# Petrial dodecahedron

Petrial dodecahedron
Rank3
TypeRegular
SpaceSpherical
Notation
Schläfli symbol${\displaystyle \{5,3\}^{\pi }}$
${\displaystyle \{10,3\}_{5}}$
${\displaystyle \left\{{\frac {10}{1,5}},3:5\right\}}$
Elements
Faces6 skew decagons
Edges30
Vertices20
Vertex figureTriangle, edge length (1+√5)/2
Related polytopes
ArmyDoe
RegimentDoe
Petrie dualDodecahedron
κ ?Great stellated dodecahedron
ConjugatePetrial great stellated dodecahedron
Convex hullDodecahedron
Abstract & topological properties
Flag count120
Euler characteristic-4
Schläfli type{10,3}
OrientableNo
Genus6
Properties
SymmetryH3
Flag orbits1
ConvexNo
Dimension vector(1,2,2)

The petrial dodecahedron is a regular skew polyhedron consisting of 6 skew decagons. The petrial dodecahedron is the Petrie dual of the dodecahedron, so therefore it shares its edges and vertices with the dodecahedron. It has an Euler characteristic of -4.

## Vertex coordinates

The vertices of the petrial dodecahedron are identical to those of the dodecahedron, being:

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

along with all permutations of

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

## Related polyhedra

The rectification of the Petrial dodecahedron is the small icosihemidodecahedron, which is uniform.