# Hemi-icosahedron

Infobox Polyhedron

Polyhedron_Type=abstract regular polyhedron
Face_List=10 triangles
Edge_Count=15
Vertex_Count=6
Symmetry_Group=A5
Vertex_List=3.3.3.3.3
Dual=hemi-dodecahedron
Property_List= non-orientable
A hemi-icosahedron is an abstract regular polyhedron, containing half the faces of a regular icosahedron. It exists on a hemisphere as a projective plane where opposite points along the boundary are connected.

It has 10 triangular faces, 15 edges, and 6 vertices. It has the same vertices and edges as the 5-dimensional polytope, the 5-simplex, but only contains half of the (20) faces.

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.

The complete graph K6

From the point of view of graph theory this is an embedding of $K_6$ (the complete graph with 6 vertices) on a projective plane. With thisembedding, the dual graph is the Petersen graph --- see hemi-dodecahedron.

*11-cell - an abstract regular polychoron constructed from 11 hemi-icosahedra.

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