Petrial great dodecahedron

Petrial great dodecahedron
Rank	3
Type	Regular
Space	Spherical
Notation
Schläfli symbol	{5,5/2}π {6,5/2}5
Elements
Faces	10 skew hexagons
Edges	30
Vertices	12
Vertex figure	Pentagram
Measures (edge length 1)
Circumradius	$\sqrt{\frac{5+\sqrt5}{8}} ≈ 0.95106$
Related polytopes
Regiment	Icosahedron
Petrie dual	Great dodecahedron
Conjugate	Petrial small stellated dodecahedron
Convex hull	Icosahedron
Abstract properties
Flag count	120
Euler characteristic	-8
Schläfli type	{6,5}
Topological properties
Orientable	No
Genus	10
Properties
Symmetry	H3, order 120
Convex	No

The petrial great dodecahedron is a regular skew polyhedron and the Petrie dual of the great dodecahedron. It consists of 10 skew hexagons, has an Euler characteristic of -8, and it shares the vertices and edges of the icosahedron.

Vertex coordinates

The vertices of a petrial great dodecahedron 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 great icosahedron is the small dodecahemicosahedron, which is uniform.

External links

Wikipedia Contributors. "Petrie dual".