# Hemi-icosahedron

Polyhedron_Type=abstract regular polyhedron

Face_List=10triangle s

Edge_Count=15

Vertex_Count=6

Symmetry_Group=A_{5}

Vertex_List=3.3.3.3.3

Dual=hemi-dodecahedron

Property_List= non-orientableA**hemi-icosahedron**is an abstract regular polyhedron, containing half the faces of a regularicosahedron . It exists on a hemisphere as aprojective 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 , thetetrahemihexahedron , 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 $K\_6$ (thecomplete graph with 6 vertices) on aprojective plane . With thisembedding, thedual graph is thePetersen graph --- seehemi-dodecahedron .**See also***

11-cell - an abstract regularpolychoron constructed from 11 hemi-icosahedra.

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