# Petrial icosahedron

Petrial icosahedron
Rank3
TypeRegular
SpaceSpherical
Notation
Schläfli symbol${\displaystyle \{3,5\}^{\pi }}$
${\displaystyle \{10,5\}_{3}}$
${\displaystyle \left\{{\frac {10}{1,5}},5:3\right\}}$
Elements
Faces6 skew decagons
Edges30
Vertices12
Vertex figurePentagon, edge length 1
Petrie polygons20 triangles
Holes10 skew hexagons
Related polytopes
ArmyIke
RegimentIke
Petrie dualIcosahedron
φ 2 Petrial great dodecahedron
κ ?Small stellated dodecahedron
ConjugatePetrial great icosahedron
Convex hullIcosahedron
Orientation double coverBlended great dodecahedron
Abstract & topological properties
Flag count120
Euler characteristic-12
Schläfli type{10,5}
OrientableNo
Genus14
Properties
SymmetryH3
Flag orbits1
ConvexNo
Dimension vector(1,2,2)

The petrial icosahedron is a regular skew polyhedron and is the Petrie dual of the icosahedron, and so it shares both of its vertices and edges with the icosahedron. It consists of 6 skew decagons and has an Euler characteristic of -12.

## Vertex coordinates

The vertices of a petrial icosahedron of edge length 1 centered at the origin are the same as those of 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 icosahedron is the small dodecahemidodecahedron, which is uniform.