[...] in a recent edit summary on Dual polyhedron, you explained that "there are other examples" [of corresponding symmetry classes containing a polyhedron / its dual].

Were you thinking of: «if a symmetry class defined as "all the polyhedra having a given symmetry group" contains a polyhedron, then it also contains the dual polyhedron», please?

"A polyhedron and its dual have the same symmetry group" should be added before the symmetry classes defined as "all the regular polyhedra", "all the isogonal polyhedra", etc; shouldn't it?

Cheers, JavBol (talk) 01:26, 22 June 2021 (UTC)[reply]

Not as the article is currently written, no. As the idea of a symmetry class is used here, a polyhedron and its dual are (unless self-dual) in different symmetry classes. However they both share the same symmetry group. In the different classes, the various orbits within the group are ascribed to different elements of the polyhedron, say to the faces of one polyhedron and to the vertices of its dual. Invoking symmetry classes in this way is not usual in such basic treatments of polyhedral duality. One normally refers simply to say "the octahedron", by which it is tacitly understood to reference any and all individual polyhedra in the octahedral class. Introducing the classes of set-based formal logic here is, in my humble opinion, a typical example of the unbalanced ownership this article (among others) suffers from. — Cheers, Steelpillow (Talk) 05:04, 22 June 2021 (UTC)[reply]

@Steelpillow: Thank you for your latest answer. However, i must say that it's a bit too subtle for me... Example: which individual polyhedra (other than the regular octahedron) are in the octahedral class, meant as you've used it, please?

Cheers, JavBol (talk) 15:15, 22 June 2021 (UTC)[reply]

I am not entirely sure what is meant. I would suppose that each individual octahedron one constructs is a member of the class. However it may mean something more subtle. For example the great stellated dodecahedron has the same symmetry orbits for its vertices as the convex dodecahedron, the same arrangement of face planes in their orbit and the same general positioning (though at right angles) of edges in their orbit. Are examples of these two polyhedra in the same class? What exactly does a "class" signify in this context? I have no idea, and this is one reason why I think that bringing in classes is a bad idea. — Cheers, Steelpillow (Talk)

@Steelpillow: OK, you've convinced me not to mess with symmetry classes... ;-)

1 last verification: you would not agree with a little addition like e.g.:

"Duality preserves the symmetries of a polyhedron. Thus, a polyhedron and its dual have the same symmetry group. Caution: symmetry group is not to be confused with symmetry class. For many classes of polyhedra defined by their symmetries, the duals belong to a related symmetry class. For example, [...]",

would you? :-P

Cheers, JavBol (talk) 21:32, 22 June 2021 (UTC)[reply]

I would agree that it is true, but I would not agree that the article should go any further down that particular rabbit-hole; it rambles too far off-topic already. Seriously, I would appreciate not discussing classes any more. — Cheers, Steelpillow (Talk) 08:40, 23 June 2021 (UTC)[reply]

@Steelpillow:

In Dual polyhedron#Kinds of duality##Topological duality, the end of the following sentence:

"An abstract polyhedron is a certain kind of partially ordered set (poset) of elements, such that adjacencies, or connections, between elements of the set correspond to adjacencies between elements (faces, edges, vertices) of a polyhedron."

should be specified, such as:

"[...] between elements (faces, edges, vertices) of a geometric polyhedron, even if the latter cannot exist.",

shouldn't it?

Besides, an example (as simple as possible) of an abstract polyhedron that cannot be geometrically realized should be added, shouldn't it? :-P

In advance, thank you very much for your answers!

Cheers, JavBol (talk) 17:57, 30 June 2021 (UTC)[reply]

That would be wrong. If a thing does not exist, then anything corresponding to it does not exist either. The point being made is that abstract polytopes are dualised by reversing the ranking, and one should not stray from that. The obvious non-simple polyhedra with unfaithful duals to introduce would be the uniform hemipolyhedra, but they would be better treated along Wenninger's lines (through polar reciprocation) before attempting any deeper remarks about the duals as only exhibiting unfaithful realizations. And I'd want to clear that on the article talk page first. — Cheers, Steelpillow (Talk) 18:30, 30 June 2021 (UTC)[reply]

Just a note, in case it is useful to you. WP:BRD is only a guideline. If the cycle goes to three reverts by the same editor, they then fall foul of the three-revert rule which is a full policy and gets strongly enforced. BRD is designed only to get people talking, 3RR is designed to check a warrior who is not engaging in discussion on the talk page (Discussion via rambling and inappropriate edit comments is against guidelines). After that, the etiquette can get complicated. For example it is always possible to push another editor to the edge of 3RR so they have to back off, but if you do that then your own failure to engage in BRD beforehand may be held against you later. While I am here, thank you for your contribution to the discussion. — Cheers, Steelpillow (Talk) 16:14, 1 July 2021 (UTC)[reply]

@Steelpillow:

In Dual polyhedron#Kinds of duality##Polar reciprocation, at the beginning of the following sentence:

"The dual of polyhedron P is often defined in terms of polar reciprocation about a sphere.",

the following specification should be added:

"In Euclidean space",

shouldn't it? :-P

@Steelpillow: Talking about lengths:

Can a polyhedron which has all edges of the same length be simply called an equilateral polyhedron (it would be very convenient e.g. in the Chamfer (geometry) article), or is it preferable to call it an equilateral-faced polyhedron, please?

In advance, thank you very much for your answers!

Cheers, JavBol (talk) 15:22, 7 July 2021 (UTC)[reply]

Hi. Please do not put discussions in edit comments, as you did here. Edit comments are to explain the reason for edit, nothing else. I am sorry, I don't study symmetry notations, so I can't help you there anyway. — Cheers, Steelpillow (Talk) 14:51, 26 May 2024 (UTC)[reply]

@Steelpillow: Thank you for your reply; I am glad that you seem to be doing well (I was a bit wondering). Sorry for having notified you in an edit summary (I recognize your intellectual rigor, here); I will not do it again. But you had not replied to my previous message to you on my talk page (just above); perhaps because the template {{U|Username}} does not work as it used to do? (The latest times, I've used [[User:Username|Username]] wikilinks.) — Cheers, JavBol (talk) 16:09, 26 May 2024 (UTC).[reply]

The best way to contact another editor is on their own user talk page: mine is User talk:Steelpillow. All the templates still work, perhaps I made some mistake but I cannot now recall.

I expect you will know by now that "equilateral" is the correct description for a polyhedron having all sides the same length. If they also lie in the same symmetry orbit then the polyhedron is also isotoxal. — Cheers, Steelpillow (Talk) 16:51, 26 May 2024 (UTC)[reply]

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