Realization

A realization or embedding of an abstract polytope refers to any way to build it in a given space. The same abstract polytope may have multiple realizations in different spaces with different properties, as for instance the Petrial tetrahedron in spherical space, or the hemicube in the projective plane.

In the most general sense, only vertices of the polytope are embedded, while edges, faces, and everything else is either inferred or treated implicitly. However, embeddings of polyhedra might also represent edges as paths on a surface, and faces as connected regions bounded by the edges. This particular type of embedding is called a map.

Realizations are often identified with the polytopes themselves. Within this community, these realizations are sometimes known as concrete polytopes, to contrast with abstract polytopes.