Hemi-dodecahedron

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.

Hemi-dodecahedron
Hemi-Dodecahedron2.PNG
Typeabstract regular polyhedron
globally projective polyhedron
Faces6 pentagons
Edges15
Vertices10
Vertex configuration5.5.5
Schläfli symbol{5,3}/2 or {5,3}5
Symmetry groupA5, order 60
Dual polyhedronhemi-icosahedron
Propertiesnon-orientable
Euler characteristic 1

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

ProjectionsEdit

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

 

Petersen graphEdit

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

See alsoEdit

ReferencesEdit

  • 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

External linksEdit