Fissary polytope

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One of the two fissary noble polyhedra. They both have irregular stellated decagram vertex figures.

An abstract polytope or compound is fissary if it is itself compound or contains compound figures, such as vertex figures, edge figures, etc. Formally, define the figure of an element e (possibly the nullitope) as the set of all proper elements incident on e and with higher rank than e. A figure F is compound iff the relation of incidence partitions F into two or more equivalence classes. The nullitope having a compound figure is equivalent to the entire poset being compound.

A (non-compound) polyhedron is fissary when it has compound vertex figures. A polychoron is fissary when it has compound vertex or edge figures. A n-polytope is fissary if it has any compound m-dimensional figures where m ranges from 0 to n-3. As a consequence of dyadicity, an n-polytope cannot have a compound figure of rank n - 2 or higher.

The term "fissary" was coined by Jonathan Bowers to classify certain polytopes found in the search for uniforms. While well-defined, the term has to date only been used in the enthusiast community.

Fissary uniforms[edit | edit source]

Cross-sections and vertex figure of Sitphi, a vertex fissary uniform polychoron.

The only fissary uniform polyhedra are compounds. The first dimension to feature fissary uniform polytopes that are not compounds is 4, such as Sitphi (vertex fissary) and Dupti (edge fissary).

Fissary uniform polytopes may or may not be excluded from the total list of uniform polytopes in a given dimension, unlike exotic ones (having compound ridges or n-2 figures). However, fissaries dominate the uniform polytopes in high dimensions if allowed.