Geometric operation
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:
![](https://upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png)
![](https://upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png)
![](https://upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png)
![](https://upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png)
![](https://upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png)
![](https://upload.wikimedia.org/wikipedia/commons/e/e9/CDel_r.png)
, ![](https://upload.wikimedia.org/wikipedia/commons/5/56/CDel_nodes_11.png)
![](https://upload.wikimedia.org/wikipedia/commons/3/32/CDel_split2.png)
![](https://upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png)
![](https://upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png)
, ![](https://upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png)
![](https://upload.wikimedia.org/wikipedia/commons/a/a1/CDel_split1.png)
![](https://upload.wikimedia.org/wikipedia/commons/5/56/CDel_nodes_11.png)
![](https://upload.wikimedia.org/wikipedia/commons/3/32/CDel_split2.png)
![](https://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:
![](https://upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png)
![](https://upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png)
![](https://upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png)
![](https://upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png)
![](https://upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png)
![](https://upload.wikimedia.org/wikipedia/commons/e/e9/CDel_r.png)
![](https://upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png)
![](https://upload.wikimedia.org/wikipedia/commons/5/58/CDel_s.png)
, ![](https://upload.wikimedia.org/wikipedia/commons/5/56/CDel_nodes_11.png)
![](https://upload.wikimedia.org/wikipedia/commons/3/32/CDel_split2.png)
![](https://upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png)
![](https://upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png)
![](https://upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png)
![](https://upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png)
, ![](https://upload.wikimedia.org/wikipedia/commons/f/fc/CDel_branch_11.png)
![](https://upload.wikimedia.org/wikipedia/commons/d/d5/CDel_3ab.png)
![](https://upload.wikimedia.org/wikipedia/commons/5/56/CDel_nodes_11.png)
![](https://upload.wikimedia.org/wikipedia/commons/3/32/CDel_split2.png)
![](https://upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png)
- For uniform n-polytopes:
.