Hemi-icosahedron
Hemi-icosahedron | |
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![]() decagonal Schlegel diagram | |
Type | abstract regular polyhedron globally projective polyhedron |
Faces | 10 triangles |
Edges | 15 |
Vertices | 6 |
Euler char. | χ = 1 |
Vertex configuration | 3.3.3.3.3 |
Schläfli symbol | {3,5}/2 or {3,5}5 |
Symmetry group | A5, order 60 |
Dual polyhedron | hemi-dodecahedron |
Properties | non-orientable |
In geometry, a hemi-icosahedron is an abstract regular polyhedron, containing half the faces of a regular icosahedron. It can be realized as a projective polyhedron (a tessellation of the real projective plane by 10 triangles), 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.
Geometry
It has 10 triangular faces, 15 edges, and 6 vertices.
It is also related to the nonconvex uniform polyhedron, the tetrahemihexahedron, which could be topologically identical to the hemi-icosahedron if each of the 3 square faces were divided into two triangles.
Graphs
It can be represented symmetrically on faces, and vertices as Schlegel diagrams:
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Face-centered |
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The complete graph K6
It has the same vertices and edges as the 5-dimensional 5-simplex which has a complete graph of edges, but only contains half of the (20) faces.
From the point of view of graph theory this is an embedding of (the complete graph with 6 vertices) on a real projective plane. With this embedding, the dual graph is the Petersen graph --- see hemi-dodecahedron.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/7/75/Complete_graph_K6.svg/240px-Complete_graph_K6.svg.png)
See also
- 11-cell - an abstract regular 4-polytope constructed from 11 hemi-icosahedra.
- hemi-dodecahedron
- hemi-cube
- hemi-octahedron
References
- 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 links
- The hemi-icosahedron