Realization

A realization of a cube on a sphere.

A realization or embedding of an abstract polytope broadly refers to any way to build it in a given space. Within this community, these realizations are sometimes known as concrete polytopes, to contrast with abstract polytopes.

Different kinds of realizations include:

• Vertex mappings, arbitrary maps from the vertex set into some other space
  • Faithful realizations, non-degenerate realizations
  • Symmetric realizations, realizations whose symmetry group is isomorphic to the polytope's auomorphism group
• Realizations as convex polytopes
• Maps on a topological surface
• Incidence complexes may be realized in complex or quaternionic space as a set of affine subspaces called flats.