Naming of 4D Catalans

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

Naming of 4D Catalans

Postby quickfur » Mon Jan 23, 2023 4:29 pm

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
Pentonian
 
Posts: 2922
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: Naming of 4D Catalans

Postby mr_e_man » Tue Jan 24, 2023 2:36 am

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

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.
ΓΔΘΛΞΠΣΦΨΩ αβγδεζηθϑικλμνξοπρϱσςτυϕφχψωϖ °±∓½⅓⅔¼¾×÷†‡• ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎
ℕℤℚℝℂ∂¬∀∃∅∆∇∈∉∋∌∏∑ ∗∘∙√∛∜∝∞∧∨∩∪∫≅≈≟≠≡≤≥⊂⊃⊆⊇ ⊕⊖⊗⊘⊙⌈⌉⌊⌋⌜⌝⌞⌟〈〉⟨⟩
mr_e_man
Tetronian
 
Posts: 472
Joined: Tue Sep 18, 2018 4:10 am

Re: Naming of 4D Catalans

Postby quickfur » Tue Jan 24, 2023 3:22 am

Oooh, the polytope wiki seems to have names for the 4D Catalans, nice! Just what I need. I'll probably adopt those names.
quickfur
Pentonian
 
Posts: 2922
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: Naming of 4D Catalans

Postby wendy » Wed Jan 25, 2023 3:04 am

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.
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 2012
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Re: Naming of 4D Catalans

Postby quickfur » Wed Mar 01, 2023 4:23 pm

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
Pentonian
 
Posts: 2922
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: Naming of 4D Catalans

Postby Klitzing » Wed Mar 01, 2023 10:01 pm

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
Pentonian
 
Posts: 1637
Joined: Sun Aug 19, 2012 11:16 am
Location: Heidenheim, Germany

Re: Naming of 4D Catalans

Postby quickfur » Wed Mar 01, 2023 11:27 pm

Ahhh that makes a lot of sense! Didn't realize join was a Conway operator. :oops:
quickfur
Pentonian
 
Posts: 2922
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North


Return to Other Polytopes

Who is online

Users browsing this forum: No registered users and 2 guests

cron