Dyadicity

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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.

Definitions[edit | edit source]

There are several equivalent definitions of the dyadicity.

Recursive[edit | edit source]

A polytope is dyadic when exactly two facets meet at every ridge, and all facets are dyadic as well.

Since points have no ridges they are vacuously dyadic and form a base case.

Sections[edit | edit source]

A polytope is dyadic when every section of rank 1 is a dyad. The Hasse diagram of such a section looks like a diamond, hence the name "diamond property".

Incidence[edit | edit source]

A polytope is dyadic when, for every (d  − 1)-element and (d  + 1)-element that are incident to one another, there are exactly two d -elements incident with both.

Weaker forms of dyadicity[edit | edit source]

Incidence complex[edit | edit source]

The complete graph is an abstractly regular polytope complex. Three edges meet at every vertex, and each edge is dyadic.

Incidence complexes are required to have a weaker form of dyadicity, such that every section of rank 1 has at least 2 proper elements.

If an incidence complex's facets are dyadic then it is a polytope complex.

Exotic polytopoids[edit | edit source]

The complex dipentagon is an exotic polygonoid with 4 edges meeting at each vertex.

A polytopoid[note 1] 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. It is not required that every ridge has the same number of incident facets. Depending on dimension, we also have the terms polygonoid, polyhedroid, etc. A polytopoid that isn't dyadic is called exotic by Jonathan Bowers.[1]

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.[2]

A crucial problem with studying non-dyadic objects 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.

See also[edit | edit source]

Notes[edit | edit source]

  1. The term polytopoid may also be used to refer to any polytope-like object.

References[edit | edit source]

  1. Bowers, Jonathan (April 2014). "Glossary". Retrieved June 23, 2021.
  2. 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