# Petrial icosahedron

Petrial icosahedron
Rank3
TypeRegular
SpaceSpherical
Notation
Schläfli symbol${\displaystyle \{3,5\}^\pi}$
${\displaystyle \{10,5\}_3}$
Elements
Faces6 skew decagons
Edges30
Vertices12
Vertex figurePentagon, edge length 1
Related polytopes
Petrie dualIcosahedron
ConjugatePetrial great icosahedron
Convex hullIcosahedron
Abstract properties
Flag count120
Euler characteristic-12
Schläfli type{10,5}
Topological properties
OrientableNo
Genus14
Properties
ConvexNo

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 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.