Throughout the years, I have found that a good system of nomenclature helps me understand and categorize objects better in my mind. Starting around 5 years ago, I got the idea of creating a new system of nomenclature for polytopes, that would be consistent and avoid the quirks in existing naming systems.
When performing a uniform polytope search by regiment enumeration, two useful things to know are (1) a polytope’s verf structure and (2) the possible Wythoffian families. In addition, convex polytopes are often described by (3) a particular Wythoffian operation applied to a base shape, distinguished by a pattern of nodes ringed in the Coxeter diagram. All of these features are therefore reasonable choices to be the basis of a naming system. In particular, when I was studying families and regiments of uniform honeycombs, I found many surprising cases where members of the same family found their way into many varied regiments. This led me to consider the Wythoffian family to be an important aspect for a naming system designed to promote this cool way of classifying polytopes.
For ease of name memorization, I also decided that the following features would be important for a nomenclature system:
• The name should be almost always uniquely derivable from the shape itself given a short set of rules. (Like the Bowers names of convex Wythoffian polytopes, but very much unlike the Bowers names of nonconvex uniform polytopes.)
• The name as generated by the rules should be unique (at least up to symmetry); even if the construction method behind the name is not unique, there should at least be a canonical one preferred.
I have been working on a new nomenclature for uniform polytopes. The basis of my system is the elemental naming scheme, a nomenclature used on the hi.gher.space wiki[1][2] and referenced on Richard Klitzing’s website[3][4][5], but extended to nonconvex shapes as well as Euclidean tesselations. I am a fan of this naming scheme for several reasons:
• Each name can be decomposed into 3 pieces for analysis: a prefix, an infix, and a suffix.
• The prefix always refers to the symmetry or to a regular polytope in the same family.
• The infix always refers to the identity of the Wythoffian operation on said regular polytope; thus, the index already indirectly indicated the verf shape.
• The suffix refers to the number of dimensions. There was even talk of using suffixes like terid, petid, ectid, etc. in place of choron, teron, peton[6] because the prefix and infix referred to the shape itself, not its facets. (It should be noted that this choice is more in line with Anton Sherwood’s original intention for this naming system, which used -choron for polyhedra and -tetron for polychora[7], but this was considered to be confusing as the suffix -choron already gave the impression of a 4D shape to almost everyone.)
• There was no historical baggage or motivation to keep the names consistent with traditional names; this allowed the scheme to be almost a blank slate for experimentation.
In the original elemental naming scheme, the names were mostly limited to shapes that resulted from Wythoffian operations on regular polytopes. The prefix represents a regular polytope (eg cosmo) or a family if the shape could be derived identically from two regular polytopes (eg rhodo). The infix represents a particular operation on the polytope (eg recti, canti, cantitomo, etc.). The suffix indicates the dimension.
I realized that, when extended to nonconvex polytopes, the infix could take two roles at once: identifying the verf shape and, assuming the shape can be constructed with a linear Dynkin diagram, identifying the operation on the base. All in all, it should be easy to interpret the infix both ways: as an identifier of the Wythoffian operation when the Coxeter diagram is linear and as an identifier of the verf shape in general. Non-Wythoffians in a Wythoffian regiment could be identified by modifying the name of one of the Wythoffians in the regiment. As this was already proving to be a very ambitious endeavor, I decided to restrict my names to the subclass of uniform polytopes that I call locally uniform.
The unfinished product is looking to have several other interesting features:
• The names are usually shorter than the common name of the same shape[8].
• Every polytope has a canonical name for each Wythoffian symmetry in which it is locally uniform.
• A polytope’s conjugate is always named by either adding or removing “quasi” from its name.
Scope of this project
This naming scheme is planned to be able to name all polytopes that are:
• Locally uniform (a sub-category of uniform; see [9] for definition)
• Either embeddable in a space of the same dimension, with a finite number of facets (spherical) or embeddable in a space of one less dimension, with an infinite number of facets (Euclidean). I decided not to make names for hyperbolic tessellations, because the number of possible symmetries (each symmetry gets an elementary name), verf shapes, and Wythoffian families would be too large.
• True polytopes under Bowers’ definition (not fissaries, coincidics, or compounds (yet)).
There should be at most one name for each symmetry, and one name for the highest Wythoffian symmetry.
Note: Naming non-Wythoffians in Wythoffian regiments seems like the most ambitious part of my project right now. They will be named based on the facets they share with Wythoffians, and I don’t expect to ever have rules for covering every Wythoffian regiment. I plan to create a partial list of rules sometime, but right now I’m mostly focusing on naming Wythoffians and hemi-Wythoffians.
The rules for naming polytopes will be posted soon.
References
1. http://hi.gher.space/wiki/Elemental_naming_scheme
2. http://hi.gher.space/wiki/List_of_uniform_polychora – try mousing over the third column of the table
3. https://bendwavy.org/klitzing/explain/product.htm
4. https://bendwavy.org/klitzing/incmats/ico.htm – look at the third name given
5. https://bendwavy.org/klitzing/incmats/pent.htm – look at the fourth name given
6. viewtopic.php?f=25&t=1510&p=18105#p18102
7. https://bendwavy.org/wp/?p=1049
8. viewtopic.php?f=25&t=1510#p15284
9. viewtopic.php?f=25&t=2440&p=27003#p27003