Polytope operations

An operation is a kind of transformation that can be applied to a polytope. It can be viewed as a function whose input and output are both polytopes, like the finding of the vertex figure.

This is an incomplete list of operations.

Wythoffian operations
These pretty much encompass all the polytopes you can get by ringing different nodes of a polytope's Coxeter-Dynkin diagram. These operations are typically done to regular polytope s and produce Wythoffian uniform polytope s.

Since a polytope's Coxeter diagram has a number of nodes equal to the polytope's dimension, an N-dimensional polytope can only use operations with "prefix numbers" (positions of ringed nodes) less than N. (Since 0 is the first one, it makes sense.) In addition, the names of some operations are typically not used until even higher dimensions than this rule suggests. For example, rectified polygons are simply called "dual" even though polygons' diagrams can be rectified, and even though node 2 of rox is ringed it is not referred to as a "birectified hi."

Note that the Coxeter diagram can also be read in the opposite direction. This is often done when most of the nodes are on the far side. The polytope's name is then based on the dual.

Dual
The dual of a polytope has vertices corresponding to the original polytope's facets, edges corresponding to the original's ridges (in 3D, edges), faces corresponding to the original's peaks (in 3D, vertices), and so on, culminating in facets corresponding to the original's vertices.

The dual of the dual is the original polytope. A polytope may be its own dual.

Petrie dual
In 3D, one can make specific paths along the edges of polyhedra that form skew polygon s. Several of these skew polygons together can constitute a polyhedron that looks much like the original, but has skew faces. Doing this is known as the "Petrie dual" or "Petrial" operation.

Like the dual, the petrial of the petrial is the original polyhedron.

This operation may generalize to other dimensions.

Blending
Loosely speaking, the blending operation "combines" two or more polytopes into one, joining the elements that coincide.

A version of blending called "augmentation" is used to create Johnson solid s and other CRF polytope s. Relatively intuitive, it is like gluing two polytopes together at a facet.

Many uniform polytope s are generated by blending smaller polytopes together. This form of blending typically takes in more than 2 polytopes and does not shy away from having them intersect one another.

Stellation
Generally, stellation is an operation that extends the facets of a polytope until they once again intersect one another. Its name comes from the star-like results it produces.

It can help to specify the dimension of the elements being affected by this operation. Thus, the following names were proposed:

Zonohedrification
The zonohedrification operation generates a new polytope based on vectors which are in turn based on the vertices of the original polytope.

The operation is not widely used, probably because it can be confusing due to its several steps and non-bijectivity (the zonohedrified tetrahedron and zonohedrified cube are both rhombic dodecahedra).