# Omnitruncation

Jump to navigation
Jump to search

In geometry, an **omnitruncation** is an operation applied to a regular polytope (or honeycomb) in 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, .
- For uniform polyhedra (3-polytopes): A cantitruncation, . (Application of both cantellation and truncation operations)
- Coxeter-Dynkin diagram:

- For uniform 4-polytopes: A runcicantitruncation, . (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}. . (Application of sterication, runcination, cantellation, and truncation operations)- Coxeter-Dynkin diagram: , ,

- For uniform n-polytopes: .

## See also[edit]

## References[edit]

- 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

- N.W. Johnson:

## External links[edit]

Seed | Truncation | Rectification | Bitruncation | Dual | Expansion | Omnitruncation | Alternations | ||
---|---|---|---|---|---|---|---|---|---|

t_{0}{p,q}{p,q} |
t_{01}{p,q}t{p,q} |
t_{1}{p,q}r{p,q} |
t_{12}{p,q}2t{p,q} |
t_{2}{p,q}2r{p,q} |
t_{02}{p,q}rr{p,q} |
t_{012}{p,q}tr{p,q} |
ht_{0}{p,q}h{q,p} |
ht_{12}{p,q}s{q,p} |
ht_{012}{p,q}sr{p,q} |