hemi-dodecahedron

{{short description|Abstract regular polyhedron with 6 pentagonal faces}}

{{Infobox polyhedron

|image=Hemi-Dodecahedron2.PNG

|caption=Decagonal Schlegel diagram

|schläfli={{math|{5,3}/2}} or {{math|{5,3}₅}}

|faces=6 pentagons

|edges=15

|vertices=10

|euler={{math|1=χ = 1}}

|symmetry={{math|A₅}}, order 60

|vertex_config={{math|5.5.5}}

}}

In geometry, 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.

{{citation | last1 = McMullen | first1 = Peter| author1-link=Peter McMullen | first2 = Egon | last2 = Schulte | chapter = 6C. Projective Regular Polytopes | title = Abstract Regular Polytopes | edition = 1st | publisher = Cambridge University Press | isbn = 0-521-81496-0 |date=December 2002 | pages = [https://books.google.com/books?id=JfmlMYe6MJgC&pg=PA162 162–165] }}