# Hemi-dodecahedron

Hemi-dodecahedron
Type abstract regular polyhedron
globally projective polyhedron
Faces 6 pentagons
Edges 15
Vertices 10
Vertex configuration 5.5.5
Schläfli symbol {5,3}/2 or {5,3}5
Symmetry group A5, order 60
Dual polyhedron hemi-icosahedron
Properties non-orientable
Euler characteristic 1

A hemi-dodecahedron is an abstract regular polyhedron, containing half the faces of a regular dodecahedron. It can be realized as a projective polyhedron (a tessellation of the real projective plane by 6 pentagons), which can be visualized by constructing the projective plane as a hemisphere where opposite points along the boundary are connected and dividing the hemisphere into three equal parts.

It has 6 pentagonal faces, 15 edges, and 10 vertices.

It can be projected symmetrically inside of a 10-sided or 12-sided perimeter:

## Petersen graph

From the point of view of graph theory this is an embedding of Petersen graph on a real projective plane. With this embedding, the dual graph is K6 (the complete graph with 6 vertices) --- see hemi-icosahedron.

The six faces of the hemi-dodecahedron depicted as colored cycles in the Petersen graph