Dyadicity

Dyadicity, also commonly known as the diamond property, is a key property of a polytope which is part of most formal definitions. It essentially states that exactly two facets must meet at any ridge. This generalizes the rule that every edge must have two vertices, that two edges must meet at a vertex in a polygon, and that two faces must meet at an edge in a polyhedron.

A polytope that isn't dyadic is called exotic by Jonathan Bowers.

Exotic polytopes
The best known example of an exotic polytope is the great disnub dirhombidodecahedron, also known as Skilling's figure. It is the single uniform polyhedron that results from relaxing dyadicity to requiring only that evenly many faces meet at each edge, while still excluding polytopes separable into compounds.

A crucial problem with studying exotic polytopes is that flag changes cannot be uniquely defined. As a consequence, concepts like volume and orientability become meaningless.