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. See p. 22, where the omnitruncation is described as a "flag graph". Because the barycentric subdivision of any polytope can be realized as another polytope, 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: .
See also Expansion (geometry) Omnitruncated polyhedron
References
Further reading Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, (pp. 145–154 Chapter 8: Truncation, p 210 Expansion) Norman Johnson Uniform Polytopes, Manuscript (1991) N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
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