Editing Abstract polytope (section)

==Duality==
Every geometric polytope has a ''[[Dual polyhedron#Dual polytopes and tessellations|dual]]'' twin. Abstractly, the dual is the same polytope but with the ranking reversed in order: the Hasse diagram differs only in its annotations. In an ''n''-polytope, 
each of the original ''k''-faces maps to an (''n''&nbsp;−&nbsp;''k''&nbsp;−&nbsp;1)-face in the dual. Thus, for example, the ''n''-face maps to the (−1)-face. The dual of a dual is ([[isomorphic]] to) the original.

A polytope is self-dual if it is the same as, i.e. isomorphic to, its dual. Hence, the Hasse diagram of a self-dual polytope must be symmetrical about the horizontal axis half-way between the top and bottom. The square pyramid in the example above is self-dual.

The vertex figure at a vertex ''V'' is the dual of the facet to which ''V'' maps in the dual polytope.