Naming Gossetic Wythoffians

Wythoffian polytopes are uniform polytopes that have a respective Coxeter diagram. They can usually be described by Wythoffian operations done on a regular polytope, with operations acting on nodes of the symmetry group of the polytope. For exmaple, the truncated cube is the truncation of the cube, with truncation being the operation that rings the first and second nodes (t_0,1) and cube referring to the Coxeter diagram x4o3o, being cubic B₃ symmetry starting from the fourfold axis. It is worth noting that applying Wythoffian operations to arbitrary polytopes does not usually produce Wythoffian polytopes, and for example the cantellated truncated cube is a thing that exists, but is not uniform and therefore not Wythoffian.

However, not all Wythoffian polytopes can be described with a Wythoffian operation. The simplest such example is the steric penteract, a 5D polyteron with demipenteractic D₅ symmetry. It might at first seem that it described by some Wythoffian operation called "steration", but it really isn't because "steration" is not a Wythoffian operation: it does not simply consist of ringing nodes of the penteractic B₅ Coxeter group. In fact, it is an operation that only applies to demicubes and rings the first and fourth (not the fifth for... reasons) nodes of the D_n Coxeter diagram beginning from one of the singular branching nodes, ignoring the other.

There is still another symmetry where this issue becomes even more relevant: the Gossetics, E_n. While some of them are named correctly, many others have been given rather dubious names. The best example of this is the trirectified 1₂₂ polypeton, which is not actually the trirectification of the 1₂₂ polypeton, and the actual trirectification of the 1₂₂ polypeton is not even isogonal. This page then attempts to create a more accurate naming system for them.

Naming system

The naming scheme works quite similar to the one used for Wythoffian derivatives of regular polytopes, which were already described in the Polytope Wiki article for Wythoffian operations.

With a branch diagram consisting of a row of nodes with a singular node branch in the middle, notated as Coxeter symbol 1_ij, ringed nodes in the row get prefixed to the main operation adjective, beginning from the closer nodes. If the row does not begin with a ringed node, the operation gets prefixed with the Latin numeral of how many nodes were skipped plus one. If the singular node branch is ringed, then the prefix demi- is added to the main operation adjective. The nomenclature for the individual operations goes as follows:

Node	Name	Prefix (standard)
1	truncal	trunci-
2	cantial	canti-
3	runcial	runci-
4	sterial	steri-
5	pential	penti-
6	hexial	hexi-
7	heptial	hepti-
8	octial	octi-

As an example, the diagram     x
x o o x x o corresponds to either demisteriruncial or demibisteritruncal, depending on which side of the row gets chosen to be the starting node. As for the previously mentioned "trirectified 1₂₂ polypeton", it would be named the sterial 2₂₁ polypeton, and the E₆ omnitruncate would be named demisteriruncicantitruncal 2₂₁ polypeton.

Multiple branch extension

While the above naming scheme does work for all finite uniform polytopes using such symmetries, it fails to account for a particularly notable sixcomb which has a diagram with three branches of two nodes each: the 2₂₂ sixcomb, a noble Euclidean sixcomb made of 2₂₁ polypeta arranged in a 1₂₂ polypetal vertex figure.

In that case, there will be another set of operators that apply to the branching nodes, written before the infix -super-, the Latin word for "above", alluding to the operations being applied "above" the main row. For example,       x o
x o x
      x x could be written as, depending on the choice of starting node and main branch (super highlighted for clarity):

• runcisupersteriruncicantial (left node, lower branch)
• steriruncisuperruncicantial (left node, upper branch)
• runcisuperstericantitruncal (lower node, lower branch)
• sterisuperruncicantitruncal (lower node, right branch)
• biruncicantisuperruncitruncal (upper node, upper branch)
• biruncisuperruncicantitruncal (upper node, right branch)

If there are multiple ringed singular node branches, the demi- is prefixed by a Greek adverbial numeral prefix (dis-, tris-, etc). If there are multiple multiple node branches, there will simply be more instances of the -super- infix, with the branches ordered in increasing number of nodes. It is worth noting though that these only really apply to hyperbolic honeycombs, which are not as interesting due to their number.