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
omnitruncated polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all o ...
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-transitive and edge-transitive, hence a step closer to regular polyhedra than the se ...
. When they are
alternated, they produce the
snub polyhedra.
All omnitruncated polyhedra are
zonohedra
In geometry, a zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in ...
. They have
Wythoff symbol
In geometry, the Wythoff symbol is a notation representing a Wythoff construction of a uniform polyhedron or plane tiling within a Schwarz triangle. It was first used by Coxeter, Longuet-Higgins and Miller in their enumeration of the uniform pol ...
''p q r , '' and
vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
s as ''2p.2q.2r''.
More generally an omnitruncated polyhedron is a bevel operator in
Conway polyhedron notation
In geometry, Conway polyhedron notation, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified by various prefix operations.
Conway and Hart extended the idea of using op ...
.
List of convex omnitruncated polyhedra
There are three
convex forms. They can be seen as red faces of one regular polyhedron, yellow or green faces of the
dual polyhedron
In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. ...
, and blue faces at the truncated vertices of the quasiregular polyhedron.
List of nonconvex omnitruncated polyhedra
There are 5
nonconvex uniform omnitruncated polyhedra.
Other even-sided nonconvex polyhedra
There are 8 nonconvex forms with mixed
Wythoff symbol
In geometry, the Wythoff symbol is a notation representing a Wythoff construction of a uniform polyhedron or plane tiling within a Schwarz triangle. It was first used by Coxeter, Longuet-Higgins and Miller in their enumeration of the uniform pol ...
s ''p q (r s) , '', and bow-tie shaped
vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
s, 2p.2q.-2q.-2p. They are not true omnitruncated polyhedra: the true omnitruncates ''p q r , '' or ''p q s , '' have coinciding 2''r''-gonal or 2''s''-gonal faces respectively that must be removed to form a proper polyhedron. All these polyhedra are one-sided, i.e.
non-orientable
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
. The ''p q r , '' degenerate Wythoff symbols are listed first, followed by the actual mixed Wythoff symbols.
General omnitruncations (bevel)
Omnitruncations are also called cantitruncations or truncated rectifications (tr), and Conway's bevel (b) operator. When applied to nonregular polyhedra, new polyhedra can be generated, for example these 2-uniform polyhedra:
See also
*
Uniform polyhedron
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 fa ...
References
*
*
*
* Har'El, Z
''Uniform Solution for Uniform Polyhedra.'' Geometriae Dedicata 47, 57-110, 1993
Zvi Har’ElKaleido software
*
Mäder, R. E.
''Uniform Polyhedra.'' Mathematica J. 3, 48-57, 1993.
{{DEFAULTSORT:Omnitruncated Polyhedron
Polyhedra