## Naming of 4D Catalans

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

### Naming of 4D Catalans

So I've started updating my website again, starting with the 3D Catalan solids, and with the plan of eventually covering all the 4D Catalans. But I ran into a little conundrum: what to name each 4D Catalan? In 3D, they have names that directly describe their geometry (rhombic dodecahedron, triakis octahedron, disdyakis hexecontahedron, etc.), but in 4D I don't know of any such names, only that they are duals to the 4D uniforms.

Is there an "official" list of 4D Catalan names? Or some kind of naming scheme I could use for them? I'd like to follow in the spirit of the 3D Catalan names, if possible.
quickfur
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### Re: Naming of 4D Catalans

The first few results from a quick search for "4D Catalan polytopes" were these three papers, and Higher Space, and your own site.

The first one says (in Section 4)
To our surprise duals of the 4D uniform polytopes have not been studied so far.
(though that was 12 years ago).

And don't forget Polyhedron Dude's 4D dice, and the Polytope Wiki.
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mr_e_man
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### Re: Naming of 4D Catalans

Oooh, the polytope wiki seems to have names for the 4D Catalans, nice! Just what I need. I'll probably adopt those names.
quickfur
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### Re: Naming of 4D Catalans

Have a look at http://www.os2fan2.com/gloss/polytope.pdf . I give names for the duals of the truncates, rectates, the runcinate (first and last node) and the omnitruncates.

Using kepler style names don't work in 4d and higher, because there really isn't a construction, but a mapping onto the CD.
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wendy
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### Re: Naming of 4D Catalans

I've been working on 4D catalans for my website (though none published yet). Tentatively, I'm using the names used on the polytope wiki. But I'm not quite understanding some of the names. For example, the dual of oxoo is called a "joined bidecachoron". What does "joined" mean? What is being joined? And why is it a bidecachoron? Would be nice to understand the logic behind these names before I start using them "officially"...
quickfur
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### Re: Naming of 4D Catalans

As for join: it is one of the Conway oparators, in fact the dual of "ambo", which in turn is better known as rectification. Thence the "dual of the rectified pentachoron" clearly is the "joined pentachoron".

The dual of the decachoron happens to be nothing but the hull of 2 mutually inverted pentachora. That might imply the bi- part of the name. However, as it is more decachoron based (as its dual, i.e. showing up doubled symmetry) and not so much pentachoron (single symmetry only) based, someone seemed to apply the bi- part to the decachoron stem instead. (That someone probably was Jonathan himself, as its OBSA is coined to be "bideca" as well.)

--- rk
Klitzing
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### Re: Naming of 4D Catalans

Ahhh that makes a lot of sense! Didn't realize join was a Conway operator.
quickfur
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