# Great dodecicosidodecahedron

Great dodecicosidodecahedron
Rank3
TypeUniform
SpaceSpherical
Notation
Coxeter diagramx5/3x5/2o3*a ()
Elements
Faces20 triangles, 12 pentagrams, 12 decagrams
Edges60+60
Vertices60
Vertex figureIsosceles trapezoid, edge lengths 1, (5–5)/2, (5–1)/2, (5–5)/2
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt{11-4\sqrt5}}{2} \approx 0.71689}$
Volume${\displaystyle 2\frac{45-17\sqrt5}{3} \approx 4.65790}$
Dihedral angles5/2–10/3: ${\displaystyle \arccos\left(-\frac{\sqrt5}{5}\right) \approx 116.56505^\circ}$
3–10/3: ${\displaystyle \arccos\left(-\sqrt{\frac{5-2\sqrt5}{15}}\right) \approx 100.81232^\circ}$
Central density10
Number of external pieces180
Level of complexity13
Related polytopes
ArmySemi-uniform Ti, edge lengths ${\displaystyle \sqrt5-2}$ (pentagons), ${\displaystyle \frac{3-\sqrt5}{2}}$ (between ditrigons)
DualGreat dodecacronic hexecontahedron
ConjugateSmall dodecicosidodecahedron
Convex coreTruncated dodecahedron
Abstract & topological properties
Flag count480
Euler characteristic–16
OrientableYes
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

The great dodecicosidodecahedron, or gaddid, is a uniform polyhedron. It consists of 20 triangles, 12 pentagrams, and 12 decagrams. One triangle, one pentagram, and two decagrams join at each vertex.

## Vertex coordinates

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

• ${\displaystyle \left(±\frac{\sqrt5-2}{2},\,±\frac12,\,±\frac12\right),}$

along with all even permutations of

• ${\displaystyle \left(0,\,±\frac{3-\sqrt5}{4},\,±\frac{5-\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{\sqrt5-1}{4},\,±\frac{\sqrt5-1}{2},\,±\frac{3-\sqrt5}{4}\right).}$

## Related polyhedra

The great dodecicosidodecahedron is the colonel of a three-member regiment that also includes the quasirhombicosidodecahedron and the great rhombidodecahedron.

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