Talk:Pentagram map

Reference form
Am having a problem with the citation (and display of the e-print link) in Schwartz, Richard Alan; Tabachnikov, Serge. Elementary Surprises in Projective Geometry. 7&amp;6=thirteen (☎) 12:37, 28 June 2011 (UTC)

✅
 * Fixed it myself.  7&amp;6=thirteen (☎) 13:16, 28 June 2011 (UTC)

Verifiability
This article is written like a survey article in a journal. It has references, but they seem more for the purpose of "giving credit where credit is due", than to provide the reader a location to verify the validity of the content. (Eppstein recently wrote Wikipedia editing for research scientists which might help one understand the differences between the Wikipedia and academic forms of writing.) Justin W Smith talk/stalk 16:34, 29 June 2011 (UTC)
 * You wrote: than to provide the reader a location to verify the validity of the content.  However, the text of almost all of the articles are linked directly, so that you can view the text yourself, and even download it.  Please reread the article, and read it and its references as an integrated whole.  You may have other valid criticisms, but certainly this aspect doesn't happen to be a winner.  You may not like the conclusions, but this is about WP:verifiable and not WP:truth.  This is absolutely verified.   7&amp;6=thirteen (☎) 16:41, 29 June 2011 (UTC)
 * The "Definition of the Map" section has exactly one two reference to a journal article that appears to be quite technical. The reader should have a better way to verify this foundational aspect of the topic. Justin W Smith talk/stalk 16:45, 29 June 2011 (UTC)
 * The references are: ^ a b c d e f g h i Schwartz, Richard Evan. "Discrete monodromy, pentagrams, and the method of condensation". journal of Fixed Point Theory and Applications (2008). Retrieved 2010-02-12.

^ a b c d e f g h i j Ovsienko, Valentin; Schwartz,, Richard Evan; Tabachnikov, Serge (2010). "The Pentagram Map, A Discrete Integrable System" (pdf). Comm. Math. Phys. 299: 409–446. Retrieved June 26, 2011. With appropriate links. As we note in the Mathematics talk page in an ongoing discussion, some editors have taken the position that this is an obscure boutique end of mathematics. This is all technical, and perhaps is beyond some readers comprehension. But that does not mean that it has to be reduced down to words of one syllable. This is written clearly and concisely, as is allowed by and consistent with the subject matter (which is inherently technical). The more technical papers are linked, so you can compare them. 7&amp;6=thirteen (☎) 16:51, 29 June 2011 (UTC)
 * I'm bothered by this article, but I don't yet have clarity as to how it should be improved. Part of my problem might be that I dislike seeing such an exhaustive article on a mostly obscure topic. It also bothers me that roughly half of the citations are to the work of Schwartz (in part b/c he introduced the concept in 1992), while Schwartz also wrote much of the content for the article. See WP:EXPERT or WP:SELFCITE. I'll follow the discussion, but I'm not sure how best to improve the article. Justin W Smith talk/stalk 17:14, 29 June 2011 (UTC)


 * Justin, I share your unease, but in an entirely different way.
 * I'm bothered by the response of parts of the Wikipedia community to this article. This is a world-recognized scholar who cleaned up an existing and wrong-headed article.  He has gotten nothing but hassle for his trouble.  We should be lauding efforts like this.  Instead of the proverbial 7th grader, we have somebody who brings to the table a unique perspective and verified credentials to back it up.  We are talking about credibility and verifiabilility.  This is like having Johnny Unitas play on an intramural football team.  I think we should be welcoming.  I would think Please do not bite the newcomers should apply, even in the case of a fully credentialed academic.  We should be supportive, not throwing brick bats.  That Wikipedia is benefiting from his contribution should not be overlooked.   <b style="color:#060">7&amp;6=thirteen</b> (<b style="color:#000">☎</b>) 17:24, 29 June 2011 (UTC)
 * I saw on Schwartz's talk page that he had been accused of vandalism, and had numerous other problems when he tried to expand/correct this article. That was unfortunate, and should've been handled differently. More needs to be done to assist academics in understanding the Wikipedia culture, and assist in their transition to a Wikipedia style of writing.
 * With that said, the issue we have now is how to improve the article. I'm certain that the content is accurate, but it needs work to be more consistent with Wikipedia standards for content. Justin W Smith talk/stalk 17:37, 29 June 2011 (UTC)
 * There was actually more to it than that. It was more like a gang attack.  Unfortunately Dr. Schwartz deleted his talk page as this stuff happened.  It's in the history, but its hard to access.
 * I agree with you that improvement of the article is the ultimate goal. The substance of the article is way over my head -- I NEVER claimed to be a mathematician -- so I can't help.  It would be great if we can involved other editors with the right stuff to lend a hand.  That is one of the reasons why I am so concerned about the isolation (WP:Orphan) of this article.  If no one knows about it, they won't be involved.  So I am pleased that we could have this problem highlighted on the Mathematics talk page.
 * My only peripheral involvement was to make the editing improvements I knew how to do, and to try to paper over the rude treatment he received. I've also tried to lend him a helping hand and teach him the ways of the Wiki; the better to make his experience be worthwhile.   <b style="color:#060">7&amp;6=thirteen</b> (<b style="color:#000">☎</b>) 18:16, 29 June 2011 (UTC)
 * The problem with an obscure topic like this is that Schwartz might be the only active Wikipedia editor very familiar with the topic. Of course, many others could learn it and then work to improve its coverage, but there's little (or no) motivation to do so. So the article will likely sit for a long time with few active editors (or readers). "Expert" editors are more valuable for their ability to improve articles on more "textbook" topics than when they work on a small specialty like this. Justin W Smith talk/stalk 18:29, 29 June 2011 (UTC)

I agree with JustinSmith that this is a pretty specialized and obscure topic. My main purpose in editing it was that the previous version was quite sketchy and inaccurate. I wanted to write something up-to-date, accurate, and useful. I was sort of in a tough spot. This topic had already been introduced as a wiki page, but it was not accurate, or useful to anyone. Would it have been better just to delete the whole thing? Is this even possible? Should I have simply left the nearly useless page as it was? (Given all my troubles with this edit, I might next time resist the temptation to meddle...)   In my efforts to make the thing accurate and useful, I may have simply written something too long. This makes it look overblown. One could certainly cut parts out, but then it would just be less complete and useful. Is the length of an article supposed to be proportional to its importance? I know that this is more or less true in traditional encyclopedias, but for something electronic, there seems room to spare.

I am mindful of the problem with self-reference and neutrality, but I'm not sure how technical subjects like this can be written by someone who isn't closely involved. Maybe these kinds of articles shouldn't be on wikipedia, but in that case perhaps there should be a mechanism for removing pages on topics that really won't fit well with wikipedia's standards. Speaking personally, if this article is going to bother folks and stick out like a sore thumb because of the combination of its obscurity and length, I would be happy just to delete the whole thing. The last thing I want is to publicly over-blow my own work. Maybe scholarpedia is a better forum for this article, and perhaps this article could be migrated there.

I'd like to make one more comment about this article. I think I was as much at fault as anyone else for the sort of unpleasant experience I had with this article. I came in and gave the thing a rapid overhaul, making quick, relentless, and sweeping changes. I'm sure that this raised some red flags in the community. Probably I should have taken a half hour to learn some editing skills and conventions before launching in. Please accept my apologies. RichardEvanSchwartz (talk) 06:18, 30 June 2011 (UTC)


 * Concerning Justin's query about verifiability, this editor has done everything that I could do to provide links to all of the articles. See the discussion hereafter.  It is exceptionally complete, and should assuage any remaining legitimate concern.  Have a great Independent Day weekend.   <b style="color:#060">7&amp;6=thirteen</b> (<b style="color:#000">☎</b>) 02:02, 1 July 2011 (UTC)


 * My concerns about the length of this article are unfounded; there's no correlation between the length and the importance of an article (e.g., the article on Fanny Crosby is several times the length of the article on Abraham Lincoln). Length of an article is just a function of the level of interest taken in a topic by some set of Wikipedia editor. Yes, there are concerns about a conflict of interest, but I truly think everything was done with the best of intentions. Richard, I also want to apologize on behalf of the Wikipedia editors who were unduly harsh.  (I've been guilty of doing that myself on a few occasions.) It does take time to become comfortable with the culture of Wikipedia.  Wikipedia editing can (and should) be a rewarding experience. Best wishes -- Justin W Smith talk/stalk 03:32, 2 July 2011 (UTC)

Some questions
Well, I love algebraic geometry, but I've never heard of the pentagram map before. I have some questions, and I suppose I could dive into some papers, but I thought it would be easier to ask here first: Ozob (talk) 02:32, 30 June 2011 (UTC)
 * 1) The article says that the map is well-defined "as long as there are enough points in the projective plane to make sense." I presume that means "make sense of the intersections", i.e., the intersections of the lines ought to be non-empty.  I'm used to defining things and then asking whether or not they have any rational points.  If we take this (more scheme-theoretic) approach, then the pentagram map should always be defined, right?  It just might not give a polygon rational over the ground field.
 * 2) I feel like there's some sort of unmentioned duality going on. First reason: The pictures all seem a little unnatural in that they are over the reals and they show line segments, not lines.  Over unordered ground fields you'd be forced to draw the lines, so I'm going to just going to assume that you have lines.  Now, an ordered collection of points is equivalent to an ordered collection of lines (draw the lines between successive points or take the intersection of successive lines, looping back at the end of the sequence).  So there's a sort of dual description of the pentagram map where you start with a collection of ordered lines, then construct a new collection of ordered lines by taking intersections and drawing new lines (as opposed to the standard description, where you draw new lines then take intersections).  Second reason: The indexing gets messed up when you apply the map once and gets fixed when you apply it twice.  That feels like there's a duality in the background.  So, what's going on?
 * 3) When you introduce coordinates on the moduli space of n-gons, there looks like there's some sort of group in the background: A point inside a line inside a plane is a complete flag variety, and I'm sure that there's some flag variety representing n-gons. So: What's this flag variety (this must be well-known. Isn't there stuff like this at the start of Lafforgue's Chirurgie des grassmanniennes?) and is there a natural description of the pentagram map in terms of this flag variety?  Something like, "send a flag to the dual flag" (which would send points to lines and vice versa) "then apply an inner product" (so that lines are sent back to points).  Given the integrable systems results below I feel like this must be understood.
 * 4) "3rd order linear ordinary differential equations"? What the heck?  How do 3rd order ODEs show up?
 * 5) There's a sentence that somehow got corrupted in the section on "Corner coordinate products": "These two functions are This observation is closely related to the 1991 paper of Joseph Zaks".

I'll try to answer these. 1. If you look at this thing over, say, the Fano plane, (the projective plane over Z/2) then there are simply not enough points to draw anything but the most trivial polygons. You will find some of the diagonals you might use to define the map coincide with each other, so it won't make sense to intersect them. More generally, even over a large field like the reals or the complex numbers, the points of a given polygon might not be in sufficiently general position. The solution I always take is just to say that the map is generically defined over (say) C.  The scheme idea sounds good but I don't know enough about schemes to pursue this.

2. Yes, of course, the pentagram map interacts very well with projective duality. The way I really think about it is that there are two kinds of polygons, PolyPoints, and PolyLines. The first is a cyclically ordered collection of points and the second is a cyclically ordered collection of lines. The act of taking diagonals maps the space of PolyPoints to the space of PolyLines, and vice versa. If you think of such a map as acting on the union of the two spaces, then it is an involution. So, the pentagram map is the composition of two such involutions. One can see a hint of this in the formula I gave for the map.

3. I don't really know enough to answer this question. Certainly you can represent a polygon as a point in a grassmannian, just by making a 3xN matrix, whose columns are representatives in $$ F^3 $$ of the points of the polygon, but I don't know how this formalism helps. We thought about this to some small extent, but never made much progress on it.

4. This approach is really due to Tabachnikov and Ovsienko, and my answer is cribbed from their book "Projective Differential Geometry: Old and New", which can be found on Tabachnikov's website http://www.math.psu.edu/tabachni/Books/Pro.pdf, Sect 2.2. Suppose you have a linear 3rd order ODE (on the line) with periodic coefficients. You take a triple of independent solutions, and interpret them as a curve in G=(G1,G2,G3) in R^3. Replacing G1 by g1(x) = a1(x)G1(x), etc., for suitable functions a1,a2,a3, you get a curve g=(g1,g2,g3) so that the det(g,g_prime,g_prime_prime)=1. (Sorry, I don't know how to write g_prime_prime, the second space derivative of g, in wiki math.) Then one has g'=a g + b g' + g.  Here a(x) and b(x) are functions which capture the geometry of the solution [G1:G2:G3], as a curve in the projective plane. This is in analogy to the (ab) coords listed in the article. 5. "are invariant". — Preceding unsigned comment added by RichardEvanSchwartz (talk • contribs) 05:45, 30 June 2011 (UTC)


 * 1. Schemes would only help if the problem was that your field wasn't algebraically closed. And come to think of it, this is never a problem: A rational curve which has a single point rational over the ground field has an entire rational curve rational over the ground field.  In more pedestrian language, if you can define the line using coordinates in the ground field, then you'll have enough points to take intersections.  Where schemes might help is in making sense of degenerate situations, but they'd only be a technical tool—you'd still need some geometric sense of what the degenerate situation ought to be doing or else you wouldn't know which scheme to pick.  See below.


 * 2. OK, that seems good!


 * 3. Right, a 3&times;N matrix would do it. That's more elementary than I was thinking last night.  So the space PolyPoints would be Gr(1, 3) &times; ... &times; Gr(1, 3) and the space PolyLines would be Gr(2, 3) &times; ... &times; Gr(2, 3).  We have maps Gr(1, 3) &times; Gr(1, 3) &rarr; Gr(2, 3) and Gr(2, 3) &times; Gr(2, 3) &rarr; Gr(1, 3) that send a pair of points to the line between them and a pair of lines to their intersection.  (These are only defined generically, of course.)  The pentagram map would be defined by ordering the Grassmannians cyclically and applying the above map to a Grassmannian and the Grassmannian two steps down.


 * In more vector-spacy language, we're either taking the span of two one-dimensional subspaces or the intersection of two two-dimensional subspaces. Degeneracy would be when the spaces coincide.  As far as defining the pentagram map in degenerate situations, here's what feels right to me at the moment: If the two spaces coincide, then we should replace each of them by an infinitesimally moved vector space.  There should be a moduli space of such things.  It feels like we're looking at a blowup of Gr(1, 3) &times; Gr(1, 3) along the diagonal.  I don't know off the top of my head whether there's a better description of it than that—there is probably some larger space than Gr(1, 3) that parametrizes "vector spaces with infinitesimal data", and the exceptional divisor of the blowup of Gr(1, 3) &times; Gr(1, 3) would sit in there.  Maybe this space is a jet space?  (This ought to be known.)


 * The span of two infinitesimally different one-dimensional subspaces would be a two-dimensional subspace, but if you throw away the infinitesimal information you end up with a one-dimensional space again. Similarly, the intersection of two infinitesimally different two-dimensional subspaces would be one-dimensional, but if you throw away the infinitesimal information you'd end up with a two-dimensional space again.  The reason these should both make sense is because the space of "vector spaces with infinitesimal data" ought to project down to the Grassmannian, and the spans and intersections of degenerate pairs of subspaces should give subspaces that are not preimages under the projection map.  When you project these down, you end up with the aforementioned dimension properties.


 * Even if you can make this construction work, you can still end up with two coincident objects in your enlarged space. There ought to be a whole hierarchy of these: I described what happens when two objects are 1st order degenerate, but there should be a similar construction to handle 2nd order degeneracies, 3rd order degeneracies, etc.  There should always be maps from the nth order thing down to the (n&minus;1)st order thing and there should be a limiting object with infinite order data (this would be very large).  Obvious question: What's this object look like, and does the pentagram map on this object have similar properties to the usual one?  (Since I'm just making this up as I go along, that's meant to be rhetorical.)


 * Another question: The pentagram map is defined by taking a Grassmannian and the Grassmannian two steps down; what happens if we choose some other pair of Grassmannians? I.e., choose a permutation &pi; and pair off Grassmannian i with Grassmannian &pi;(i).  Clearly there are degenerate situations (like the identity permutation), and if &pi; is the cycle (0 1 2 ...) then iterating this &pi;-pentagram map just moves our indexes around the polygon.  The page mentions that hexagons under the standard pentagram map have simple behavior; is this because 2 divides 6, and similarly, if &pi; is (0 k 2k 3k ...), do kn-gons have simple behavior?


 * 4. OK, that's very cool! By the way, to make a prime mark, you use the HTML entity  .  Two apostrophes turn on italics and three turn on bolding; this is why your equation g&prime;&prime;&prime;=a g&prime;&prime; + b g&prime; + g didn't turn out so well.  Oh, also, you can sign your posts with four tildes, as in ~.


 * Ozob (talk) 11:34, 30 June 2011 (UTC)

Hi Ozob. In response to #3: That's an interesting idea. I guess that you are trying to explicitly compactify the space on which the pentagram map acts -- Is this like the Mumford compacification? I thought a bit about ideas like this, but never got anything to work. I think that the recent work of Soloviev, mentioned in the page, shows that over C there really is this larger space on which the P-map is completely defined. This space is a fiber bundle, where the fibers are abelian varieties. (As I write this I realize that this discussion probably isn't going to be pleasing to people who want less rather than more technical detail!) RichardEvanSchwartz (talk) 12:43, 30 June 2011 (UTC)


 * Yes, I suppose it started out as an attempt to explicitly compactify the space. It failed when I realized that there would be higher-order degeneracies.  But Soloviev's approach is very different!  I glanced through his paper and it seems interesting.  I have to read more about this stuff, I am quite curious.  Thank you for taking the time to answer my questions!  Ozob (talk) 00:53, 1 July 2011 (UTC)

Archival copy of an important article
It took some perseverance, because it would not come up in google. But I managed to come up with this, which is located at the Journal of Experimental Math home page, and their archive of back copies, I think. I therefore probably have not cited it correctly. Nevertheless, here it is: <b style="color:#060">7&amp;6=thirteen</b> (<b style="color:#000">☎</b>) 01:22, 1 July 2011 (UTC)
 * Another article had similar problems. I think it is an abstract, and it is in some peculiar formats.  I doubt that I got the form of the citation right.  But here is what I've got: *
 * My goal was to try to find links for every one of the cited articles. Can't seem to find one for the 1945 article, however.  I was trying to address the putative "verifiability" issue.  I would note that the articles and links are all here, they say what they say, and they are consistent with the article.  In short, 'this dog don't hunt.'     <b style="color:#060">7&amp;6=thirteen</b> (<b style="color:#000">☎</b>) 01:56, 1 July 2011 (UTC)

Motzkin paper
Despite my best efforts, I can't find an on line copy of: Motzkin, Theodore (1945). "The pentagon in the projective plane and a note on Napier's rule". Bulletin A.M.S. If anybody has some suggestions, or better yet, can find it, please assist. <b style="color:#060">7&amp;6=thirteen</b> (<b style="color:#000">☎</b>) 21:01, 6 July 2011 (UTC)
 * AMS 1945 bulletin is about as close as I could get.  <b style="color:#060">7&amp;6=thirteen</b> (<b style="color:#000">☎</b>) 21:18, 6 July 2011 (UTC)
 * I found it here:
 * Justin W Smith talk/stalk 22:51, 6 July 2011 (UTC)
 * ✅ Thanks.   <b style="color:#060">7&amp;6=thirteen</b> (<b style="color:#000">☎</b>) 03:12, 7 July 2011 (UTC)
 * ✅ Thanks.   <b style="color:#060">7&amp;6=thirteen</b> (<b style="color:#000">☎</b>) 03:12, 7 July 2011 (UTC)