Omnitruncation

In geometry, an omnitruncation of a convex polytope is a simple polytope of the same dimension, having a vertex for each flag of the original polytope and a facet for each face of any dimension of the original polytope. Omnitruncation is the dual operation to barycentric subdivision.[1] Because the barycentric subdivision of any polytope can be realized as another polytope,[2] the same is true for the omnitruncation of any polytope.

When omnitruncation is applied to a regular polytope (or honeycomb) it can be described geometrically as a Wythoff construction that creates a maximum number of facets. It is represented in a Coxeter–Dynkin diagram with all nodes ringed.

It is a shortcut term which has a different meaning in progressively-higher-dimensional polytopes:

  1. ^ Matteo, Nicholas (2015), Convex Polytopes and Tilings with Few Flag Orbits (Doctoral dissertation), Northeastern University, ProQuest 1680014879 See p. 22, where the omnitruncation is described as a "flag graph".
  2. ^ Ewald, G.; Shephard, G. C. (1974), "Stellar subdivisions of boundary complexes of convex polytopes", Mathematische Annalen, 210: 7–16, doi:10.1007/BF01344542, MR 0350623