# Great dodecicosidodecahedron

Great dodecicosidodecahedron
Rank3
TypeUniform
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${\displaystyle {\frac {\sqrt {11-4{\sqrt {5}}}}{2}}\approx 0.71689}$
Volume${\displaystyle 2{\frac {45-17{\sqrt {5}}}{3}}\approx 4.65790}$
Dihedral angles5/2–10/3: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$
3–10/3: ${\displaystyle \arccos \left(-{\sqrt {\frac {5-2{\sqrt {5}}}{15}}}\right)\approx 100.81232^{\circ }}$
Central density10
Number of external pieces180
Level of complexity13
Related polytopes
ArmySemi-uniform Ti, edge lengths ${\displaystyle {\sqrt {5}}-2}$ (pentagons), ${\displaystyle {\frac {3-{\sqrt {5}}}{2}}}$ (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

• ${\displaystyle \left(\pm {\frac {{\sqrt {5}}-2}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right),}$

along with all even permutations of

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

## 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 ()