Catalan Naming Scheme

Date posted: 3/7/2026

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The normal naming scheme for the convex non-multiprismatic dual uniforms ("Catalan polytopes") are just... bad. Except for dual truncates and rectates, they are all the same: [facet name] [facet count]tope. This leads to some... issues, like how the Polytope Wiki uses the exact same names for the dual prismatorhombates. So in this first post, I'll describe a naming scheme for Catalans alike that of their dual Archimedeans: an operation on a polytope.

So... Where To Start?

First off, the names for dual truncates and rectates will stay. They are already an operation on a polytope. As for the rest, well as it turns out Jonathan Bowers himself has already gotten a head start on us! The names for decaic and contic isogonals and isochorics all have identical structure: bi[operation name]decachoron/tetracontoctachoron. From this, we can reason through the following:

The last one is noticably different, so we should try to avoid it. What about higher dimensions? Well, Bowers hasn't coined names yet, so we'll have make our own names.

The System As It Currently Stands

General Rules

Generally, we take the Bowers long name of the uniform and then take the dual of each component. The duals of the operators are as follows:

As is, I don't have names for dual great cellation, teration, petation, and exation, so although this scheme can name every Catalan up to 8 dimensions, it is technically incomplete.

Symmetric Diagrams

Symmetric diagrams should show all relevant ends of the diagram. If it is actually supersymmetric, use the name for the relevant supersymmetry This is hard to explain with words, so take some examples:

• Dual tesseractihexadecachoron: no supersymmetry - mesoapiculatotesseractihexadecachoron
• Dual small prismatodecachoron: supersymmetric with order 2 - cystallobidecachoron
• Dual small rhombated rectified-pentacontatetrapetic hexacomb: supersymmetric with order 6 - orthotrisbipentacontatetrapetic hexacomb

Dual Demicubics

For dual demicubics, there are two possible options: either use as described or use as follows. Instead of simply describing an operation on the demistellated orthoplex, replace "stellated" with, not just the operator in question, but specifically the operation that you do on a hypercube before taking an alternated faceting. In other words, instead of starting with the Bowers long name (e.g. small rhombated demipenteract), you start with the Johnson name (e.g. runcic penteract).

This (should) be everything needed to describe all convex dual Wythoffian polytopes.

List of Names Up To Six Dimensions

Anything in italics is subject to change due to missing operator prefixes
Anything in bold is non-Wythoffian and thus has a special name.

Three Dimensions

Four Dimensions

Five Dimensions

Coming eventually

Six Dimensions

Coming eventually