Hemi-octahedron

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

Hemi-octahedron
Type	abstract regular polyhedron globally projective polyhedron
Faces	4 triangles
Edges	6
Vertices	3
Vertex configuration	3.3.3.3
Schläfli symbol	{3,4}/2 or {3,4}3
Symmetry group	S4, order 24
Dual polyhedron	hemicube
Properties	non-orientable Euler characteristic 1

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:

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