Omnitruncation Knowpia

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, $t_{0,1}\{p\}=t\{p\}=\{2p\}$ $t_{0,1}\{p\}=t\{p\}=\{2p\}$ .
  - Coxeter-Dynkin diagram
- For uniform polyhedra (3-polytopes): A cantitruncation, $t_{0,1,2}\{p,q\}=tr\{p,q\}$ $t_{0,1,2}\{p,q\}=tr\{p,q\}$ . (Application of both cantellation and truncation operations)
  - Coxeter-Dynkin diagram:
- For uniform polychora: A runcicantitruncation, $t_{0,1,2,3}\{p,q,r\}$ $t_{0,1,2,3}\{p,q,r\}$ . (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}. $t_{0,1,2,3,4}\{p,q,r,s\}$ $t_{0,1,2,3,4}\{p,q,r,s\}$ . (Application of sterication, runcination, cantellation, and truncation operations)
  - Coxeter-Dynkin diagram: , ,
- For uniform n-polytopes: $t_{0,1,...,n-1}\{p_{1},p_{2},...,p_{n}\}$ .

References edit

^ 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

External links edit

Weisstein, Eric W. "Expansion". MathWorld.

Polyhedron operators
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t
e
Seed	Truncation	Rectification	Bitruncation	Dual	Expansion	Omnitruncation	Alternations


$t 0 {p, q} {p, q}$	$t 01 {p, q} t{p, q}$	$t 1 {p, q} r{p, q}$	$t 12 {p, q} 2t{p, q}$	$t 2 {p, q} 2r{p, q}$	$t 02 {p, q} rr{p, q}$	$t 012 {p, q} tr{p, q}$	$ht 0 {p, q} h{q, p}$	$ht 12 {p, q} s{q, p}$	$ht 012 {p, q} sr{p, q}$

[1] 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".

[2] 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

[1]

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