Page "Dual polyhedron" Paragraph 37

A self-dual polyhedron must have the same number of vertices as faces.

We can distinguish between structural ( topological ) duality and geometrical duality.

The topological structure of a self-dual polyhedron is also self-dual.

Whether or not such a polyhedron is also geometrically self-dual will depend on the particular geometrical duality being considered.

For example, every polygon is topologically self-dual ( it has the same number of vertices as edges, and these are switched by duality ), but will not in general be geometrically self-dual ( up to rigid motion, for instance ) – regular polygons are geometrically self-dual ( all angles are congruent, as are all edges, so under duality these congruences swap ), but irregular polygons may not be geometrically self-dual.