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:
Uniform polytope truncation operators
For regular polygons: An ordinary truncation, .
Coxeter-Dynkin diagram
For uniform polyhedra (3-polytopes): A cantitruncation, . (Application of both cantellation and truncation operations)
Coxeter-Dynkin diagram:
For uniform polychora: A runcicantitruncation, . (Application of runcination, cantellation, and truncation operations)
Coxeter-Dynkin diagram: , ,
For uniform polytera (5-polytopes): A steriruncicantitruncation, t0,1,2,3,4{p,q,r,s}. . (Application of sterication, runcination, cantellation, and truncation operations)
Coxeter-Dynkin diagram: , ,
For uniform n-polytopes: .
^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".
^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
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...
until edges disappear), and cantellation (cutting edges and vertices). Omnitruncation is an operation that combines truncation and cantellation. Snubbing...
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can have faces that are not simplices. This is the dual operation to omnitruncation. The vertices of the barycentric subdivision correspond to the faces...
are applied at once, the operation can be more generally called an omnitruncation. One special operation, called alternation, removes alternate vertices...
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symbols are listed first, followed by the actual mixed Wythoff symbols. Omnitruncations are also called cantitruncations or truncated rectifications (tr),...