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, ${\displaystyle t_{0,1}\{p\}=t\{p\}=\{2p\}}$.
    • 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: ${\displaystyle t_{0,1,...,n-1}\{p_{1},p_{2},...,p_{n}\}}$.

• Expansion (geometry)
• Omnitruncated polyhedron

• 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

• 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}