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.
^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
omnitruncation, geometry, omnitruncation, convex, polytope, simple, polytope, same, dimension, having, vertex, each, flag, original, polytope, facet, each, face, dimension, original, polytope, dual, operation, barycentric, subdivision, because, barycentric, su. 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 t 0 1 p t p 2 p displaystyle t 0 1 p t p 2p Coxeter Dynkin diagram For uniform polyhedra 3 polytopes A cantitruncation t 0 1 2 p q t r p q displaystyle t 0 1 2 p q tr p q Application of both cantellation and truncation operations Coxeter Dynkin diagram For uniform polychora A runcicantitruncation t 0 1 2 3 p q r displaystyle t 0 1 2 3 p q r 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 t 0 1 2 3 4 p q r s displaystyle t 0 1 2 3 4 p q r s Application of sterication runcination cantellation and truncation operations Coxeter Dynkin diagram For uniform n polytopes t 0 1 n 1 p 1 p 2 p n displaystyle t 0 1 n 1 p 1 p 2 p n Contents 1 See also 2 References 3 Further reading 4 External linksSee also EditExpansion geometry Omnitruncated polyhedronReferences Edit 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 0350623Further reading EditCoxeter H S M Regular Polytopes 3rd edition 1973 Dover edition ISBN 0 486 61480 8 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 1966External links EditWeisstein Eric W Expansion MathWorld Polyhedron operators vte Seed Truncation Rectification Bitruncation Dual Expansion Omnitruncation Alternations nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp t0 p q p q t01 p q t p q t1 p q r p q t12 p q 2t p q t2 p q 2r p q t02 p q rr p q t012 p q tr p q ht0 p q h q p ht12 p q s q p ht012 p q sr p q Retrieved from https en wikipedia org w index php title Omnitruncation amp oldid 1148759024, wikipedia, wiki, book, books, library,
