structural:expansion, structural:summarization, stylistic:style_transfer
Duality

For every convex polyhedron, there exists a dual polyhedron having

The dual of a convex polyhedron can be obtained by the process of polar reciprocation. Dual polyhedra exist in pairs, and the dual of a dual is just the original polyhedron again. Some polyhedra are self-dual, meaning that the dual of the polyhedron is congruent to the original polyhedron.

Abstract polyhedra also have duals, obtained by reversing the partial order defining the polyhedron to obtain its dual or opposite order. These have the same Euler characteristic and orientability as the initial polyhedron. However, this form of duality does not describe the shape of a dual polyhedron, but only its combinatorial structure. For some definitions of non-convex geometric polyhedra, there exist polyhedra whose abstract duals cannot be realized as geometric polyhedra under the same definition.