{{short description|Geometric operation}} 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.<ref>{{citation|title=Convex Polytopes and Tilings with Few Flag Orbits|type=Doctoral dissertation|last=Matteo|first=Nicholas|publisher= Northeastern University|year=2015|id={{ProQuest|1680014879}}}} See p. 22, where the omnitruncation is described as a "flag graph".</ref> Because the barycentric subdivision of any polytope can be realized as another polytope,<ref>{{citation | last1 = Ewald | first1 = G. | last2 = Shephard | first2 = G. C. | doi = 10.1007/BF01344542 | journal = Mathematische Annalen | mr = 350623 | pages = 7–16 | title = Stellar subdivisions of boundary complexes of convex polytopes | volume = 210 | year = 1974}}</ref> 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, <math>t_{0,1}\{ p \} = t\{ p\} = \{ 2p\}</math>. *** Coxeter-Dynkin diagram {{CDD|node_1|p|node_1}} ** For uniform polyhedra (3-polytopes): A cantitruncation, <math>t_{0,1,2}\{ p,q \} = tr\{ p,q\}</math>. (Application of both cantellation and truncation operations) *** Coxeter-Dynkin diagram: {{CDD|node_1|p|node_1|q|node_1}} ** For uniform polychora: A runcicantitruncation, <math>t_{0,1,2,3}\{ p,q,r \}</math>. (Application of runcination, cantellation, and truncation operations) *** Coxeter-Dynkin diagram: {{CDD|node_1|p|node_1|q|node_1|r|node_1}}, {{CDD|nodes_11|split2|node_1|p|node_1}}, {{CDD|node_1|split1|nodes_11|split2|node_1}} ** For uniform polytera (5-polytopes): A steriruncicantitruncation, t<sub>0,1,2,3,4</sub>{p,q,r,s}. <math>t_{0,1,2,3,4}\{ p,q,r,s \}</math>. (Application of sterication, runcination, cantellation, and truncation operations) *** Coxeter-Dynkin diagram: {{CDD|node_1|p|node_1|q|node_1|r|node_1|s|node_1}}, {{CDD|nodes_11|split2|node_1|p|node_1|q|node_1}}, {{CDD|branch_11|3ab|nodes_11|split2|node_1}} ** For uniform n-polytopes: <math>t_{0,1,...,n-1}\{ p_1, p_2,...,p_n \}</math>.
== See also == * Expansion (geometry) * Omnitruncated polyhedron
== References == {{reflist}}
==Further reading== * Coxeter, 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, 1966
== External links == * {{mathworld | urlname = Truncation | title = Truncation}}
{{Polyhedron_operators}}
Category:Polyhedra Category:Truncated tilings Category:Uniform polyhedra