Hemi-dodecahedron

Hemi-dodecahedron
	decagonal schlegel diagram
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.

Projections

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

From the point of view of graph theory this is an embedding of the 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

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McMullen, Peter; Schulte, Egon (December 2002), "6C. Projective Regular Polytopes", Abstract Regular Polytopes (1st ed.), Cambridge University Press, pp. 162–165, ISBN 0-521-81496-0

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