# Hemi-icosahedron

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.

Hemi-icosahedron
Typeabstract regular polyhedron
globally projective polyhedron
Faces10 triangles
Edges15
Vertices6
Vertex configuration3.3.3.3.3
Schläfli symbol{3,5}/2 or {3,5}5
Symmetry groupA5, order 60
Dual polyhedronhemi-dodecahedron
Propertiesnon-orientable
Euler characteristic 1

## 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:

## 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 ${\displaystyle K_{6}}$  (the complete graph with 6 vertices) on a real projective plane. With this embedding, the dual graph is the Petersen graph --- see hemi-dodecahedron.

The complete graph K6 represents the 6 vertices and 15 edges of the hemi-icosahedron