Talk:Fano plane

Citation needed
Hi everybody, long time no see ! 6 years ?

This section is about references on configurations.

Having studied ( I think) everything written on species, today I am able to count configurations in Fano plane. Here is how it works, following the references list. Below I show for lines and triangles.

1) in BLL book, page 108, exercise 3.31.b.iii take Fano pg for the G species and for F the S_3 acting on a 3-set. It results a species cycle index.

2) By using Conway-Hulpke-Mckay appendix A and Maple, identify the candidates for stabilizers.

3) Come back from species to stabilizers via a Yeh article. Stabilizers will give the size of the orbits.

4) I would add a Labelle article that insists on fix(sigma), the power piece of this approach.

So we have a book and three articles that work together, none of them explicitly referring to Fano plane configuration.

If not accepted the above references, we will have to wait until some reputed author will write something on the cycle index of a group action.

Thanks for your eventual opinion !

and yes, there are 42 Kleins (rectangles) in the Fano plane :) Nboyku (talk) 13:45, 2 June 2018 (UTC)


 * Well I won't go as far as to say welcome back, but ...
 * some things have not changed. Since none of your sources actually talks about the Fano plane, you are not paraphrasing their content. What you are doing is using the content to come up with your own arguments and this is considered to be WP:SYNTH a form of WP:OR and not allowed. It also appears that all of the objections that were raised in the past are still valid. In particular, none of this fancy machinery is needed to reach the conclusions of this simple single example and trying to incorporate it can only lead to confusion for our readers, especially when you attempt to bend standard terminology to fit your particular viewpoint. --Bill Cherowitzo (talk) 19:19, 2 June 2018 (UTC)
 * Thank you for taking your time. I have done my mile and I really appreciate it, more than always. This is the best hit I have. Right now I really doubt that someone will write on configuration counting why ? sincerely speaking configurations are species and nothing else, and the job is already done. Another thing that has not change is that I still really need the Fano plane in order to explain species to a third party. But there is a good part, something that have changed -> it's me, today I know how to count configurations :) Anyway if I ever fall on some literature more suitable to your vision I will bring it here. Nboyku (talk) 21:04, 2 June 2018 (UTC)
 * Before counting rectangles, one has to identify lines (given the collineation group only). In order to do this we have to go to (1937, Carmichael) and see the link between double-transitivity and lines. Coming back, points of a triplet are doubly-transitive permuted by the collineation group. Then, by evaluating the action on triples, the lines reveal themselves : 35 triples = 7 lines + 28 triangles.
 * Lines being identified, we may talk about rectangles. For each rectangle, two edges point toward the first point 0 on the horizon line, and two edges point toward the second point 1 of the horizon line. There are seven (horizon) lines, each being determined by two points, a 0 and a 1 that may be chosen in 6 ways. Therefore, there are 7x6=42 rectangles.Hubby56 (talk) 10:09, 22 January 2021 (UTC)

Fano 3-space
Recent additions to this page involving Fano 3-space, while greatly appreciated, do not seem to fit the topic of this article. I propose that this material be taken out and a new article be created with that subject matter. The title of the proposed new article is a little problematic. It would naturally be called "PG(3,2)" for that is what it is almost universally called by geometers. I dislike titles that consist solely of specialized notation as this gives readers unfamiliar with the subject absolutely no clue as to what the article is about. Alternatives, such as "Fano 3-space", are just not widely used and would thus go counter to COMMONNAME. Thoughts? --Bill Cherowitzo (talk) 19:09, 24 January 2019 (UTC)
 * See Kirkman's schoolgirl problem for a 1910 reference and doi — Rgdboer (talk) 03:13, 25 January 2019 (UTC)
 * The point is not whether PG(3,2) is notable. It's whether it should be covered so extensively in an article about a different space. Surely you don't think, for instance, that there should be a detailed history of the presidency of Donald Trump in our article about Barack Obama, merely because one was the immediate successor of the other? —David Eppstein (talk) 06:13, 25 January 2019 (UTC)
 * Lifting mathematical patterns to living situations allows more people to visualize those patterns. The creativity of Thomas Kirkman in 1850 brought PG(3,2) to the daily outings of school children, and his problem is a notable chapter in combinatorics reflected by the "problem". The link was made here in Talk to show PG(3,2) is already illustrated by that article on a classic instantiation. My contribution should not be interpreted as support for inclusion of PG(3,2) in Fano plane. — Rgdboer (talk) 02:03, 26 January 2019 (UTC)
 * I am a bit confused by the point you are trying to make. It doesn't seem to be related to the issue I raised. The connection between Kirkman's schoolgirl problem and PG(3,2) is also rather tenuous. There are 7 non-isomorphic solutions to the 15 schoolgirl problem and not all of them can be embedded in PG(3,2). The solutions were all known by Woolhouse in 1862, but the realization of some of these solutions as packings of PG(3,2) did not come about until Conwell gave it in 1910.--Bill Cherowitzo (talk) 03:52, 26 January 2019 (UTC)
 * The phrase Fano three-space is not notable, and leaving it on the page is deceptive for students looking for correct terminology. It is an example of a three-dimensional space so the section could be moved there. Further, it exemplifies finite geometry, where Kirkman has already been mentioned. But as the Kirkman problem does not correspond exactly with PG(3,2) (as noted), the colorful scenario in that article can not cover it. An article titled binary geometry covering PG(n,2) would have Fano plane for n=2, and n=3 for "Fano 3-space" as described here. Checking for notability, Binary geometry turned up at finitegeometry.org! — Rgdboer (talk) 03:35, 27 January 2019 (UTC)
 * As Cullinane points out in that blog, there is no field called binary geometry and he then goes on to to say that it would therefore be a good name for the kinds of things that he is interested in, namely, sets having 2n elements. The projective geometries of characteristic two in any dimension do not have that number of elements. These are Galois geometries and at the present time it would probably be better to add a section in that article about the geometries of characteristic 2. This does not preclude an article on the smallest three dimensional Galois geometry. --Bill Cherowitzo (talk) 06:24, 27 January 2019 (UTC)
 * I reverted your proposed section header since it did not describe PG(3,2). In fact, all PG(n, 2) for n > 1 have planes that are all Fano planes.--Bill Cherowitzo  (talk) 05:26, 29 January 2019 (UTC)

And why, exactly, is this discussion not appearing in the "Level 5 Mathematics" discussion index?
Is it perhaps because the Fano plane does not focus or bear on prime numbers, the Riemann hypothesis, or the Goldbach conjecture? It's like, mathematical obessive-compulsive disorder has regard for only prime numbers as a flavoring. Danshawen (talk) 14:01, 5 October 2021 (UTC)danshawen
 * This page is for discussing improvements to the article to which it is attached. --JBL (talk) 14:12, 5 October 2021 (UTC)

Collineations
It is a bit tricky to distinguish the two conjugacy classes with 7-cycles. Currently the article describes them in this style: " A maps to B, B to C, C to D. Then D is on the same line as A and B. " There is also a way to do this without referring to the geometry of the Fano plane. It can be seen in cycle images like those on the right, and expressed in the following formula:


 * $$p$$ is the cycle of permutation $$P$$.   $$n$$ is any integer.    $$\oplus$$ is the bitwise XOR.
 * Then $$P$$ is in conjugacy class $$\begin{cases}

\text{7a}, ~ \text{if} ~ p(n \bmod 7) ~=~ p((n+2) \bmod 7) ~\oplus~ p((n+3) \bmod 7) \\ \text{7b}, ~ \text{if} ~ p(n \bmod 7) ~=~ p((n-2) \bmod 7) ~\oplus~ p((n-3) \bmod 7) \end{cases} $$.

I find this easier, when the conjugacy classes have names, and I chose 7a and 7b. See here for details. Maybe someone wants to use this in the article. I don't know if there is a source for the current description, or if one could be found for this one.

(When half arrows are interpreted as edges, these images show the Heawood graph.) Greetings, Watchduck (quack) 00:36, 21 January 2022 (UTC)

Wrong permutation in generating set
Currently the article states, that the group is generated by (1432657), (162)(374), (14)(27), (17)(24) and (17)(24)(36). But the last permutation must be wrong. None of the described conjugacy classes has this cycle type. You have added this example four years ago. Is that example from the given source? (Pisanski & Servatius 2013, p. 171) I don't get why the generating set is so large. As far as I can see, two generators are enough. I tried (23)(4657) and (14)(36). (Gray code and bit-reversal permutation.) Greetings, Watchduck (quack) 17:39, 26 July 2022 (UTC)


 * I agree that the last element of the generating set is incompatible with the rest of the information in the section. The first four elements generate a group of order 168 -- but in fact that group is already generated by the 7-cycle and any one of the following three elements.  --JBL (talk) 19:37, 26 July 2022 (UTC)


 * I removed the generators from the article. I think all these group details would make more sense in PSL(2,7). BTW, here is the Cayley graph generated by Gray code and bit-reversal permutation. (It is a rather complicated graph, related to the .) --Watchduck (quack) 13:53, 26 August 2022 (UTC)