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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, 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, ${\displaystyle t_{0,1}\{p\}=t\{p\}=\{2p\}}$.
    • Coxeter-Dynkin diagram
  • For uniform polyhedra (3-polytopes): A cantitruncation, ${\displaystyle t_{0,1,2}\{p,q\}=tr\{p,q\}}$. (Application of both cantellation and truncation operations)
    • Coxeter-Dynkin diagram:
  • For uniform polychora: A runcicantitruncation, ${\displaystyle t_{0,1,2,3}\{p,q,r\}}$. (Application of runcination, cantellation, and truncation operations)
    • Coxeter-Dynkin diagram: , ,
  • For uniform polytera (5-polytopes): A steriruncicantitruncation, 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: ${\displaystyle t_{0,1,...,n-1}\{p_{1},p_{2},...,p_{n}\}}$.

See also

• Expansion (geometry)
• Omnitruncated polyhedron

References

1. 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".
2. 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

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

• Weisstein, Eric W. "Expansion". MathWorld.

Polyhedron operators

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Seed	Truncation	Rectification	Bitruncation	Dual	Expansion	Omnitruncation	Alternations
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}

• Polyhedra
• Truncated tilings
• Uniform polyhedra