# Petrial dodecahedron

Petrial dodecahedron Rank3
TypeRegular
SpaceSpherical
Notation
Schläfli symbol$\{5,3\}^\pi$ $\{10,3\}_5$ Elements
Faces6 skew decagons
Edges30
Vertices20
Vertex figureTriangle, edge length (1+√5)/2
Related polytopes
Petrie dualDodecahedron
ConjugatePetrial great stellated dodecahedron
Convex hullDodecahedron
Abstract properties
Flag count120
Euler characteristic-4
Schläfli type{10,3}
Topological properties
OrientableNo
Genus6
Properties
ConvexNo

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:

• $\left(\pm\frac{1+\sqrt{5}}{4},\,\pm\frac{1+\sqrt{5}}{4},\,\pm\frac{1+\sqrt{5}}{4}\right),$ along with all permutations of

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

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