Hemi-octahedron

{{short description|Abstract regular polyhedron with 4 triangular faces}}

{{no footnotes|date=November 2014}}

{{Infobox polyhedron

|image=Hemi-octahedron2.png

|type=abstract regular polyhedron
globally projective polyhedron

|schläfli={{math|{3,4}/2}} or {{math|{3,4}₃}}

|faces=4 triangles

|edges=6

|vertices=3

|euler={{math|1=χ = 1}}

|symmetry={{math|S₄}}, order 24

|vertex_config={{math|3.3.3.3}}

|dual=hemicube

|properties= non-orientable

}}

In geometry, a hemi-octahedron is an abstract regular polyhedron, containing half the faces of a regular octahedron.

It has 4 triangular faces, 6 edges, and 3 vertices. Its dual polyhedron is the hemicube.

It can be realized as a projective polyhedron (a tessellation of the real projective plane by 4 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 four equal parts. It can be seen as a square pyramid without its base.

It can be represented symmetrically as a hexagonal or square Schlegel diagram:

:160px

It has an unexpected property that there are two distinct edges between every pair of vertices – any two vertices define a digon.