In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, 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
In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes). It describe ...
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,
.
***
Coxeter-Dynkin diagram
** For
uniform polyhedra (3-polytopes):
A cantitruncation,
. (Application of both
cantellation
In geometry, a cantellation is a 2nd-order truncation in any dimension that bevels a regular polytope at its edges and at its vertices, creating a new facet in place of each edge and of each vertex. Cantellation also applies to regular tiling ...
and truncation operations)
*** Coxeter-Dynkin diagram:
** For
uniform polychora
In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.
There are 47 non-prismatic convex uniform 4-polytopes. There ...
:
A runcicantitruncation,
. (Application of
runcination
In geometry, runcination is an operation that cuts a regular polytope (or honeycomb) simultaneously along the faces, edges, and vertices, creating new facets in place of the original face, edge, and vertex centers.
It is a higher order truncatio ...
, cantellation, and truncation operations)
*** Coxeter-Dynkin diagram: , ,
** For
uniform polytera (5-polytopes):
A steriruncicantitruncation, t
0,1,2,3,4.
. (Application of
sterication, runcination, cantellation, and truncation operations)
*** Coxeter-Dynkin diagram: , ,
** For
uniform n-polytopes:
.
See also
*
Expansion (geometry)
In geometry, expansion is a polytope operation where facets are separated and moved radially apart, and new facets are formed at separated elements ( vertices, edges, etc.). Equivalently this operation can be imagined by keeping facets in the ...
*
Omnitruncated polyhedron
References
*
Coxeter, H.S.M. ''
Regular Polytopes'', (3rd edition, 1973), Dover edition, (pp.145-154 Chapter 8: Truncation, p 210 Expansion)
*
Norman Johnson ''Uniform Polytopes'', Manuscript (1991)
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966
External links
*
{{Polyhedron_operators
Polyhedra
Uniform polyhedra