Wythoffian operation

A Wythoffian operation is a polytope operation that can be represented on a conventional Coxeter-Dynkin diagram.

Applied to the regular polytope s in a given dimension, the Wythoffian operations produce the "Archimedean polytopes" of that dimension (named as an extension of the Archimedean solids - convex, finite, non-prismatic, and maintaining the regulars' symmetry). Wythoffian operations can also be applied to some uniform polytope s to produce other uniform polytopes, although this may produce degenerate cases such as multiple covers.

Shorthand and naming
Each named Wythoffian operation (except for loose cases like "expanded" or "omnitruncated") corresponds to a specific ringing of nodes and how far it is from the terminus of a Coxeter diagram, regardless of the diagram's size. That is, rectification always means node #1 is ringed, and bitruncation always means nodes #1 and #2 are ringed. (This may become confusing on branched and looped diagrams - for instance, the birectified pentacontatetrapeton has two #2 nodes ringed because the indexing starts at the single-node branch in the middle.)

Which nodes are ringed in a Coxeter diagram can be represented by numbers corresponding to their indices in the diagram, starting at #0. Thus we can say "t012" instead of "the #0, #1, and #2 nodes are ringed" or drawing out the entire diagram.

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. (It makes sense since 0 is included in those numbers.) In addition, the names of some operations are typically not used until even higher dimensions than this rule may suggest. For example, rectified polygons are simply called "dual" even though a polygon's diagram can technically be rectified, and even though node 2 of the rectified hexacosichoron is ringed it is not referred to as a "birectified hecatonicosachoron."

When the ringing of a Coxeter diagram has a symmetry to it (ignoring the weights of the connections), the name for the resulting figure can be a combination of the regular and its dual. Examples include the icosidodecahedron (3D, ) and the hexeractihexacontatetrapeton (6D, ).

Examples of Wythoffian operations and their CDs
The Johnson/Ruen adjectives are meant to describe the "distances" between the first ringed node and each other ringed node. A prefix indicates which node is the first one ringed, then the largest distance between rings is listed (the distance between the first and last rings), followed by each other ring's distance from the first.

When the Johnson/Ruen adjectives differ from the Bowers adjectives, the former is more descriptive of the Coxeter diagram.

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. When this is done, the polytope's name is based on the dual. An operation with three or more rings is reduced to multiple two-ring operations that all start at the first ringed node.

There can, of course, be bi-, tri-, or even further versions of these from the terminus of the diagram. Some operations do not correspond to an exact number and position of ringed nodes, but are well-defined nonetheless.

It would not make sense for a polytope to be bidualed, triexpanded, or even biomnitruncated. If one wanted to describe the polytope ... as "biomnitruncated" instead of writing out all the operations in full, they could just call it a "t1,2,3,...,n-simplex,".

Abstract definition
Wythoffian operations may be generalized to abstract polytopes, which need not be regular. The simplest version of this generalization takes as input a rank-$$n$$ abstract polytope $$\mathcal{P}$$ and a set $$T \subseteq \{1, \ldots, n\}$$ of unringed nodes in the Coxeter diagram, which is assumed to be linear and numbered left to right. For example, $$T=\{1, \ldots, n-1\}$$would represent the dual, as the only ringed node would be node $$n$$. The unringed nodes are given rather than the ringed ones because, instead of performing Stott expansion on the original polytope by "adding" its various rectates, this definition begins with the omnitruncate (where all nodes are ringed) and then "collapses" elements to lower dimensions based on the unringed nodes.

To illustrate how this collapse works, consider the truncated icosahedron, with CD diagram. However, the orientation of the diagram in this case would suggest a bitruncated dodecahedron instead, where $$n=3$$, $$\mathcal{P}=$$ doe, and $$T=\{1\}$$. The omnitruncate of doe is grid, which may be constructed abstractly from doe as follows.


 * First, the elements of grid are defined to be the chains of doe which do not include the "improper" top and bottom elements. In this case:
 * the top element of grid is the empty set (which is vacuously a chain of doe);
 * the faces of grid are chains containing a single proper element of doe (the 20 hexagons are vertices, the 30 squares are edges, and the 12 decagons are faces);
 * the edges of grid are chains with two proper elements of doe (namely 60 vertex-edge pairs, 60 vertex-face pairs, and 60 edge-face pairs);
 * and the vertices of grid are 3-element chains i.e. flags of doe (minus the improper elements), of which there are 120.
 * Next, the chains are ordered by reverse inclusion, i.e. for all chains $$A$$, $$B$$ of doe, $$A$$ contains $$B$$ in grid if and only if $$B \subseteq A$$.


 * Finally, a bottom element is added to obtain the full abstract polytope of grid. Although the bottom element may be any set other than the aforementioned chains, the set of all elements of doe is a good choice, as the ordering definition naturally extends to include it.