# Great dodecahedron

Great dodecahedron
Rank3
TypeRegular
Notation
Coxeter diagramo5/2o5x ()
Schläfli symbol
• {5,5/2}
• {5,5∣3}[1]
Elements
Faces12 pentagons
Edges30
Vertices12
Vertex figurePentagram, edge length (1+5)/2
Petrie polygons10 skew hexagons
Holes20 triangles
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {5+{\sqrt {5}}}{8}}}\approx 0.95106}$
Edge radius${\displaystyle {\frac {1+{\sqrt {5}}}{4}}\approx 0.80902}$
Inradius${\displaystyle {\sqrt {\frac {5+{\sqrt {5}}}{40}}}\approx 0.42533}$
Volume${\displaystyle {\frac {5+3{\sqrt {5}}}{4}}\approx 2.92705}$
Dihedral angle${\displaystyle \arccos \left({\frac {\sqrt {5}}{5}}\right)\approx 63.43495^{\circ }}$
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
SkeletonIcosahedral graph
Properties
SymmetryH3, order 120
Flag orbits1
ConvexNo
NatureTame

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 the icosahedron, its regiment colonel.

## Related polytopes

### Alternative realizations

The great dodecahedron and the small stellated dodecahedron are conjugates. Thus they are both faithful symmetric realizations of the same abstract regular polytope, {5,5∣3}. This abstract polytope is a quotient of the order-5 pentagonal tiling which tessellates Bring's surface. There are in total 6 faithful symmetric realizations of the underlying abstract polytope. The great dodecahedron and the small stellated dodecahedron are the only pure faithfully symmetric realizations, the others are the results of blending those two along with {5,5∣3}/2.

Faithful symmetric realizations of {5,5∣3}
Dimension Components Name
3 Great dodecahedron Great dodecahedron
3 Small stellated dodecahedron Small stellated dodecahedron
6
8
8
11

### Compounds

Two uniform polyhedron compounds are composed of great dodecahedra:

### 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} ${\displaystyle {\frac {{\sqrt {5}}-1}{2}}}$
Faceted hexacosichoron {3,5,5/2} ${\displaystyle 1}$