Talk:Fano plane/Archive 1

Numbering, please

Section added. —Nils von Barth (nbarth) (talk) 07:58, 25 November 2009 (UTC)

Please add a fano plane with numbering.

http://math.ucr.edu/home/baez/octonions/img113.gif

http://math.ucr.edu/home/baez/octonions/fano.gif

This website is a good introduction model - Please include it as an external link http://math.ucr.edu/home/baez/octonions/node4.html

Please add a reference to an incidence matrix. This website is a good introduction model of an incidence matrix - http://mathworld.wolfram.com/IncidenceMatrix.html GeMiJa 13:27, 12 June 2007 (UTC)

Bigger ones

You can construct a family of finite geometries of this form for each prime number, where this seven-element plane is te geometry for the number 2. —Preceding unsigned comment added by 207.169.186.10 (talk) 04:35, 5 October 2007 (UTC)

You mean projective planes over various finite fields, correct?

I’ve added such a note – thanks!

—Nils von Barth (nbarth) (talk) 07:58, 25 November 2009 (UTC)

Combinatorial description

Fano" = X.Klein

Since two points determine a line, after labeling any two points in the Fano plane another point is settled. The relabeling liberty for the rest of the four remaining points is described by the Klein Group.

The Maple permutation group for Fano plane is 7T5.

The e.g.f. is

$30.{x^{7} \over 7!}$

hence there are 30 ways to label the plane.

Dear Joel, the comment is already worthwile to include. It is also worthwile for you and to take a look to the Combinatorial species where this language is studied. Please believe me, to justify the equation is not a short task and it is not the case to embed hundreds of pages in the Fano article.

Yes, that simple task of choosing a point, labeling it, then choosing another one, applying a second label, and then observing the 1+4 points that remain, is described by the very precice equation Fano"=X.Klein.

Where Klein is the Klein group, acting on four points.

Hoping we will not spend the rest of our lives writing and erasing the equation Fano"=X.Klein in Wikipedia, Regards, Nicolae-boicu (talk) 01:43, 5 July 2012 (UTC)

Let's start with the basics: (1) What question do you think this section is addressing? (2) What does the notation you are using mean? (3) Is this your own work, or is it something for which you could produce a reference if asked? Right now, questions (1) and (2) are totally unaddressed in the section you've written, and surely must be clarified before the section you've written could be used. The third question is relevant because of WP:OR. --JBL (talk) 02:57, 5 July 2012 (UTC)

Let me extend my comment somewhat to actually tell you why I don't like the content you're adding, rather than just giving you hoops to jump through:

• it seems like a not-very-interesting piece of trivia about the Fano plane, and contains no motivation whatsoever
• it expresses some simple facts about a single concrete example in a language that makes them seem obscure
• it is largely redundant with the section that precedes it, and to the extent that it is not it should be worked in to that section
• it is poorly formatted (e.g., the use of "." instead of "\cdot" for multiplication in LaTeX, the unexplained bold text)
• the language could use polishing
• the reference to Maple is completely out of place

I'd be happy to work with you to address these issues if you want. --JBL (talk) 03:08, 5 July 2012 (UTC)

I fully support everything that Joel (JBL) has said above. I am particularly concerned with the WP:OR issue, but a simple reference would put that to rest. I have been in a seminar, run by Joyal, on the foundations of this concept of combinatorial species (albeit quite a while ago). In the intervening 30 years, as an active combinatorialist, I have never had the occaision to need to use this material. I find it to be very pretty theoretically, but of little practical use. Although I am biased against the "abstraction for abstraction's sake" POV of some mathematicians, I am willing to put that aside and help you make a contribution here, provided it is notable. Bill Cherowitzo (talk) 04:55, 5 July 2012 (UTC)

Dear Bill and dear Joel

This kind of formula occurs in the area of transitive groupe of permutations. There are not too many of such formulas, and they based on the existence of simple, double, or triple transitive groups. (The maximum degree of transitivity are 4 an 5 for the Mathieu groups). 30 years ago, the classification of transitive groups was not finished. The clasification of multiple tranisive groups is a consequence of the classification of simple groups (Dixon and Mortimer) and is a recent event.

Thus, the structures like Fano plane is among the very few ones where you will see the Joyal's language at work. Nicolae Nicolae-boicu (talk) 07:04, 5 July 2012 (UTC)

Dear Nicolae, your comment addresses at most one of the several significant issues raised, and even then it is not constructive engagement. I assure you that the paragraph you are adding will never have a permanent place in the article verbatim (for the reasons that I have mentioned); if you engage constructively then perhaps the content can be used. --JBL (talk) 13:06, 5 July 2012 (UTC)

Also, please see WP:3RR. --JBL (talk) 13:08, 5 July 2012 (UTC)

Dear Joel, it seems we will spend a lot of time together at the morning coffee, you erasing and me derasing my contribution(s). This will happen because wherever there is a structure to be labeled, there is place for a combinatorial description, that include an equation/definition, some permutation groups and the associated e.g.f.

the Fano plane as the subject of labeling. There are, as shown in the article, 30 distinct ways to label this structure

In fact I feel encouraged by your request of constructivism. I took it very serious, and I am working on it. My question is now, how to do it, without placing a bunch of species stuff in a geometrical subject ?

The combinatorial equation I have wrote describes the relation between the projective Fano plane and the affine Klein "plane". A similar thing happens when completing a line with the infinity point, or a field with the infinity symbol. One obtains a projective line in the first (geometrical) case and a Moebius field in the algebraical case. This operation of adding a point corresponds to integration of species (the converse of derivation).

The deep reason that I cannot be very constructive is that equations like Fano"=X.Klein or Field'=Cyc represents the End of the Theory of Species. The translation from permutation groups to species ends at the normal subgroup concept, that becomes almost group(group). The language of species is a very limited one. So let's the Theory of Species RIP and the language of the four operations live...

I sincerely hope to learn something from our morning coffee dialog. Nicolae-boicu (talk) 14:08, 5 July 2012 (UTC)

Hello Gandalf61 - welcome in the club ! — Preceding unsigned comment added by Nicolae-boicu (talk • contribs) 14:12, 5 July 2012 (UTC)

Perhaps you can start by explaining what you mean when you say there are "30 ways to label the plane". As explained in the article, the symmetry group of the Fano plane has order 7 x 6 x 4 = 168, so it's not clear to me where you get 30 from, or what you are counting here. Gandalf61 (talk) 14:31, 5 July 2012 (UTC)

That would be 30 different ways counting symmetries as the same i.e. 7!/(7.6.4) = 30 Dmcq (talk) 14:50, 5 July 2012 (UTC)

Okay, that's a possibility. But, if that is the case, then Nicolae's exposition is spectacularly obscure. Gandalf61 (talk) 15:07, 5 July 2012 (UTC)

Hi Dmcq, thank you for the 30 orbits. Let me start with the affine correspondent plane. Say X Y, Z and T are four points, slots or whatever, that should be labeled or filled with distincts objects.

Than one has XYZT = YXTZ = ZTXY = TZYX - the translation group is the Klein group acting transitively on the four positions.

Let a, b, c, d four distinct labels or other objects. Than there are 4!/4 = 6 distinct ways to place this objects in the affine Klein plane. (eg. aX, bY, cZ,dT) The e.g.f. is 6.x^4/4! and the permutation group is 4T2. Nicolae-boicu (talk) 15:29, 5 July 2012 (UTC)

• About OR : this is not more OR than saying the 5-12-13 are phytagoreic numbers when everybody knows that 3-4-5 are phytagoreic numbers. Imagine that I am thinking to build some special box to describe labeled structures. Something to contain the equation, the group and the e.g.f. What do you think ? would this be OR ?Nicolae-boicu (talk) 15:46, 5 July 2012 (UTC)

Do you have a reference which uses the notation:

• Fano" = X.Klein

? And, even if Combinatorial species were generally useful, its application here is questionable, at best. You seem to be saying that, defining an "affine plane" to be 4 points, no three on a line (commonly known as an oval (projective plane), that (1) any 3 non-collinear points are contained in a unique oval. (2) Any map from one oval to another can be extended uniquely to a permutation of the Fano plane. But I don't see how K_4, 4T2, or 7T5 gets into into it. — Arthur Rubin (talk) 17:28, 5 July 2012 (UTC)

Gee men, you are good. Last time I was explaining species I presented them as under-arithmetics. Then the moderators gave me free way, no OR and other restrictions, since there was about logical and arithmetical operations, like the sum and the multiplication. I was just free to say whatever I think ! But you really quick pushed me to the ultimate corner : Klein'=X.X.X.

Imagine a sheet of paper on a table that one cand translate only, and not rotate or something else. The transformation group is sharply transitive. Then, with only one thumbpin someone could fix the entire paper.

Group'=Lin One needs only one fixed point to coordinate a space having a translation group.

here is my highest academic reference :: OEIS: A000001 Nicolae-boicu (talk) 20:42, 5 July 2012 (UTC)

Fano plane as labeled structure

Fano" = X.Klein

Since two points determine a line, after labeling any two points in the Fano plane another point is settled. The relabeling liberty for the rest of the four remaining points is described by the Klein Group. (here Klein'=X.X.X)

The Maple permutation group for Fano plane is 7T5.

The e.g.f. is

$\int \int x.6.{x^{4} \over 4!}=30.{x^{7} \over 7!}$

hence there are 30 ways to label the plane. Here 6 represents the six distinct ways of labeling the affine (Klein) corresponding plane.

this is the last variant that has been erased, pls leave it as it is, I am planning to derase it ASAP. — Preceding unsigned comment added by Nicolae-boicu (talk • contribs) 16:10, 5 July 2012 (UTC)

Please do not "derase" it ("reinstate" is the correct word for what you mean, I think). You have been told by several editors that your proposed addition is badly written, poorly formatted and very obscure. Repeating it over and over again does not help. You have not addressed the objections listed by JBL above. One of the cornerstones of Wikipedia is collaborative editing, but I am afraid you are showing few signs of being open to collaboration. Gandalf61 (talk) 16:28, 5 July 2012 (UTC)

I am working on it ! Let's talk about the objections 1) 3) and 6)

1) I think is ok, there is nothing trivia to label structures, fano or not fano.

3) Yes, Math itself is a juge redundant discipline and there are many ways to aproach a subject. That is why I like math : there are alwayse more solutions to one problem. And this is why I love Wikipedia, for its tolerance to multiplicity.

6) I think the very tight relation between groups of "relabelings" and permutation groups has been revealed. :)

if you think the above objections are OK, I will focus on the objections 2), 4) and 5) obscurity, formating and poor language. I will try do it all myself, since no one helps me by adding or correcting.

Nicolae-boicu (talk) 17:05, 5 July 2012 (UTC)

to Arthur -> I read your comment, and I think that you have written the equivalent proposition of my Fano"=X.Klein Thank you so much. To introduce coordinates in the Fano plane one must choose an oval / three non colinear points. This means once choosed the three right points, all the others are uniquelly determined (settled). well, the two apostrofes in my equation means to label any two points. The third is determined (the X) so this one needs no more coordination. It is a function of the two first. To coordinate the remaining four points, one needs to label/fix only one of them, since they form a group structure. — Preceding unsigned comment added by Nicolae-boicu (talk • contribs) 17:51, 5 July 2012 (UTC)

Is there anything more nontrivial here than the basic fact of arithmetic that 7!/168 = 30? I.e. there are 7! assignments of distinct labels to the points of the plane, 168 symmetries of the plane, so (up to symmetry) 7!/168=30 combinatorially distinct labelings? In any case, basic as it may be, this fact needs a reliable source, not so much for verifiability as because we need some evidence that mathematicians have found it to be significant. —David Eppstein (talk) 17:59, 5 July 2012 (UTC)

Dear David, hello ! I never imagined I would polarize so much attention ! never imagined, so I am little ovewhelmed. I entered Wikipedia when on friend of mine, whom is engineer, asked me about some geometry issues that were abused by matematicians. It was about homotethies, plans, tranlations. After a while, here I am explaining why geometrical lines are so strictly related to algebraical fields.

Anyway, this is the very first time that I learn that the matematicians opinion should be relevant in Wikipedia. I will take this as a seventh requierement: 7) — Preceding unsigned comment added by Nicolae-boicu (talk • contribs) 18:29, 5 July 2012 (UTC)

Nicolae, this is getting a bit tiresome. So far, you have made no attempt to improve your text in a way that is responsive to any of the complaints I or anyone else have made. Because there are so many things wrong with the section you've written, this would be very easy to do; for example, you could find a reference where this computation is made, or you could add relevant links so as to give a reader some hope of deciphering your notation, or you could find an appropriate way to work the things that interest you into the much clearer existing section that already deals with the symmetries of the Fano plane, etc. The fact that you've done none of these things even though five or six other editors have expressed objections is not encouraging :(. --JBL (talk) 19:29, 5 July 2012 (UTC)

Let me derivate one more time the Fano"=X.Klein. This means to see what happens if someone chose three distinct points in the Fano plane.

Fano"'= X'.Klein + X.Klein' = Klein + XXXX

Klein + XXXX means that either the three point are colinear and the plane was not well coordinated, or XXXX, the coordination succeded.

The missing piece here is Klein'=X.X.X.

Imagine a sheet of paper on a table that one cand translate only, and not rotate or something else. The transformation group is sharply transitive. Then, with only one thumbpin someone could fix the entire paper.

Group'=Lin One needs only one fixed point to coordinate a space having a translation group.

here is my highest academic reference :: OEIS: A000001 Since there is no A000000 sequence in OEIS, I cannot bring a higher one. Groups are primitives of Lin species, or, in other words, one need to fix exactly one point to coordinate a translation group.

Once accepted a such thing, very easily comes Field'=Cyc and ProjectiveLine'=Field. They are so trivial !

Anyway, if you really believe that further references are needed, tell me so and I will work to obtain something.

@Joel, as I said, I am working on it. Why are you in such a hurry ? pls. first read above to understand what Klein and a group is. Of course, this is not a substitute to a serious study of species. Nicolae-boicu (talk) 19:43, 5 July 2012 (UTC) @Joel I have added (here Klein'=X.X.X) to my section. As I said befor, this talk page is a unique ocasion to improve my section. Nicolae-boicu (talk) 20:06, 5 July 2012 (UTC)

2) regarding the point two, the obscurity. Yes it is true. The Theory of Species is only a very small and obscure chapter that is learned in only several Universities, in Canada, France and Autrich. My question is, what do you expect to find in a free Wikipedia article, some info about labeling structures or desynapsating afirmations like  :

There are 42 ordered pairs of points, and again each may be mapped by a symmetry onto any other ordered pair.

The "obscurity" is not an argument to clean the Wikipedia from alternate opinions. Wanna produce the Ultimate Truth ? Join the French Academy. Or maybe the White House, not Wikipedia.

me, I start to think to #4 and #5 issues. — Preceding unsigned comment added by Nicolae-boicu (talk • contribs) 21:03, 5 July 2012 (UTC)

Your latest version is extremely marginally better written, but otherwise has not changed to address any of the issues raised. Asserting that you disagree with the consensus position does not constitute addressing the issues. Maybe once your block is over this can continue in a less silly way. --JBL (talk) 00:25, 6 July 2012 (UTC)

revised : Fano plane as labeled structure

Let temporarily, just for better reading,

Fano = X^7/PSL(2,7)a : the species that correspond to the collineation group of Fano plane

and

Klein = P₄bic = the species that correspond to the Klein group acting on itself.

then

Fano" = X.Klein ( where Klein' = X.X.X )

Since two points determine a line, after labeling any two points in the Fano plane another point is settled. The relabeling liberty for the rest of the four remaining points is described by the Klein Group.

By derivating the above, one obtains :

Fano"' = X'.Klein + X.Klein' = Klein + X.X.X.X

After labeling three distinct points in the Fano plane, exactly two situation may occure :

- Either the points are coliniar and the remaining labeling liberty is described by Klein

- Or the points form an oval and the whole plane is coordinated : X.X.X.X

The e.g.f. is

$\int \int x.6.{x^{4} \over 4!}=30.{x^{7} \over 7!}$

hence there are 30 ways to label the plane. Here 6 represents the six distinct ways of labeling the affine (Klein) corresponding plane.

Bibliography Yves Chiricota, CLASSIFICATION DES ESPÈCES MOLÉCULAIRES DE DEGRÉ 6 ET 7 Ann. Sci. Math. Québec 17 (1993), no. 1, 1 l-37.

dear authors, thank you so much for your remarks. I have worked these day on subject to fulfill all the requierements. If you want to judge again, please remember that Wiki is not the French Academia and we are not Godel, Wittgenstein or whoever. — Preceding unsigned comment added by Nicolae-boicu (talk • contribs) 16:22, 7 July 2012 (UTC)

OK, I have place it, my last attempt, is promised. For me is math, like the math of the Rubik cube or math of Sudoku. — Preceding unsigned comment added by Nicolae-boicu (talk • contribs) 17:19, 7 July 2012 (UTC)

Even if this was significant, you're still using non-standard notation, even among the combinatorial species people. I mean, "P₄bic". Really? — Arthur Rubin (talk) 19:22, 7 July 2012 (UTC)

This is officially verified. Polygons with alternating two colors edges, or the container that holds as symmetry group the Klein group. These infrastructures were not referred as container(klein) (for example) but received their own names even they were one-to-one with the permutation groups. Why ? if you remain inside the normal theory of sets and groups, maintaining the already known names, you do not have enough copies of the empty set. You have only one copy, the unique empty set, that does not suffice. Species are not sets but they are something else. To avoid confusions, the group terminology was removed. As a collateral damage, my equation Groupe'=Lin was missed 30 years ago. And the very abstract definition of a group. Nicolae-boicu (talk) 09:42, 8 July 2012 (UTC)

Using phrases like "my equation" makes this sound more and more like original research. Even if it is not, your explanation is so abstract that it cannot be understood without a knowledge of combinatorial species. This makes it unsuitable for an article such this one, which should be accessible to a reader with a general mathematical background. Maybe it could be added to the combinatorial species article as an example, although even then it would need a lot of cleaning up first. Gandalf61 (talk) 12:29, 8 July 2012 (UTC)

Original research ? oh, this is wallet found on street, not OR !!!. Let me rephrase what I have understood. The subject of the article belongs to the general math, that has exactly one empty set, without copies. Meanwhile, the combinatorial Fano plane described by Fano"=blabla belongs to another math, that has more copies of the empty set.

Even the in the real world there are many empty boxes, it is incredibly complicate to describe them in a rigorous mathematical fashion.

Overall, the section that I have proposed would point even an advanced user (e.g. a college teacher) to a "twilight zone" and would be rather confusing, contrarilly to Wiki project intentions.

Right ? Nicolae-boicu (talk) 14:20, 8 July 2012 (UTC)

ADAOS I have start the work in my sandbox for an eventualy new section in Combinatorial species I have remembered my course several years ago... To take notices, I was spliting my paper sheets verticaly :

$x=x$	$x\neq x$

in order to not make confusions. — Preceding unsigned comment added by Nicolae-boicu (talk • contribs) 15:41, 8 July 2012 (UTC)

My dear geometers wish me luck Combinatorial_species#A_simple_example_-_labeling_the_Fano_plane ! I have placed my section into the original species article. Someone wrote above in that article : Differentiation is an operation that makes more sense for series than it does for species. Maybe my example will bring some sense into differentiation issue. Thanks for your patience. Never now ! maybe one day we will see these section also placed in the geometrical article about Fano plane.Nicolae-boicu (talk) 07:54, 9 July 2012 (UTC)

the cycle index of the Fano plane

dear editors, how about this ? a weaker variant for Fano"=X.Klein; it works only because in these case the cycle index uniquely determine the group actions.

$Z_{Fano}(x_{1},x_{2},x_{3},x_{4},x_{7})={1 \over 168}[x_{1}^{7}+21x_{1}^{3}x_{2}^{2}+42x_{1}x_{2}x_{4}+56x_{1}x_{3}^{2}+48x_{7}]$

after two differentiation to x₁ one gets:

$Z_{Fano}''=x_{1}.{1 \over 4}[x_{1}^{4}+3x_{2}^{2}]=x_{1}Z_{Klein}$

These polynomials are useful when it comes to draw something on the Fano plane.

For example, there are 10 distinct drawings that one could make on the Fano plane using two colors. $Z_{Fano}(2,2,2,2,2)={1 \over 168}[2^{7}+21.2^{5}+42.2^{3}+56.2^{3}+48.2]=10$

add The Fano plane has kleins in the very same way that the Cube has squared faces. The shortest way (not the simplest) to write this information is Fano"=X.Klein. The corresponding cube equation is Cube' = X.Square in the cube case (rotations acting on oriented faces). I am still hoping you will not deny this fact until the End of Internet.

aaa ! What is a klein ? Take four letters. Then build a set using them. Or build a sequence of lenght four. Or a cycle of four letters.

The very same way one could form a klein of four letters. This means a klein is a species. Nicolae-boicu (talk) 19:51, 27 July 2012 (UTC)

well, again the user

User:Wcherowi reverted my cycle index -this time- as "inappropriate". Why to list the permutations is appropriate but adding them is not ? I will publish the equation Fano"=X.Klein on his talk page.

This is absurd. Listing all that cycles and not writing their index is like staying in a cat food warehouse and pretending there are no cats.

And the OR accuse is ABSURD. The very first cycle index that I have derivated in my life was the above, yesterday.I never studied the Polya aspects, those being not so interesting for me. Yesterday I have found that one must take the derivative in raport with the first variable, by reading the BLL book. I have applied a writen method that I have read. I have never imagined myself how to derivate that polynomial in many variables. Until yesterday I only considered the e.g.f (useful to color the Fano plane in 7 colors) and nothing deeperNicolae-boicu (talk) 00:10, 28 July 2012 (UTC)

I have edited your contribution in accordance with my summary when I reverted it. The cycle index material has been placed in the section on symmetries since the cycle index is a codification of the structure of the automorphism group in its action on the plane. This is the appropriate place for that material. I have removed the rest as a violation of OR. I do not believe that you have fully come to grips with the meaning of WP:NOR. Original research in the sense of Wikipedia articles is different from the way we would use the term as mathematicians. If I were to read something in, say, Joe Smith's research article and then worked out a simple example for myself that illustrated a point he was making and then took this example and put it into a Wikipedia article – that would be considered OR. What I would be adding to the article is not something I found in Joe Smith's paper, but rather something I created (even though it was trivially done and could be checked by anyone). There is another problem with this scenario. Since Joe's article is a primary source, by deciding to put this example of his concept into WP I am making a judgement call – I am deciding that this material is notable ... important enough to be included in this encyclopedia. WP takes the stance that this is not the role for an editor ... these decisions should be made by someone else (an authority in the specific field is preferred) and so the information should be coming from secondary sources where such people have already made that decision.

These limitations on editors (and every contributor is an editor) are self-imposed, but those of us who believe in what Wikipedia is trying to become realize how important the strictures are. Without them WP degenerates into just another open forum, with no reason for anyone to trust what is written.

To avoid having your contribution labelled as OR, you would need to cite a secondary source which makes the statements you want to make. You would have to use the notation of the source and not make up your own. You can add explanatory material that you think will help readers make sense of the statements. You also may not make claims about how wonderful, beautiful, brilliant, etc. your contribution is in the subject, as this would violate yet another stricture, WP:NPOV, keeping a neutral point of view.Bill Cherowitzo (talk) 17:36, 28 July 2012 (UTC)

Thank you for your detailed explanations. Independently on my hypothetical claim on the brilliancy of my absent contribution that you have noticed, there are seven kleins lying in the Fano plane and laughing at our bla-bla. To convince yourself just aggregate the so called "appropriate" exhaustive list of cycles, compose them into the cycle index and differentiate twice, as any mathematician has the duty to do. It is an honor duty to differentiate a polynomial, not original research as you claim. You will convince your-self of their existence, and (this is OR) they are laughing at us.

I think you, right now, are just exercising your agenda on my contribution. I will not play your game. Remember that at your first revert, the upper moderation level counted 4-0. O (zero) was for my opinion. You already won this round with 1-0, you get it ? I just hoped you will not revert the information I have added after following the gazillion of recommendations I received. I have rolled back the species, replacing them with the weaker and more known Polya polynomial.

To turn this, I already have to go up 2, 3 or more levels of moderation. Fano"=X.Klein 66.130.132.143 (talk) 21:18, 28 July 2012 (UTC)

Gee Bill, thanks. Those 10 ways to color shows that because of their tough symmetry, the Fano structures are almost sets. For an informatician is a clear sign not to stock bynary information in such kinds of things.

I am still thinking at that differentiation, if is or not OR. Simple arithmetic additions or substractions are not OR. Should Math project follow the general reccomandations, or shoud it raise the acceptable level of non-OR operations ? Best Regards, Nicolae-boicu (talk) 21:53, 28 July 2012 (UTC)

I am right here Bill, watching; How do you plug the components (subgroups acting on subsets) within respect to OR ! Malhereusement, groups entered in math by the Algebra door. Bring back the groups to Geometry !

look at this published paragraph :

• There are 42 ordered pairs of points, and again each may be mapped by a symmetry onto any other ordered pair. For any ordered pair there are 4 symmetries fixing it.

and read :

• There are 42 ordered pairs of points, and again each may be mapped by a symmetry onto any other ordered pair. For any ordered pair there is a klein laughing at us. Since a symmetry fixing two points also fixes the entire line, for every fixed line there is a klein laughing at us, i.e. seven kleins.Nicolae-boicu (talk) 00:05, 30 July 2012 (UTC)

just in case; By digging libraries, I have found a big piece of the puzzle : 1937, Carmichael, ch XIII shows how to recover a field starting with a sharply double tranzitive group; this grants the species equation Field'=Cyc.

And my greatest wiki-rollback was when I changed the title of my Romanian article on species from "Species of Numbers" to "Species (mathematics)". There are Cardinal Numbers (ens), Ordinal Numbers (lin), but also there are Cyclic numbers, kleins, or whatever trees. Not enough background. I need kleins and the their fanoplanes.Nicolae-boicu (talk) 04:25, 31 July 2012 (UTC)

I got a new one

: d²Fano = X.Square Here {2, 3} depicts d²Fano , 1 depicts X and 4567 depicts Square

d²Fano = X.Square

this time is about upper differentiation, i.e. fixing an un-oriented pair. One third point is also fixed, and the remaining liberty is the liberty of re-labelling a square. Nicolae-boicu (talk) 02:43, 16 August 2012 (UTC)

PG(3,2)

I've moved these suggested additions to the listing of properties of the Fano 3-space here.

• every trio of noncollinear points is in exactly one plane
• every pair of intersecting lines is in exactly one plane

These are true, but generic (satisfied by all projective spaces of dimension at least 2). Where do we draw the line about what to include in this section? I would suggest that generally only properties specific to the Fano 3-space should be included with the only exceptions being those properties which are needed to show that this space has certain features (like the last two which establish that the space is projective). Bill Cherowitzo (talk) 17:50, 22 December 2015 (UTC)

Must have been late sorry. But how about the following remarks:

• each line is contained in 3 planes
• each line intersects or is containt in every plane
• each line is intersected by 18 other lines

was thinking about making a seperate page about the 3 dimensional and 4 dimensional Fano plane (is that GP(4,2)? note the article nowhere mentions what GP the 2 dimensional Fano plane is), but I fear that would breach wp: original research.WillemienH (talk) 07:23, 24 December 2015 (UTC)

I need to think about how useful these counts are. The list can't be exhaustive and long lists quickly lose their effectiveness in conveying information. I've taken care of your point about PG(2,2) ... this is standard notation so there is no concern about OR. Bill Cherowitzo (talk) 17:16, 24 December 2015 (UTC)

I did add the first 2 was wondering what is the dual of "each line intersects or is contained in every plane", thanks for the edit of the lead. WillemienH (talk) 23:35, 24 December 2015 (UTC)

The dual statement is "Given a line and a point, either the line contains the point or there is a unique plane containing the point and line." This can be seen more easily by writing the original statement in a less abbreviated manner. "Given a line and a plane, either the line is contained in the plane or there is a unique point on both the line and the plane." Bill Cherowitzo (talk) 05:15, 26 December 2015 (UTC)

Edited again think the properties can be devided in the following two groups:

Special properties of projective geometry of the Fano 3 space (mentionable)

• Each point is contained in 7 lines and 7 planes
• Each line is contained in 3 planes and contains 3 points
• Each plane contains 7 points and 7 lines
• Each plane is isomorphic to the Fano plane
• Every pair of distinct planes intersect in a line
• A line and a plane intersect in at least one point

General properties (no need to mention)

• Every pair of distinct points are contained in exactly one line
• every trio of noncollinear points is in exactly one plane
• every pair of intersecting lines is in exactly one plane
• Every pair of distinct planes intersects in exactly one line (important is they do intersect it is always in maximum one line)

I am not sure about "Each plane is isomorphic to the Fano plane" seems to be a general truth as well and could be removed. WillemienH (talk) 13:39, 26 December 2015 (UTC)

I thought that the last point needed to be stated more clearly so I made a change in the article. In what sense is the statement about isomorphic planes to be thought of as general? You need to make a number of assumptions before that holds in any generic sense. Bill Cherowitzo (talk) 21:03, 26 December 2015 (UTC)

Good question, but how can the lower dimensional structures of a higher dimensional version of that structure not be isomorphic to the lower dimensional version? I cannot really answer this question so i will leave it in the article. As rule of thumb my idea is only to mention properties that differ from euclidean space geometry. I think we made the section better, Thanks, but still a section on PG(4,2) would be nice :) WillemienH (talk) 13:37, 27 December 2015 (UTC)