sqrt {2{\sqrt {5}}-2}}}{8}}},\,\pm {\sqrt {\frac {{\sqrt {5}}-1+2{\sqrt {{\sqrt {5}}-2}}}{8}}}\right).}$

## Related polyhedra

o5/3o5/2o3*a truncations
Name OBSA CD diagram Picture
Great complex icosidodecahedron (degenerate, sissid+gike) gacid x5/3o5/2o3*a ()