.gif)
Omnitruncation (geometry)
Encyclopedia
In geometry
, an omnitruncation is an operation applied to a regular polytope
(or honeycomb
) in 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:
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 flags, 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...
(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
In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction.- Construction process :...
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 truncationTruncation (geometry)In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex.- Uniform truncation :...
, t0,1{p}={2p}.- Coxeter-Dynkin diagramCoxeter-Dynkin diagramIn geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...
- Coxeter-Dynkin diagram
- For uniform polyhedraUniform polyhedronA uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive...
(3-polytopes): A cantitruncation, t0,1,2{p, q}. (Application of both cantellation and truncation operations)- Coxeter-Dynkin diagram:
- For uniform polychoraUniform polychoronIn geometry, a uniform polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedra....
(4-polytopes): A runcicantitruncation, t0,1,2,3{p, q,r}. (Application of runcinationRuncinationIn geometry, 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 polyteraUniform polyteronIn geometry, a uniform polyteron is a five-dimensional uniform polytope. By definition, a uniform polyteron is vertex-transitive and constructed from uniform polychoron facets....
(5-polytopes): A steriruncicantitruncation, t0,1,2,3,4{p, q,r, s}. (Application of sterication, runcination, cantellation, and truncation operations)- Coxeter-Dynkin diagram: , ,
- For uniform n-polytopesUniform polytopeA uniform polytope is a vertex-transitive polytope made from uniform polytope facets of a lower dimension. Uniform polytopes of 2 dimensions are the regular polygons....
: t0,1,...,n-1{p1,p2,...,pn}.
- For regular polygons: An ordinary truncation