# Great dodecahedron

Great dodecahedron Rank3
TypeRegular
Notation
Coxeter diagram     Schläfli symbol$\{5,5/2\}$ $\{5,5\mid 3\}$ Elements
Faces12 pentagons
Edges30
Vertices12
Vertex figurePentagram, edge length (1+5)/2 Petrie polygons10 skew hexagons
Holes20 triangles
Measures (edge length 1)
Circumradius${\sqrt {\frac {5+{\sqrt {5}}}{8}}}\approx 0.95106$ Edge radius${\frac {1+{\sqrt {5}}}{4}}\approx 0.80902$ Inradius${\sqrt {\frac {5+{\sqrt {5}}}{40}}}\approx 0.42533$ Volume${\frac {5+3{\sqrt {5}}}{4}}\approx 2.92705$ Dihedral angle$\arccos \left({\frac {\sqrt {5}}{5}}\right)\approx 63.43495^{\circ }$ Central density3
Number of external pieces60
Level of complexity3
Related polytopes
ArmyIke
RegimentIke
DualSmall stellated dodecahedron
Petrie dualPetrial great dodecahedron
φ 2 Icosahedron
ConjugateSmall stellated dodecahedron
Convex coreDodecahedron
Abstract & topological properties
Flag count120
Euler characteristic–6
Schläfli type{5,5}
SurfaceBring's surface
OrientableYes
Genus4
Properties
SymmetryH3, order 120
ConvexNo
NatureTame
History
Discovered byJohannes Kepler[note 1]
First discovered1613

The great dodecahedron, or gad, is one of the four Kepler-Poinsot solids. It has 12 pentagons as faces, joining 5 to a vertex in a pentagrammic fashion.

It is in the same regiment as the icosahedron, and comes from using the icosahedron's vertex figure pentagons as the faces.

It is the second stellation of the dodecahedron.

Great dodecahedra appear as cells in two star regular polychora, namely the great hecatonicosachoron and great grand hecatonicosachoron.

## Vertex coordinates

Its vertices are the same as those of its regiment colonel, the icosahedron.

## In vertex figures

The great dodecahedron appears as a vertex figure of two Schläfli–Hess polychora.

Name Picture Schläfli symbol Edge length
Grand stellated hecatonicosachoron {5/2,5,5/2} ${\frac {{\sqrt {5}}-1}{2}}$ Faceted hexacosichoron {3,5,5/2} $1$ ## Related polyhedra

Abstractly the great dodecahedron is a quotient of the order-5 pentagonal tiling. Specifically it is $\{5,5\mid 3\}$ , a tessellation of Bring's surface. It is also abstractly equivalent to its conjugate, the small stellated dodecahedron.