Talk:Duality (projective geometry)

First talk

• "Then the unique great circle perpendicular to point P3 is line L3 in S1."

How can a circle be perpendicular to a point?

• "L1 is the unique great circle in a plane perpendicular to the line through the pair of points P1"

Why is P1 referred to as a pair of points when it was originally defined as a single point? Is the author referring to the antipodal equivalent as well here? If so, then a definate article for "the line" doesn't make sense since there is no unique line through antipodal points.

Update: I think I understand what you were trying to say now. By line, I think you mean a line through euclidian 3D space, although earlier you defined a line to be a great circle. I will submit a re-wording of your example if I am successful in coming up with a more clearly stated revision.

The footnote

>>*¹ When we say a line through pair of points P, or simply a line through P, we refer to the three dimensional euclidian line that passes through the antipodal points represented by P. When we say that a geodesic line, or a great circle, L is perpendicular to the line passing through the pair of points P, we mean that L lies on the plane that is perpendicular to, and intersects at the midpoint of, the straight line segment in euclidian space that connects the antipodal points that is represented by P. In other words, L is the set of points equidistant in euclidian space to the antipodal points represented by P. L is unique for P. <<

is exactly right: it is precisely what was meant! Apparently there was no conscious awareness of just how the meanings of "line" and "circle" were being overloaded. --AugPi 00:55, 18 Feb 2005 (UTC)

Points and lines in the plane

It seems to me this section makes it harder to understand than is really necessary, since using (m, b) to represent a line with slope m and intercept b misses some lines, and also loses the symmetrical quality of projective space. I'm inclined to think it needs a massive rewrite, starting from the point of view of the section I added on duality in terms of vector spaces. Gene Ward Smith 14:38, 5 May 2006 (UTC)

I suggest moving most of that section to Duality (projective geometry)/Proofs, in the same fashion as has been done in Category:Article proofs. Such busy-body expositions may be useful to some, but do make the article much more opaque. linas 14:31, 12 December 2006 (UTC)

While I would normally prefer to edit this section so that it was at least correct, I am finding this section hard to deal with in that way. Firstly, the statement is wrong as there is no duality in affine planes. The section ignores vertical lines. Secondly, the formulas seem to show the existence of a duality, but this is due to ignoring possible divisions by zero. To fix this, one would need to remove lines from the plane without removing the points on them leaving a structure which is not that familiar, or build the affine plane up to a projective plane by adding points at infinity - but then you have nothing that hasn't been said in earlier sections. So, with heavy heart, in my next edit I will remove this section. Bill Cherowitzo (talk) 21:21, 27 September 2011 (UTC)

Self dual

I think that the main article needs a good definition what self-dual means in the context of projective geometry. The definition in dual (mathematics) is not clear enough.

TomyDuby (talk) 08:02, 14 February 2009 (UTC)

Duality mapping defined

Three questions to the first paragraph of this section:

1. The word reciprocal needs to be explained.
2. Is metrics defined in this space?
3. Does the definition depend on the choice of origin?

TomyDuby (talk) 11:09, 28 February 2009 (UTC)

Re 1, I think the word reciprocal has its ordinary meaning, and doesn't need a definition presented here.

Re 2, what's being presented is actually a particular model of projective geometry. Projective geometry is a nonmetrical theory. This model, however, is a model in the Euclidean plane, which has a metric.

Re 3, it clearly does, but that's a feature of the model only. No special point is singled out by the axioms of projective geometry.--76.167.77.165 (talk) 23:14, 4 September 2009 (UTC)

I think this section, and the pretty, colorful figure, are sweet, and I love their concreteness. I would like to see it stay. However, it was clearly thrown in here by someone who was naive about its significance. Its presentation as if it was more than a model is a symptom of a broader lack of organization in the article.--76.167.77.165 (talk) 23:14, 4 September 2009 (UTC)

Problems with this article

Duality is a property of projective geometry that IMHO makes the subject "beautiful". You could not however get that impression from reading this article. It is poorly written, overly restrictive and incoherent at times. Most of the article makes it seem like the only projective plane is the real projective plane. One of the reasons that this is a bad idea is that it is only the self-dual property of the real projective plane that makes some statements true - these are not generally true statements for all projective planes. One has to be careful with planes since there are projective planes that are not self-dual. The problem does not arise in higher dimensional projective spaces. The only reference given (to MathWorld), contains the same error and makes blatantly false statements about duality. There is also no clear statement about the Principle of Duality vs. duality as a mapping in this article. So unless you want to change this to Duality (real projective plane), I'd suggest that we start working on fixing this article. Wcherowi (talk) 20:47, 10 September 2011 (UTC)

Seeing no takers I've started the process myself. Bill Cherowitzo (talk) 06:27, 25 September 2011 (UTC)

Mistake with Self-Duality

In the article, it was claimed that projective planes PG(2,K) for any division ring K are self-dual. This, however, is not true. It follows easily from Second Fundamental Theorem of Projektive Geometry that this is true if and only if the division ring is isomorphic to its dual (the division ring obtained by reversing the two arguments in the multiplication). Thus, the statement remains true, if one additionally requires the plane to be of finite order (because this automatically means that $K$ is a field). The mistake was also noticed on the math research forum MathOverflow (see http://mathoverflow.net/questions/114305/is-wikipedia-correct-about-desarguesian-projective-planes-being-self-dual). I have corrected this. 89.199.235.94 (talk) 18:37, 6 January 2013 (UTC)

"Metamathematical"?

The first sentence says, "duality is the formalization of this metamathematical concept". I think this should be "duality is the formalization of this informal concept", unless the literature describes symmetry as metamathematics (which I've never seen). Metamathematics ordinarily means the study of models of the semantics and syntax of axiom systems using mathematics "one level up". 108.49.114.34 (talk) 14:20, 26 October 2014 (UTC)

Sounds reasonable. I think what was meant here is "protomathematical" rather than "metamathematical". Tkuvho (talk) 14:33, 26 October 2014 (UTC)

Agreed. I removed it. Bill Cherowitzo (talk) 17:42, 26 October 2014 (UTC)

Redirect?

Should plane duality redirect here? — Preceding unsigned comment added by 70.247.174.22 (talk) 13:12, 22 December 2015 (UTC)

Yes, that would make sense since a new page with that title would only repeat what is on this page. Bill Cherowitzo (talk) 17:09, 22 December 2015 (UTC)

Someone looking for that title might equally well be looking for the duality of plane graphs (in this context, "plane" is used to mean that a planar graph has a specific planar embedding already given). Our article on that topic is at dual graph. So I think that if plane duality links anywhere it should be to a disambiguation page. —David Eppstein (talk) 17:27, 22 December 2015 (UTC)

Oops. You're right. I guess I was thinking like a geometer again. Bill Cherowitzo (talk) 17:59, 22 December 2015 (UTC)

Graph duality

I assume that the notion of graph duality is captured by the notion of plane duality mentioned in this article. I wish that was clearer from the article, however. — Preceding unsigned comment added by 70.247.174.22 (talk) 13:47, 22 December 2015 (UTC)

No, these are different concepts with a tenuous connection (via polyhedra). Perhaps a hatnote would be in order, but the page Duality (mathematics) would not be a good choice to send folks interested in this topic. Dual graph is the appropriate page. Bill Cherowitzo (talk) 17:19, 22 December 2015 (UTC)

Permutation of the subspaces

under "general projective spaces" what is meant by "A duality δ of a projective space is a permutation of the subspaces of PG(n, K)", specifically, "permutation of the subspaces"? Blackjanek (talk) 15:39, 14 March 2023 (UTC)