Omnitruncation
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, t0,1{p} = t{p} = {2p}.
- For uniform polyhedra (3-polytopes): A cantitruncation, t0,1,2{p,q} = tr{p,q}. (Application of both cantellation and truncation operations)
- Coxeter-Dynkin diagram:
- Coxeter-Dynkin diagram:
- For Uniform 4-polytopes: A runcicantitruncation, t0,1,2,3{p,q,r}. (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}. (Application of sterication, runcination, cantellation, and truncation operations)
- Coxeter-Dynkin diagram:
, , 
- Coxeter-Dynkin diagram:
- For uniform n-polytopes: t0,1,...,n-1{p1,p2,...,pn}.
References
- 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
| Seed | Truncation | Rectification | Bitruncation | Dual | Expansion | Omnitruncation | Alternations | ||
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| 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} |
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