Talk:Fano plane

message: 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 Wikipedia:COMMONNAME. Thoughts? --Bill Cherowitzo  (talk) 19:09, 24 January 2019 (UTC)
replies: text: See Kirkman's schoolgirl problem#Galois geometry for a 1910 reference and doi — Rgdboer (talk) 03:13, 25 January 2019 (UTC)
replies: text: 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)

text: 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)
replies: text: 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)

text: 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)
replies: text: 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)

text: 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)
thread_title:  Fano 3-space 

message: 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)
replies: text: This page is for discussing improvements to the article to which it is attached. --JBL (talk) 14:12, 5 October 2021 (UTC)
thread_title:  And why, exactly, is this discussion not appearing in the "Level 5 Mathematics" discussion index? 

message: It is a bit tricky to distinguish the two conjugacy classes with 7-cycles. Currently the article describes them in this style: "
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)
thread_title:  Collineations 

message: 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)
replies: text: 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)
replies: text: I removed the generators from the article. I think all these group details would make more sense in PSL(2,7). BTW,  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)
thread_title:  Wrong permutation in generating set