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
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...
(or
honeycomb
A honeycomb is a mass of Triangular prismatic honeycomb#Hexagonal prismatic honeycomb, hexagonal prismatic Beeswax, wax cells built by honey bees in their beehive, nests to contain their larvae and stores of honey and pollen.
beekeeping, Beekee ...
) in a
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
A facet is a flat surface of a geometric shape, e.g., of a cut gemstone.
Facet may also refer to:
Arts, entertainment, and media
* ''Facets'' (album), an album by Jim Croce
* ''Facets'', a 1980 album by jazz pianist Monty Alexander and his tri ...
. 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 polygon
In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex p ...
s:
An ordinary truncation,
.
***
Coxeter-Dynkin diagram
** For
uniform polyhedra
In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent.
Uniform polyhedra may be regular (if also ...
(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:
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
In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons (the definition is different in 2 dimensions to exclude vert ...
, 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 In geometry, an omnitruncated polyhedron is a truncated quasiregular polyhedron
In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-tran ...
References
*
Coxeter, H.S.M. ''
Regular Polytopes
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...
'', (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