Omnitruncation
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, [math]\displaystyle{ t_{0,1}\{ p \} = t\{ p\} = \{ 2p\} }[/math].
- Coxeter-Dynkin diagram
- Coxeter-Dynkin diagram
- For uniform polyhedra (3-polytopes): A cantitruncation, [math]\displaystyle{ t_{0,1,2}\{ p,q \} = tr\{ p,q\} }[/math]. (Application of both cantellation and truncation operations)
- Coxeter-Dynkin diagram:
- Coxeter-Dynkin diagram:
- For uniform polychora: A runcicantitruncation, [math]\displaystyle{ t_{0,1,2,3}\{ p,q,r \} }[/math]. (Application of runcination, cantellation, and truncation operations)
- Coxeter-Dynkin diagram:
, 
, 
- Coxeter-Dynkin diagram:
- For uniform polytera (5-polytopes): A steriruncicantitruncation, t0,1,2,3,4{p,q,r,s}. [math]\displaystyle{ t_{0,1,2,3,4}\{ p,q,r,s \} }[/math]. (Application of sterication, runcination, cantellation, and truncation operations)
- Coxeter-Dynkin diagram:
, , 
- Coxeter-Dynkin diagram:
- For uniform n-polytopes: [math]\displaystyle{ t_{0,1,...,n-1}\{ p_1, p_2,...,p_n \} }[/math].
- For regular polygons: An ordinary truncation, [math]\displaystyle{ t_{0,1}\{ p \} = t\{ p\} = \{ 2p\} }[/math].
See also
References
- ↑ Matteo, Nicholas ( 2015), Convex Polytopes and Tilings with Few Flag Orbits, 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
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
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