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Omnitruncation (geometry)
 
 
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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, an omnitruncation is an operation applied to a regular polytope
Regular polytope

In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flag , thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension = n....
 (or honeycomb
Honeycomb (geometry)

In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions....
) in a Wythoff construction
Wythoff construction

File:Wythoffian_construction_diagram.pngIn geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling....
 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.








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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, an omnitruncation is an operation applied to a regular polytope
Regular polytope

In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flag , thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension = n....
 (or honeycomb
Honeycomb (geometry)

In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions....
) in a Wythoff construction
Wythoff construction

File:Wythoffian_construction_diagram.pngIn geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling....
 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.

See also

  • Uniform polytope#Truncation_operators
    Uniform polytope

    A uniform polytope is a vertex-transitive polytope made from uniform polytope Facet . A uniform polytope must also have only regular polygon faces....
    • For regular polygons: An ordinary truncation
      Truncation (geometry)

      In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new Facet in place of each vertex....
      , t0,1=.
      • Coxeter-Dynkin diagram
        Coxeter-Dynkin diagram

        In geometry, a Coxeter-Dynkin diagram is a Graph with labelled edges. It represents the spatial relations between a collection of mirrors , and describes a Kaleidoscope construction....
         ** For uniform polyhedra
        Uniform polyhedron

        A Uniform polytope polyhedron is a polyhedron which has regular polygons as Face and is transitive on its vertex . It follows that all vertices are Congruence , and the polyhedron has a high degree of reflectional and rotational symmetry....
         (3-polytopes): A cantitruncation
        Uniform polyhedron

        A Uniform polytope polyhedron is a polyhedron which has regular polygons as Face and is transitive on its vertex . It follows that all vertices are Congruence , and the polyhedron has a high degree of reflectional and rotational symmetry....
        , t0,1,2. (Application of both cantellation and truncation operations)
      • Coxeter-Dynkin diagram: ** For uniform polychora
        Uniform polychoron

        In geometry, a Uniform polytope polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedron.This article contains the complete list of 64 non-prismatic convex uniform polychora, and describes two infinite sets of convex prismatic forms....
         (4-polytopes): A runcicantitruncation
        Uniform polychoron

        In geometry, a Uniform polytope polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedron.This article contains the complete list of 64 non-prismatic convex uniform polychora, and describes two infinite sets of convex prismatic forms....
        , t0,1,2,3. (Application of runcination
        Runcination

        In geometry, a runcination is an operation that cuts a regular polytope simultaneously along the faces, edges and vertices, creating new facets in place of the original face, edge, and vertex centers....
        , cantellation, and truncation operations)
      • Coxeter-Dynkin diagram: ** For uniform polytera
        Uniform polyteron

        In geometry, a uniform polyteron is a five-dimensional uniform polytope. By definition, a uniform polyteron is vertex-transitive and constructed from uniform polychoron Facet ....
         (5-polytopes): A steriruncicantitruncation
        Uniform polychoron

        In geometry, a Uniform polytope polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedron.This article contains the complete list of 64 non-prismatic convex uniform polychora, and describes two infinite sets of convex prismatic forms....
        , t0,1,2,3,4. (Application of sterication, runcination, cantellation, and truncation operations)
      • Coxeter-Dynkin diagram: ** For uniform n-polytopes
        Uniform polytope

        A uniform polytope is a vertex-transitive polytope made from uniform polytope Facet . A uniform polytope must also have only regular polygon faces....
        : t0,1,...,n-1.
  • Expansion (geometry)
    Expansion (geometry)

    In geometry, expansion is a polytope operation where Facet are separated and moved radially apart, and new facets are formed at separated elements ....