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:
{{citation}}
: Check date values in: |year=
(help)CS1 maint: year (link) See p. 22, where the omnitruncation is described as a "flag graph".Seed | Truncation | Rectification | Bitruncation | Dual | Expansion | Omnitruncation | Alternations | ||
---|---|---|---|---|---|---|---|---|---|
![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() |
![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() |
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} |