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 is dyadic whenever exactly two facets meet at each ridge, and all facets are dyadic as well. Any point is considered dyadic by default.
An equivalent reformulation states that a polytope is dyadic whenever, for every (d − 1)-element and (d + 1)-element that are incident to one another, there are exactly two d-elements incident to both. That is to say, a polytope is dyadic when every section of rank 1 has exactly four elements. The Hasse diagram of such a section looks like a diamond, hence the name "diamond property."
A polytopoid is a polytope-like object where any positive even number of facets may meet at each ridge, although facets are still required to be dyadic. Depending on dimension, we also have the terms polygonoid, polyhedroid, etc. A polytopoid that isn't dyadic is called exotic by Jonathan Bowers.
Some properties and classifications of polytopes, such as uniformity, still apply to polytopoids. The best known example of an exotic polytopoid is the great disnub dirhombidodecahedron, also known as Skilling's figure. It is the single uniform polyhedroid 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 polytopoids is that flag changes cannot be uniquely defined. As a consequence, concepts like volume and orientability become meaningless. For cases where an even number of facets meet at a ridge, one can solve this problem by treating multiple otherwise identical polytopoids as different based on winding/volume, though this method requires consideration of non-combinatorial properties.
- ↑ Bowers, Jonathan (April 2014). "Glossary". Retrieved June 23, 2021.
- ↑ Skilling, John (1975), "The complete set of uniform polyhedra", Philosophical Transactions of the Royal Society A, 278 (1278): 111–135, doi:10.1098/rsta.1975.0022