Petrial great dodecahedron

Petrial great dodecahedron
Rank	3
Type	Regular
Space	Spherical
Notation
Schläfli symbol	$\{5,5/2\}^{\pi }$ $\{6,5/2\}_{5}$ $\left\{{\frac {6}{1,3}},{\frac {5}{2}}:5\right\}$
Elements
Faces	10 skew hexagons
Edges	30
Vertices	12
Vertex figure	Pentagram
Petrie polygons	12 pentagons
Holes	6 skew decagons
Measures (edge length 1)
Circumradius	${\sqrt {\frac {5+{\sqrt {5}}}{8}}}\approx 0.95106$
Related polytopes
Army	Ike
Regiment	Ike
Petrie dual	Great dodecahedron
φ 2	Petrial icosahedron
Conjugate	Petrial small stellated dodecahedron
Convex hull	Icosahedron
Orientation double cover	Blended icosahedron
Abstract & topological properties
Flag count	120
Euler characteristic	-8
Schläfli type	{6,5}
Orientable	No
Genus	10
Skeleton	Icosahedral graph
Properties
Symmetry	H3, order 120
Flag orbits	1
Convex	No
Dimension vector	(1,2,2)

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 those of 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".
• Hartley, Michael. "{6,5}*120c".
• Wedd, N. N10.5′