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Types of configurations?
Nice article, David, and I agree with your edit summary that the notability is obvious. Two things: First, I am bit confused about the phrase "types of configurations". Is the type of a configuration a formal notion that needs defining, or are you just informally indicating how configurations are categorized? If the latter, is "types of" really needed here? Second, and speaking of categorization, I added the Projective geometry category, which I assume is OK with you. I wonder if the Euclidean plane geometry category is really appropriate though, considering the nature of, for instance, the Fano plane. Michael Kinyon 12:13, 29 September 2006 (UTC)
- Thanks! Re types: what I meant was that duality takes e.g. a Pappus configuration to another Pappus configuration. But not necessarily to the same Pappus configuration: at least, I'm not sure offhand if all Pappus configurations are projectively equivalent, and certainly all (5252)'s aren't. Re categories: yes, projective geometry makes more sense than Euclidean as the main cat for this one. I should have checked that it existed... I'll take off Euclidean as it's less relevant, though I think complete quadrangle should stay in both. —David Eppstein 14:59, 29 September 2006 (UTC)
This notation is incomplete
Excerpt from this article:
- A configuration is denoted by the notation (pγlπ), where p denotes the number of points, l denotes the number of lines, γ denotes the number of lines per point, and π denotes the number of points per line. These numbers necessarily satisfy the equation
- pγ = lπ
- as this product is the number of point-line incidences.
- [ snip ]
- * (103103), Desargues configuration.
This could leave the impression that the notation (pγlπ) determines the configuration up to incidence isomorphism. But in fact there exists at least one (pγlπ) configuraation that is not incidence-isomorphic to the Desargues configuration (I'm going to see if I can find it on the web and post a picture here). Since that one example is all I know, I'll let others see if they can add some material to the article on this point before I do. Michael Hardy 18:11, 29 September 2006 (UTC)
- Problem 7.7 on page 77 of this reference suggests a pair of mutually inscribed pentagons as another (103103). I'll see if I can produce a drawing of this. —David Eppstein 18:37, 29 September 2006 (UTC)
- I've just tried for a half-hour to view that page; my browser just spins its wheels. But "pair of mutually inscribed pentagons" is at least suggestive of the same sort of thing that I remember seeing. Michael Hardy 19:04, 29 September 2006 (UTC)
I've added this paragraph to the article:
- The notation (pγlπ) does not determine a projective configuration up to incidence isomorphism. In particular, there exist (103 103) configurations that are not incidence-isomorphic to the Desargues configuration.
Michael Hardy 19:13, 29 September 2006 (UTC)
- I changed what you added to use Pappus instead of Desargues, to match what Hilbert wrote — he mentions the three different 93 configurations but doesn't seem to say what else is possible for 103. Also, it turns out (p.127) that the two mutually inscribed pentagons is just Desargues again. —David Eppstein 20:25, 29 September 2006 (UTC)
- I see that someone (you?) has put a picture of two "mutually inscribed pentagons" into the article titled Desargues' theorem. I surmise that that may be the example you're referring to here. But it's different from the one I had in mind. Do you happen to know whether any theorem states that any two 103103 configurations must be incidence-isomorphic? Michael Hardy 20:12, 12 October 2006 (UTC)
- I'm not certain, but I think it's likely that there are multiple nonisomorphic 103103's. Certainly there are multiple 9393's, as the current version of the entry and Hilbert both state. The mutually inscribed pentagons turned out to be isomorphic to the Desargues configuration (as Hilbert also writes), which is why I added them to that entry rather than here.—David Eppstein 20:22, 12 October 2006 (UTC)
The OEIS entry for the sequence that was just added says:
|“||A combinatorial configuration of type (n_3) conists of an (abstract) set of n points together with a set of n triples of points, called lines, such that each point belongs to 3 lines and each line contains 3 points.||”|
That sounds as if they're not actually worrying about whether the "configuration" thus defined is actually realizable in some projective space. Are the 10 nonisomorphic configurations of type 103103 realizable in real projective space? Maybe in the real projective plane? How much does the answer change if one replaces "real" by other fields of scalars? Michael Hardy 19:39, 16 October 2006 (UTC)
- I don't know, but the article already mentions the issue of abstract configurations not being realizable in the projective plane, when it discusses the Fano plane. —David Eppstein 20:17, 16 October 2006 (UTC)
Does anyone know of a published reference for the spatial configuration of nine points (six vertices of a triangular prism, and three centroids of its square sides) and nine planes (six through a vertex and opposite edge of the prism, and the three square sides)? It has type (9595). If it's well known, I'd like to mention it, but I can't find it in Hilbert or elsewhere and don't want to commit WP:OR. —David Eppstein 23:13, 16 October 2006 (UTC)
Title of article
Most books that I have read tend to refer simply to "configurations" rather than "projective configurations". Certainly the theory of configurations is at its most complete in projective spaces. Though I do not have a copy of Hilbert and Cohn-Vossen to hand, ISTR that it (briefly) mentions them in a more general context before addressing the projective variety. I'd suggest that this article is moved to "Configuration (geometry)". -- Steelpillow (talk) 19:22, 8 April 2008 (UTC)
- The title of the section of Hilbert and Cohn-Vossen on this subject (in the English translation I have at hand) is "projective configurations". They start in a less general context, the Euclidean plane. But they call these things "configurations" in many places without qualifying them as projective. I don't see any reason to object to your proposed change. —David Eppstein (talk) 20:17, 8 April 2008 (UTC)
Ernst Steinitz wrote his dissertation on the subject in 1894, and they were popularized by Hilbert...
- The subject of this entire paragraph is configurations and there is no ambiguity here. Steinitz's dissertation was on the construction of n3 configurations (in German of course). Configurations play a dominant role in Hilbert's book. The continuous repetition of the word "configuration" would make this paragraph sound very stilted and sophmoric. Bill Cherowitzo (talk) 21:17, 10 February 2016 (UTC)