Omnitruncation
Geometric operation
title: "Omnitruncation" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["polyhedra", "truncated-tilings", "uniform-polyhedra"] description: "Geometric operation" topic_path: "general/polyhedra" source: "https://en.wikipedia.org/wiki/Omnitruncation" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
::summary 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. Because the barycentric subdivision of any polytope can be realized as another polytope,{{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}} 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} = { 2p}.
- For uniform polyhedra (3-polytopes): A cantitruncation, 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 }. (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 }. (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 }.
References
References
- Matteo, Nicholas. (2015). "Convex Polytopes and Tilings with Few Flag Orbits". Northeastern University.
::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::