Degenerate polytope

A polytope is informally dubbed degenerate when it satisfies certain undesirable properties. These might possibly include coplanar, or coincident elements, skew elements, non-dyadicity, or being fissary.

Definitions

There is currently no single definition for what it means for a shape to be degenerate. As such, this term is ambiguous in formal discussion, unless the specific cases for which the term will be used are agreed upon beforehand.

McMullen and Schulte

In their study of abstract polytopes McMullen and Schulte define a polytope 𝓟 to be degenerate if an adjacent pair of its distinguished generators commute. That is if there is a pair of generators $\rho _{i}$ and $\rho _{i+1}$ such that $\rho _{i}\rho _{i+1}\rho _{i}\rho _{i+1}=1$.

For regular polytopes this is exactly those polytopes that have a 2 in their Schläfli type.

The Atlas of Small Regular Polytopes also uses this definition.

Bibliography

• McMullen, Peter (2007). "Four-Dimensional Regular Polyhedra" (PDF). Discrete & Computational Geometry.