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,
.
- For uniform polyhedra (3-polytopes): A cantitruncation,
. (Application of both cantellation and truncation operations)
- For uniform polychora: A runcicantitruncation,
. (Application of runcination, cantellation, and truncation operations)
- Coxeter-Dynkin diagram:
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png)
![](//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png)
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png)
![](//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png)
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png)
![](//upload.wikimedia.org/wikipedia/commons/e/e9/CDel_r.png)
, ![](//upload.wikimedia.org/wikipedia/commons/5/56/CDel_nodes_11.png)
![](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_split2.png)
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png)
![](//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png)
, ![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png)
![](//upload.wikimedia.org/wikipedia/commons/a/a1/CDel_split1.png)
![](//upload.wikimedia.org/wikipedia/commons/5/56/CDel_nodes_11.png)
![](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_split2.png)
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png)
- 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:
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png)
![](//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png)
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png)
![](//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png)
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png)
![](//upload.wikimedia.org/wikipedia/commons/e/e9/CDel_r.png)
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png)
![](//upload.wikimedia.org/wikipedia/commons/5/58/CDel_s.png)
, ![](//upload.wikimedia.org/wikipedia/commons/5/56/CDel_nodes_11.png)
![](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_split2.png)
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png)
![](//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png)
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png)
![](//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png)
, ![](//upload.wikimedia.org/wikipedia/commons/f/fc/CDel_branch_11.png)
![](//upload.wikimedia.org/wikipedia/commons/d/d5/CDel_3ab.png)
![](//upload.wikimedia.org/wikipedia/commons/5/56/CDel_nodes_11.png)
![](//upload.wikimedia.org/wikipedia/commons/3/32/CDel_split2.png)
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png)
- For uniform n-polytopes:
.
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".
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
- 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
More information Seed, Truncation ...
Polyhedron operators
Seed |
Truncation |
Rectification |
Bitruncation |
Dual |
Expansion |
Omnitruncation |
Alternations |
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png) ![](//upload.wikimedia.org/wikipedia/commons/6/6d/CDel_node_n1.png) ![](//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png) ![](//upload.wikimedia.org/wikipedia/commons/2/20/CDel_node_n2.png) |
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png) ![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) |
![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png) ![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) |
![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png) ![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png) ![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) |
![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png) ![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) |
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png) ![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) |
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png) ![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png) ![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) |
![](//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png) ![](//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) |
![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png) ![](//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png) ![](//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png) ![](//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png) |
![](//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png) ![](//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png) ![](//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png) ![](//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png) ![](//upload.wikimedia.org/wikipedia/commons/2/28/CDel_node_h.png) |
![](//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Uniform_polyhedron-43-t0.svg/50px-Uniform_polyhedron-43-t0.svg.png) |
![](//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Uniform_polyhedron-43-t01.svg/50px-Uniform_polyhedron-43-t01.svg.png) |
![](//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Uniform_polyhedron-43-t1.svg/50px-Uniform_polyhedron-43-t1.svg.png) |
![](//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Uniform_polyhedron-43-t12.svg/50px-Uniform_polyhedron-43-t12.svg.png) |
![](//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/Uniform_polyhedron-43-t2.svg/50px-Uniform_polyhedron-43-t2.svg.png) |
![](//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Uniform_polyhedron-43-t02.png/50px-Uniform_polyhedron-43-t02.png) |
![](//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Uniform_polyhedron-43-t012.png/50px-Uniform_polyhedron-43-t012.png) |
![](//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/Uniform_polyhedron-33-t0.png/50px-Uniform_polyhedron-33-t0.png) |
![](//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Uniform_polyhedron-43-h01.svg/50px-Uniform_polyhedron-43-h01.svg.png) |
![](//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Uniform_polyhedron-43-s012.png/50px-Uniform_polyhedron-43-s012.png) |
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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