Talk:Duality (mathematics)

How would you call this duality?

16×16 matrix of 1×4 matrices

Below the dual 1×4 matrix
of 16×16 matrices

When you have an m×n matrix of p×q matrices, it is possible to define a dual p×q matrix of m×n matrices. Compare: v:User:Watchduck/hat#Dual_matrix I often use this to take a closer look at the binary digits, when the elements of a matrix are binary numbers. That can be seen in the nimber multiplication table on the right, with the permuted binary Walsh matrices below. Does anyone know, how this kind of duality is called? It's not limited to matrices. Watchduck (talk) 22:10, 17 October 2011 (UTC)

It sounds to me that you really have a tensor of dimensions (m,n,p,q) and you are taking a transpose of it. —David Eppstein (talk) 22:01, 17 July 2019 (UTC)

@David Eppstein: When the nested objects are matrices, that is a way to see it. But I don't think it is helpful to talk about the multiplication table on the right as something with 3 dimensions (or about this Cayley table as something with 4). For this graph "containing" 2×2 matrices the equivalent of your tensor would be a graph product, and the dual is a 2×2 matrix containing colored graphs. How would you call the equivalent to the transpose in this case?

What I was looking for when I asked this question was the right way™ to say something like:

The four elements of the dual matrix are permuted Walsh matrices. Watchduck (quack) 21:58, 22 July 2019 (UTC)

I don't know offhand, but I suspect the right answer is that if you find a reliable source talking about permuting this specific matrix in this way you can use whatever terminology they use, and if not then it's WP:OR and off-topic here. —David Eppstein (talk) 22:37, 22 July 2019 (UTC)

Dual graphs

Dual graphs are discussed in the article, but only in the case where the embedding surface is a plane (or equivalently a sphere). There is no mention of dual graphs in other surfaces, although the term is used in WP articles, e.g. Dyck graph. In view of this omission, it is not surprising that there is also no mention of Petrie duals. Maproom (talk) 22:21, 2 January 2015 (UTC)

Dual cone

I think the recent addition of the "Dual cone" section, and particularly its position near the top of the "Introductory examples", is unfortunate:

• It is not an example of duality: the relation is not symmetric.
• It is obscure. Dual polytopes, and dual graphs, will be more familiar to most readers.
• It is poorly explained. The text does not make it sufficiently clear that the position of the cone depends on the origin of the space. The diagram does not make it clear which region constitutes the dual cone.

Maproom (talk) 10:40, 24 July 2015 (UTC)

I disagree with you regarding all points. It is the very purpose of mentioning this duality that not every duality is involutive, such as the one on vector spaces, for example. To properly define dual polytopes you need quite some space (as opposed to hand-waving where the vertices of the dual polytope lie). Likewise for dual graphs (look up what precise assumptions you need to even define the dual graph). The origin of the space is implicit in talking about points in R^2. Other than that it is also entirely irrelevant, any choice of origin will do. Finally, the picture does display the dual cone in red.

The purpose of this introductory list is to give examples of dualities whose structural properties appear again and again. Moreover, the rationale behind this list is to increase their generality and mathematical complexity. The dual cone is just fitting right there, I think. Jakob.scholbach (talk) 12:24, 24 July 2015 (UTC)

If the dual cone example started by saying that the example is not involutive, I might agree with your first point. t doesn't. It is therefore likely to confuse the readers of this "introductory example". Yes, for dual graph, you need a space, but not one with a metric and an origin. The picture displays the dual cone in red, but the reader may fail to guess this, the label C^* is in the lowest of the three triangles into which the red area is divided. You say "any choice of origin will do"; it seems to me that this is false, the position of the cone is determined by the origin, its apex is at the origin.

For an expert like yourself who already knows about dual cones and the more familiar properties of duality, the example may be a good one. But in a list of "introductory examples", I believe it is confusing. It would be better to have an example which most readers are likely to understand, for example the way that the octahedron is the dual of the cube. Maproom (talk) 16:32, 24 July 2015 (UTC)

What exactly do you consider confusing about the dual cone?

Actually I just learned about the dual cone while working on this article. I chose it as a prototypical example, because it is completely elementary, and the closely related to the duality in optimization. I plan to work on the further sections to make this clearer.

My comment about the origin was intended to say that the precise choice of the origin does not matter. Choosing one origin of course does matter. Jakob.scholbach (talk) 21:39, 26 July 2015 (UTC)

I am sorry, Jacob, but Maproom is right: the "dual cone" as presented is not an example of duality in the sense of the article, and the picture does more harm than good. I suggest that you delete this subsection. There is a standard dual cone construction for convex cones (related to polarity for convex sets containing the origin) that qualifies far better under this rubric; whether or not it is an introductory example is arguable. Arcfrk (talk) 05:11, 27 July 2015 (UTC)

Well, we can certainly tweak the section and emphasize it is a duality in the narrow sense on cones. If you dislike the picture, maybe you can create a better one? Jakob.scholbach (talk) 15:45, 27 July 2015 (UTC)

The key word is "convex". Applying this construction to an arbitrary set (even a cone) suffers from the same defect as taking orthogonal complement to an arbitrary subset C of a linear space: namely, the result depends only on the span of C; only in the case of polar duality, convex span replaces the span. In effect, you compose a genuine duality between convex sets (or linear subspaces) with a "forgetful map" from general sets/cones (respectively, general subsets) to convex (respectively, linear) ones — this is what makes the construction asymmetric. And yes, the convex duality is more sophisticated than the linear one, so it would be logical to treat it later in the article. Arcfrk (talk) 07:14, 28 July 2015 (UTC)

There is no formal definition of "duality" in the article; so it should cover all mathematical concepts which are called "dual". Apparently, "dual cone" is such "an example of duality in the sense of the article". I don't understand the problems that Maproom and Arcfrk have with it, except that it wasn't familar to them (it was neither to me), and its explanation could be improved (I'd like to contribute when this dispute is settled) . - Jochen Burghardt (talk) 16:25, 27 July 2015 (UTC)

As I said before, the purpose of the "Introductory examples" section should be to introduce the subject of the article to readers, using as far as possible examples that may already be familiar to them. Complement of a subset is good for this; so would dual polyhedron be. That section should not be used to teach unfamiliar material to them. I don't object to "dual cone" being explained later in the article, particularly now that that subsection is rather clearer. Maproom (talk) 17:09, 27 July 2015 (UTC)

Ok, I see. Do you suggest another non-involutionary introductory example, or do you think we don't need one (in the introduction)? - Jochen Burghardt (talk) 17:24, 27 July 2015 (UTC)

I am still not convinced that the dual polyhedron is actually easier than the dual cone. As I said before, precisely even defining the dual polyhedron requires quite some space. It is not a very prototypical example. For example, it is harder (it seems) to understand the fact that in practically all dualities you have a reversal of morphisms.

@Arcfrk: of course every cone is convex here (and closed). I guess the every day meaning of a cone presupposes this, but I had already added a footnote to make this precise. This is a technical point which deserves to be briefly mentioned, but it is not a fundamental difficulty.

I suggest we all spend more time on the article itself than on the talk space. Jakob.scholbach (talk) 08:04, 28 July 2015 (UTC)

The question is not whether "dual polyhedron is easier than the dual cone", it's whether it is more familiar to the typical reader. The introductory section is not there to present unfamiliar and interesting examples of duality, it's there to help the reader grasp what duality is. Maproom (talk) 08:21, 28 July 2015 (UTC)

Exactly. I think the reader will not benefit from an early mention of the dual polyhedron in the sense that he / she will not grasp better what duality is about in genereal. The duality of polyhedrons is simply not a prototypical duality, therefore it does not help that much to understand what the general phenomenon of duality is. You can talk about it using elementary language, yes, but the same is true for the dual cone. The dual cone is easier to precisely define. The dual cone highlights the fact that dualities are not in all cases involutions. In short, the dual cone is a better example than the dual polyhedron.

If you are still not convinced, I heartily invite you to write an introductory section (possibly starting from what we have in the article) about the dual polyhedron; maybe here in the talk space. You will see that making it precise is not that easy; the space you will spend is better invested in other dualities. Jakob.scholbach (talk) 12:29, 29 July 2015 (UTC)

I believe that the article would be improved by removing the "dual cone" example from the introductory section. The essence of duality is that it's reflexive. Ok, the concept is extended to non-reflexive subjects such as dual vector spaces. But the introductory section should be addressed to readers who are not yet familiar with the concept. Using an introductory example which is obscure, illustrated by a diagram that's hard to understand, and above all is not reflexive and does not even mention that it's not reflexive, is likely to induce readers to give up and look for some other explanation of duality. Maproom (talk) 19:00, 17 November 2016 (UTC)

I'm not sure that the dual cone example should be removed, but I am certain it should be moved down. It is not an introductory example, and it should not be the second example on the page. An introductory example should be something familiar to the reader, which gives the correct general idea, to be refined later. The dual cone construction may be a good example to show that not every duality is an involution, but if so it does not make sense to explain that before the reader has a clear idea what a duality is. The dual cone construction is unfamiliar even to many mathematicians. I will shortly move it downwards. —Mark Dominus (talk) 21:41, 18 November 2016 (UTC)

OK, move it down then. Jakob.scholbach (talk) 18:57, 19 November 2016 (UTC)

I've changed my mind. I think the dual cone and Galois group sections should both be moved down, but the way the article is structured at present there isn't a good place to move them to. And I think some of the more familiar examples should be moved up. But I don't want to undertake a wholesale reorganization of the article's examples, so I'm going to leave it alone, at least for now. —Mark Dominus (talk) 20:45, 29 November 2016 (UTC)

Missing Link

The order-theoretic duals link is missing Chris2crawford (talk) 15:18, 22 September 2015 (UTC)

Flow-colouring duality

... in the sense of 6 Flows, 6.5 Flow-colouring duality, textbook *Graph Theory* by Reinhard Diestel. Is this one covered in this article, plus in List of dualities § Mathematics? I am a student yet, but I think this duality is omitted in both. --Szozdakosvi (talk) 13:49, 9 March 2021 (UTC)