Omnitruncation Explained

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, 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\} = \{ 2 p\} .

• • • • • Coxeter-Dynkin diagram 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, t_0,1,2,3,4.

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} \} .

See also

• Expansion (geometry)
• Omnitruncated polyhedron

Further reading

• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, (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

Notes and References

1. See p. 22, where the omnitruncation is described as a "flag graph".