# Petrial great dodecahedron

Petrial great dodecahedron
Rank3
TypeRegular
SpaceSpherical
Notation
Schläfli symbol{5,5/2}π
{6,5/2}5
Elements
Faces10 skew hexagons
Edges30
Vertices12
Vertex figurePentagram
Measures (edge length 1)
Circumradius${\displaystyle \sqrt{\frac{5+\sqrt5}{8}} ≈ 0.95106}$
Related polytopes
RegimentIcosahedron
Petrie dualGreat dodecahedron
ConjugatePetrial small stellated dodecahedron
Convex hullIcosahedron
Abstract properties
Flag count120
Euler characteristic-8
Schläfli type{6,5}
Topological properties
OrientableNo
Genus10
Properties
SymmetryH3, order 120
ConvexNo

The petrial great dodecahedron is a regular skew polyhedron and the Petrie dual of the great dodecahedron. It consists of 10 skew hexagons, has an Euler characteristic of -8, and it shares the vertices and edges of the icosahedron.

## Vertex coordinates

The vertices of a petrial great dodecahedron of edge length 1 centered at the origin are the same as the icosahedron, being all cyclic permutations of:

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

## Related polyhedra

The rectification of the petrial great icosahedron is the small dodecahemicosahedron, which is uniform.