Talk:Dual polyhedron

message: YES
# Tom Ruen 02:43, 18 February 2006 (UTC)
replies: text: What about putting the polytopes stuff (including any polytope-specific generalizations) at the end of the article, after the elementary stuff? The concept of dual polytopes is a generalization of dual polyhedra, after all. We can add a section, maybe entitled "Generalization to higher dimensions", and put the polytope-related material under it.—Tetracube (talk) 20:38, 29 October 2008 (UTC)
replies: text: The difficulty here is that the theory of topological and abstract dualities is primarily worked out for arbitrary dimensionality, i.e. for polytopes generally. Here, it would be better to treat polyhedra as a follow-on topic. There is a Wikipedia policy which allows some repetition of material, where it enables each article to stand on its own. In the present case, the treatment of the common material would tend to diverge in the two articles - the one becoming a little simpler, and the other more advanced. -- Cheers, Steelpillow (Talk) 20:56, 29 October 2008 (UTC)

text:  In passing, I'm amused at the term polytope-specific, since polytope is the general case and polyhedron is specific. —Tamfang (talk) 07:24, 30 October 2008 (UTC)
thread_title:  Consensus to rename to [[Dual polytope]]? 

message: There is much in this article at an elementary level, such as the important Dorman Luke construction. To bury this in a discussion of polytopes generally is IMHO unhelpful to the majority of students who wish only to find out about dual polyhedra. If you guys move it, I will have to pull the elementary stuff back here - so by all means move or copy across the relevant stuff, but please save me the trouble of recreating this article over. -- Cheers, Steelpillow (Talk) 20:30, 29 October 2008 (UTC)
replies: text: What about putting the polytopes stuff (including any polytope-specific generalizations) at the end of the article, after the elementary stuff? The concept of dual polytopes is a generalization of dual polyhedra, after all. We can add a section, maybe entitled "Generalization to higher dimensions", and put the polytope-related material under it.—Tetracube (talk) 20:38, 29 October 2008 (UTC)
replies: text: The difficulty here is that the theory of topological and abstract dualities is primarily worked out for arbitrary dimensionality, i.e. for polytopes generally. Here, it would be better to treat polyhedra as a follow-on topic. There is a Wikipedia policy which allows some repetition of material, where it enables each article to stand on its own. In the present case, the treatment of the common material would tend to diverge in the two articles - the one becoming a little simpler, and the other more advanced. -- Cheers, Steelpillow (Talk) 20:56, 29 October 2008 (UTC)

text:  In passing, I'm amused at the term polytope-specific, since polytope is the general case and polyhedron is specific. —Tamfang (talk) 07:24, 30 October 2008 (UTC)
thread_title:  Please, no 

message: The Dorman Luke construction of the dual essentially makes use of the fact that the facets of the dual polytope are precisely the dual of the vertex figures suitably enlarged. For polyhedra, it does not really matter whether it's the vertex figure or its dual, at least not for regular/uniform polyhedra, since regular polygons are self-dual. In higher dimensions, however, this quickly becomes obvious (e.g., the vertex figure of a 24-cell is a cube, yet its dual has octahedral cells; similarly, the vertex figure of the 600-cell is an icosahedron, but its dual has dodecahedral cells). Even in 3D, however, we do see a subtle hint that it's not simply the vertex figures, but the dual of the vertex figures, that form the dual polyhedron: take the vertex figures of the cube, for example. They are triangles, but oriented in the dual position to the orientation of the faces of the octahedron. Taking their duals gives us the faces of the octahedron in the correct orientation.
By this, it should be obvious that the Dorman Luke construction of the polyhedral dual is easily generalized to higher dimensions: given an n-polytope, finding its dual amounts to finding the duals of its vertex figures. Since vertex figures are (n-1)-dimensional, we simply recursively apply this process until we reach the trivial case (polygons).
I'd add this to the article, except that I'm not sure if this violates original research or not. :-)—Tetracube (talk) 20:36, 6 August 2008 (UTC)
replies: text: None of this is OR. However, one needs to be very careful in generalising - Luke's construction was originally described only for highly symmetrical polyhedra. Wenninger generalised it to some extent, but it does not apply to polyhedra in general. It is in fact a special case of the more general reciprocity (also called polarity or, incorrectly in this instance, projective duality) of any polyhedron about any sphere. Yes the Dorman Luke construction can be generalised to higher dimensions, but at this level of difficulty it is more productive to generalise polar reciprocation. FYI, see my essay on Vertex Figures. -- Cheers, Steelpillow (Talk) 21:21, 7 August 2008 (UTC)
thread_title:  Generalization of the Dorman Luke construction to ''n'' dimensions 

message: Has anyone noticed that all references, citations, and Wiki links to my transverse homonym are missing from these articles? Even his/her/their biography(ies) is (are) missing from WikipediA. Laburke (talk) 19:48, 1 November 2011 (UTC)
replies: text: I guess that's because he was (IIRC) an otherwise undistinguished school teacher, whose construction was only ever published by other authors. — Cheers, Steelpillow (Talk) 20:54, 1 November 2011 (UTC)
replies: text: Too bad, but people do seem to go on about his "generalization". Was he so undistinguished that they don't want to honor him by raising it to a "conjecture"? Perhaps his own schools were too undistinguished for that ;) As for school teachers in general, Eriugena was a distinguished philosopher yet he was killed by the quills of his students. Thanks for answering so quickly and BTW, I "see" polyhedra as central point(s) joining apices. It's a chemist sort of thing. Laburke (talk) 03:33, 2 November 2011 (UTC)
replies: text: The generalisation of his construction seems to have been first described by Wenninger. Both Luke's original and Wenninger's generalisation are simplifications of construction methods in projective geometry that had been known for many years - specifically the polarisation of a polygon in a conic section (here a circle). The appealing feature of these simplifications is that they are suited to the classroom, not that they have any theoretical novelty. BTW, Socrates was a distinguished philosopher but killed himself because he was condemned to do so by his fellow citizens. — Cheers, Steelpillow (Talk) 22:42, 2 November 2011 (UTC)
thread_title:  Transverse Homonym 

message: A sentence in Square pyramid prompts me to wonder: What's the simplest convex polyhedron which has V=F but is not self-dual? —Tamfang (talk) 05:33, 21 May 2010 (UTC)
replies: text: Gyrobifastigium looks like a good candidate. Tom Ruen (talk) 21:25, 1 November 2011 (UTC)
replies: text: First off, I realised that the solutions must come in pairs - dualising any solution obtains a second solution (because V and F dualise to each other). I then asked around. Someone pointed out that in the table of heptahedra, the 4th from the left in row 5 and the 2nd from the left in the last row are dual solutions. I checked the rest, then the simpler hexahedra, etc, and his solution appears to be correct. — Cheers, Steelpillow (Talk) 22:30, 2 November 2011 (UTC)
thread_title:  self and not 

message: Under Polar reciprocation it says:
:If r0 is the radius of the sphere, and r1 and r2 respectively the distances from its centre to the pole and its polar, then:
:r12 · r22 = r02 .
:Shouldn't the formula read:
:r1 · r2 = r02 ,
:in accordance with Inversive geometry#Circle inversion?
--Episcophagus (talk) 17:25, 2 December 2014 (UTC)
replies: text: Yes indeed. Now fixed. — Cheers, Steelpillow (Talk) 17:46, 2 December 2014 (UTC)
thread_title:  ''r''<sub>1</sub> · ''r''<sub>2</sub> = ''r''<sub>0</sub><sup>2</sup> 

message: Substituted at 19:56, 1 May 2016 (UTC)
thread_title: Assessment comment

message: An editor cannot summarily remove say a clarification or citation tag placed in good faith, without first dealing with it either in the article or on the talk page. I tagged "Grünbaum (2007) argues that any proper definition of a non-convex polyhedron should include a notion of a dual polyhedron." because the remark stands alone, without explanation, as the conclusion to a subsection and I fail to see any encyclopedic value in it. Is there any? — Cheers, Steelpillow (Talk) 10:32, 11 February 2017 (UTC)
replies: text: Here's the original edit in the self-dual section and part of a longer sentence.  Tom Ruen (talk) 10:50, 11 February 2017 (UTC)
replies: text: I had removed the bit about the excavated dodecahedron because any discussion of a definitional muddle (long resolved), between it and Brückner's icosahedron of the same outward appearance, belongs in that or the polyhedron article and not here. This leaves the Grünbaum quote with nothing to amplify. Hence my tagging it to see if anything worth amplifying came up. My tag was reverted without discussion, which breaches our behavioural guidelines (You are letting your tea go cold, David), so I opened a discussion. — Cheers, Steelpillow (Talk) 11:26, 11 February 2017 (UTC)
replies: text: I don't have any strong feelings about the inclusion or not of the excavated dodecahedron, but if there is a reasonable definition of geometric duality for non-convex polyhedra (that we have not yet presented in the article, because we only define abstract duality and have a sentence about polarity running into difficulties) then it might make a reasonable example for that. It at least has a well-defined vertex figure (as the link of each vertex lies entirely within the halfspace perpendicular to the center) whose dual polygon could be used as the face shape of a geometric dual. —David Eppstein (talk) 18:10, 11 February 2017 (UTC)
thread_title: Non-convexity

message: I have reported David at WP:ANI. Until that is resolved, I would be grateful if folks could regard this discussion as on hold. — Cheers, Steelpillow (Talk) 19:29, 11 February 2017 (UTC)
replies: text: "On hold" means that you get your way — the article stays in its misleading-to-readers state. Why should that be the default outcome of your drama-mongering? —David Eppstein (talk) 20:05, 11 February 2017 (UTC)
replies: text: In case other editors are worried by that, you are of course welcome to keep editing the article as usual. I am only suggesting that this discussion be put on hold for a bit. — Cheers, Steelpillow (Talk) 20:40, 11 February 2017 (UTC)

text: Now that the ANI issue is basically settled, I will return to these discussions once I know where they are taking place: both the duality and the "what is a polyhedron?" discussions have been forked to Talk:Polyhedron#Duality. May I suggest that the "what is a polyhedron?" definitional issue should be addressed there, but that the subsequent duality issue is best addressed here, where it began? It is not helpful to have parallel discussions on the same issue. If folks have a problem with that, please reply at the higher-level article discussion rather than fork the discussion about the fork! — Cheers, Steelpillow (Talk) 15:56, 13 February 2017 (UTC)
thread_title: Pause?

message: I received this message on my talk page and copy it here:
Hi!
I'm asking you just because you are the last user who edited the WikiVisually article entitled "Dual polyhedron", which i can not edit...
In its 1.1 paragraph, entitled "Polar reciprocation", is a very unclear mathematical writing; just after this equation of sphere:
x^2 + y^2 + z^2 = r^2,
the dual of a polyhedron P is expressed as:
P° = {y € R^3 / y.x =< r^2 for all x € P}
where y.x denotes the dot product of y and x.
Please, could you improve this writing into something like:
P° = {N € R^3, N // M, & 0 =< N.M =< r^2}(M € P)
?
In advance, thank you very much! :-)
I'm so sorry for bothering you... Entschuldigung! :-P
RavBol ()
:N and M are not good choices of variables for 3d vectors. —David Eppstein (talk) 17:17, 31 August 2020 (UTC)
replies: text: Having just defined x, y and z coordinates, the current x and y are probably even worse. — Cheers, Steelpillow (Talk) 17:55, 31 August 2020 (UTC)
replies: text: True. But other than in changing the variable names, I don't think the proposed formula is an improvement. —David Eppstein (talk) 18:09, 31 August 2020 (UTC)

text: There's no point in specifying that the point in P should be the parallel one with positive dot product, because the same inequality is valid for all the other points in P as well. The added specification just complicates the formula for no good reason. —David Eppstein (talk) 18:33, 31 August 2020 (UTC)

text: It wasn't a waste of time. The resulting change in variable names was a clear improvement. —David Eppstein (talk) 21:45, 31 August 2020 (UTC)

text: Sources written in languages other than English are allowed. However, they must be reliable sources (usually meaning, published by reputable journals or book publishers). Web pages are usually not reliable in this sense. —David Eppstein (talk) 19:12, 8 October 2020 (UTC)
thread_title:  Unclear formula in ''Polar reciprocation''? 

message: The Dorman Luke example of going from the cuboctahedron to its dual, the rhombic dodecahedron, needs a better explanation — especially of why the  vertex figure is drawn as a 1:√2 rectangle.216.161.117.162 (talk) 01:41, 26 September 2020 (UTC)
replies: text: if the reader cannot see that from the drawing of the cut apex, is there any point in explaining it? — Cheers, Steelpillow (Talk) 08:51, 26 September 2020 (UTC)
thread_title:  Dorman Luke example needs better explanation 

message: The MOS says: "Articles should avoid common blackboard abbreviations such as wrt (with respect to), wlog (without loss of generality), and iff (if and only if), as well as quantifier symbols ∀ and ∃ instead of for all and there exists."
Within a sentence in English: OK.
But within a math formula, like in Dual polyhedron#Kinds of duality##Polar reciprocation, replacing "∀" with "for all" results in mixing math symbols and English words...
Moreover, it would be consistent to replace "|" with "such that", and "{...}" with "the set of ...", too...
Suggestion: in a math formula, using the necessary math symbols, but bluelinking each of them to an explanation article.
: what do you think of this suggestion, please? --JavBol (talk) 01:29, 27 June 2021 (UTC)
replies: text: In general, it is better to spell out quantifiers in English words rather than using symbols, even within set-builder formulas as in this case. There are occasional exceptions where one is specifically discussing quantified formulas (see the first example displayed formula in true quantified Boolean formula) but they are rare and do not apply in this article. For the same reason, I would also strongly consider using the English word "in" in place of the set membership operator \in in the same formula. We need to keep in mind WP:TECHNICAL and the fact that many readers of this article are likely to be non-mathematicians; words are easier to read than unfamiliar notation. —David Eppstein (talk) 05:50, 27 June 2021 (UTC)
replies: text: Right.  In other words: a fake consistency does not serve readers well, and serving readers is our job.  Links in formulas are confusing and unclear, as well, and in 95% of cases should be removed, usually by rewriting to give an in-text gloss. —JBL (talk) 11:47, 27 June 2021 (UTC)

text: I think maybe it's better without the \in\mathbb{R}^3 altogether. It's unambiguous which set q comes from, so I think it just makes the formula unnecessarily complicated. —David Eppstein (talk) 16:28, 27 June 2021 (UTC)

text: P^\circ = \{q in Euclidean space \mid~q \cdot p \leq r^2, for all p in P\} ?
--JavBol (talk) 23:41, 30 June 2021 (UTC)
thread_title:  ALWAYS avoid quantifier symbols ∀ and ∃? 

message: Hello,
I tried Dorman Luke construction for many uniform polyhedra, It doesn't seem to work when the vertex figure is self-intersecting. Are you sure it doesn't mean the face of a face-transitive polyhedron is dual to it's dual polyhedron vertex-figure? And are you sure it's written in the source «Uniform polyhedron» not «Archimedean solid»?
Thanks, Hirbod Foudazi2 (talk) 15:26, 15 September 2021 (UTC)
replies: text: You will find many, if not all, of the constructions in Wenninger's book Dual Models (Cambridge University Press, 1983). It worked for him. — Cheers, Steelpillow (Talk) 16:17, 15 September 2021 (UTC)
replies: text:  No, It worked for convex examples for him. I saw page 30, it is written:"The easiest way to find the faces of convex duals (i.e., The Archimedean or semi-regular duals) is  to use the so-called Dorman Luke construction."  Thank you, Hirbod Foudazi2 (talk) 16:46, 15 September 2021 (UTC)
replies: text: See also paragraph on Pages 32-5 and the whole of Chapter IV. Indeed, I strongly recommend studying his entire text and accompanying drawings before jumping to conclusions. — Cheers, Steelpillow (Talk) 17:21, 15 September 2021 (UTC)
replies: text: Hello, I don't have access to the pages you mentioned (I found p.30 in Google books). So as you say it works for any uniform dual. Could you please show me how it works to find the faces of great triambic icosahedron for example? I might have understood it wrongly. Thanks for your attention,  Hirbod Foudazi2 (talk) 07:48, 16 September 2021 (UTC)
replies: text: The great triambic icosahedron is the dual of the great ditrigonal icosidodecahedron (W87). Its Dorman Luke construction is shown on Page 49. If you enable user email in your preferences and send me an off-wiki message, I can send you a photo of the page. — Cheers, Steelpillow (Talk) 08:22, 16 September 2021 (UTC)

text: I have moved it across and left behind a small subsection, which others of you may or may not wish to rearrange. — Cheers, Steelpillow (Talk) 08:28, 20 September 2021 (UTC)

text: See two last paragraphs of the page I mentioned in particular. Hirbod Foudazi2 (talk) 06:05, 23 September 2021 (UTC)
replies: text: I would assume that Steelpillow is already familiar with Cundy & Rollett's coverage of this topic. —David Eppstein (talk) 06:50, 23 September 2021 (UTC)
replies: text: I did not say he/she doesn't. Hirbod Foudazi2 (talk) 08:37, 23 September 2021 (UTC)

text: Indeed I am. It is the original published source of the Dorman Luke construction. But I am not sure what point is being made here. Nor am I clear that I am worth arguing over. — Cheers, Steelpillow (Talk) 08:41, 23 September 2021 (UTC)
thread_title:  Dorman Luke construction for all uniform polyhedra? 

message: Wenninger uses the Dorman Luke construction for many if not all of the nonconvex uniform duals. While it would appear to work for all polyhedra with a midsphere, including canonical dual pairs, I know of no reliable source stating this. So, while Wenninger's use also clearly makes the article on the convex Catalan solids an inappropriate home, I do not think that any argument can be sustained against moving it to that on the Uniform dual polyhedra. I personally dislike my own suggestion, as I am sure that one day we will have to move it again, but for now it does seem to be the home that best conforms to Wikipedia's policies. — Cheers, Steelpillow (Talk) 15:52, 19 September 2021 (UTC)
replies: text: I have moved it across and left behind a small subsection, which others of you may or may not wish to rearrange. — Cheers, Steelpillow (Talk) 08:28, 20 September 2021 (UTC)

text: See two last paragraphs of the page I mentioned in particular. Hirbod Foudazi2 (talk) 06:05, 23 September 2021 (UTC)
replies: text: I would assume that Steelpillow is already familiar with Cundy & Rollett's coverage of this topic. —David Eppstein (talk) 06:50, 23 September 2021 (UTC)
replies: text: I did not say he/she doesn't. Hirbod Foudazi2 (talk) 08:37, 23 September 2021 (UTC)

text: Indeed I am. It is the original published source of the Dorman Luke construction. But I am not sure what point is being made here. Nor am I clear that I am worth arguing over. — Cheers, Steelpillow (Talk) 08:41, 23 September 2021 (UTC)
thread_title: Proposed section move 

message: In the passage "For example, the dual of a regular tetrahedron", there should be a link from 'tetrahedron' to the Wikipedia article on that topic. Kontribuanto (talk) 00:34, 2 March 2024 (UTC)
thread_title:  missing link