Duality
Every polytope has a dual, a polytope in which the partial order is reversed: the Hasse diagram of the dual is that of the original turned upside-down. In an n-polytope, each of the original k-faces maps to an (n − k − 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.
Read more about this topic: Abstract Polytope