Petrial icosahedron

Petrial icosahedron
Rank	3
Type	Regular
Space	Spherical
Notation
Schläfli symbol	$\{3,5\}^\pi$ $\{10,5\}_3$
Elements
Faces	6 skew decagons
Edges	30
Vertices	12
Vertex figure	Pentagon, edge length 1
Related polytopes
Petrie dual	Icosahedron
Conjugate	Petrial great icosahedron
Convex hull	Icosahedron
Abstract properties
Flag count	120
Euler characteristic	-12
Schläfli type	{10,5}
Topological properties
Orientable	No
Genus	14
Properties
Convex	No

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:

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

Related polyhedra

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

External links

• Wikipedia Contributors. "Petrie dual".
• Hartley, Michael. "{10,5}*120b".

Wikipedia Contributors. "Petrie dual".