A hemi-octahedron is an abstract regular polyhedron

Abstract polytope

In mathematics, an abstract polytope, informally speaking, is a structure which considers only the combinatorial properties of a traditional polytope, ignoring many of its other properties, such as angles, edge lengths, etc...

, containing half the faces of a regular octahedron

Octahedron

In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....

.

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

Dual polyhedron

In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...

 is the hemicube.

It can be seen as an square pyramid

Square pyramid

In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it will have C4v symmetry.- Johnson solid :...

 without its base.It can be realized as a projective polyhedron

Projective polyhedron

In geometry, a projective polyhedron is a tessellation of the real projective plane. These are projective analogs of spherical polyhedra – tessellations of the sphere – and toroidal polyhedra – tessellations of the toroids....

 (a tessallation of the real projective plane

Real projective plane

In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold, that is, a one-sided surface. It cannot be embedded in our usual three-dimensional space without intersecting itself...

 by 4 triangles), which can be visualized by constructing the projective plane as a hemisphere

Hemisphere

Hemisphere may refer to:*Half of a sphereAs half of the Earth:*Any half of the Earth, see Hemispheres of the Earth, see:...

 where opposite points along the boundary are connected and dividing the hemisphere into four equal parts.

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

Digon

In geometry, a digon is a polygon with two sides and two vertices. It is degenerate in a Euclidean space, but may be non-degenerate in a spherical space.A digon must be regular because its two edges are the same length...

.