The standard polytope naming scheme sucks. As far as I can tell, there are a set of prefixes for the rank of a polytope, and a system to go from an integer to a prefix. These are then attached to get the platonic solid names. For archimedeans and other such things, you can do [operation polytope], but what's more often used is the prefix for the number of sides, and then the suffix for the rank of the polytope, preceeded by the name of its facet.
For example, the triangular antitegmatic icosachoron. This means "the 20 sided polychoron with triangular antitegum sides". That's a very long name for such a simple shape! Also, it having 20 sides isn't that important. I mean it's one of the first things I want to know about a polytope, but so is vertex count. That leads me nicely into the first thing I want to talk about, which is "what do we care about in a naming scheme?", but I want to shit on some more examples first.
The rhombicuboctahedron is a convex uniform polyhedron, and yet it is given a heptasyllabic name. This name tells you it has something to do with rhombuses, cubes, and octahedra. This name is actively bad as it has nothing to do with rhombuses. I have learned that the reason it has "rhombi" in its name is because it has square faces that correspond to the faces of a rhombic dodecahedron, and that's just plain silly! I knew what a rhombic dodecahedron was and couldn't reason that out myself.
Well, our primary goal should be making most shape names short and easy to pronounce. Our second most important goal is making our names comprehensible. Our third most important goal is making them easy to generate and expand on your own. But those goals are vague and broad.
Refining them further, what I really want is for the symmetry group and type to be the most obvious things, and then each polytope type will have its own naming scheme. To improve this naming scheme, I can only make it apply to certain things to focus my efforts. So this naming scheme will only apply to uniforms, uniform duals and regulars.
I also want to minimize the amount of memorization. Uniforms and uniform duals will be named after their operation, and regulars will be named after their schlafli symbol.
Regular shapes are the simplest so lets go over them first. The regular polygons are named as so:
| 3 Sides | 4 Sides | 5 Sides | 6 Sides | 7 Sides | 8 Sides | 9 Sides | 10 Sides | 11 Sides | 12 Sides |
| Triangle | Square | Pentagon | Hexagon | Heptagon | Octagon | Nonagon | Decagon | Undecagon | Dodecagon |
| Star | Hexagram | 2-Heptagram | 2-Octagram | 2-Nonagram | 2-Decagram | 2-Undecagram | 2-Dodecagram | ||
| 3-Heptagram | 3-Octagram | 3-Nonagram | 3-Decagram | 3-Undecagram | 3-Dodecagram | ||||
| 4-Nonagram | 4-Decagram | 4-Undecagram | 4-Dodecagram | ||||||
| 5-Undecagram | 5-Dodecagram |
After those, we have the rank 3 polytopes. Just reading out their schlafli symbol, while tempting, isn't the best idea. These will be used a lot so we can simplify them to codes.
| {3, 3} | {3, 4} | {4, 3} | {3, 5} | {5, 3} | {3, 5/2} | {5/2, 3} | {5, 5/2} | {5/2, 5} |
| Pyra | Octa | Cube | Ico | Doe | Gico | Stedo | Grod | Smod |
As you can see, these are all lazily derived from their normal counter parts. Ico conflicting with 24-cell will surely never bring me any problems!
These platonic solids all have names that use the system, but it makes more sense to use the short names for other things. The way the system works is that you use shortened versions and you use latin for the first part of the symbol and greek for the second part. Those names are tresa, quad, quin, and tri, tetra, penta, and finally 5/2 is stella.
| {3, 3} | {3, 4} | {4, 3} | {3, 5} | {5, 3} | {3, 5/2} | {5/2, 3} | {5, 5/2} | {5/2, 5} |
| Tresatrihedron | Tresatetrahedron | Quadtrihedron | Tresapentahedron | Quintrihedron | Tresastellahedron | Stellatrihedron | Quinstellahedron | Stellapentahedron |
Pretty obvious why I want you to memorize the 9 terms instead of these longer names. Now for polychora.