Petrial dodecahedron

Petrial dodecahedron
Rank	3
Type	Regular
Space	Spherical
Notation
Schläfli symbol	$\{5,3\}^\pi$ $\{10,3\}_5$
Elements
Faces	6 skew decagons
Edges	30
Vertices	20
Vertex figure	Triangle, edge length (1+√5)/2
Related polytopes
Petrie dual	Dodecahedron
Conjugate	Petrial great stellated dodecahedron
Convex hull	Dodecahedron
Abstract properties
Flag count	120
Euler characteristic	-4
Schläfli type	{10,3}
Topological properties
Orientable	No
Genus	6
Properties
Convex	No

The petrial dodecahedron is a regular skew polyhedron consisting of 6 skew decagons. The petrial dodecahedron is the Petrie dual of the dodecahedron, so therefore it shares its edges and vertices with the dodecahedron. It has an Euler characteristic of -4.

Vertex coordinates

The vertices of the petrial dodecahedron are identical to those of the dodecahedron, being:

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

along with all permutations of

• $\left(\pm\frac{3+\sqrt{5}}{4},\,\pm\frac{1}{2},\,0\right).$

Related polyhedra

The rectification of the Petrial dodecahedron is the small icosihemidodecahedron, which is uniform.

External links

• Wikipedia Contributors. "Petrie dual".
• Hartley, Michael. "{10,3}*120b".