omnitruncation

{{short description|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.{{citation|title=Convex Polytopes and Tilings with Few Flag Orbits|type=Doctoral dissertation|last=Matteo|first=Nicholas|publisher= Northeastern University|year=2015|id={{ProQuest|1680014879}}}} See p. 22, where the omnitruncation is described as a "flag graph". Because the barycentric subdivision of any polytope can be realized as another polytope,{{citation

| last1 = Ewald | first1 = G.

| last2 = Shephard | first2 = G. C.

| doi = 10.1007/BF01344542

| journal = Mathematische Annalen

| mr = 350623

| pages = 7–16

| title = Stellar subdivisions of boundary complexes of convex polytopes

| volume = 210

| year = 1974}} 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, $t\_\{0,1\}\backslash \{\; p\; \backslash \}\; =\; t\backslash \{\; p\backslash \}\; =\; \backslash \{\; 2p\backslash \}$.
• Coxeter-Dynkin diagram {{CDD|node_1|p|node_1}}
• For uniform polyhedra (3-polytopes): A cantitruncation, $t\_\{0,1,2\}\backslash \{\; p,q\; \backslash \}\; =\; tr\backslash \{\; p,q\backslash \}$. (Application of both cantellation and truncation operations)
• Coxeter-Dynkin diagram: {{CDD|node_1|p|node_1|q|node_1}}
• For uniform polychora: A runcicantitruncation, $t\_\{0,1,2,3\}\backslash \{\; p,q,r\; \backslash \}$. (Application of runcination, cantellation, and truncation operations)
• Coxeter-Dynkin diagram: {{CDD|node_1|p|node_1|q|node_1|r|node_1}}, {{CDD|nodes_11|split2|node_1|p|node_1}}, {{CDD|node_1|split1|nodes_11|split2|node_1}}
• For uniform polytera (5-polytopes): A steriruncicantitruncation, t_0,1,2,3,4{p,q,r,s}. $t\_\{0,1,2,3,4\}\backslash \{\; p,q,r,s\; \backslash \}$. (Application of sterication, runcination, cantellation, and truncation operations)
• Coxeter-Dynkin diagram: {{CDD|node_1|p|node_1|q|node_1|r|node_1|s|node_1}}, {{CDD|nodes_11|split2|node_1|p|node_1|q|node_1}}, {{CDD|branch_11|3ab|nodes_11|split2|node_1}}
• For uniform n-polytopes: $t\_\{0,1,...,n-1\}\backslash \{\; p\_1,\; p\_2,...,p\_n\; \backslash \}$.