# Great dodecicosidodecahedron

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.

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

## 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 (   )
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 (   )