Talk:E8 (mathematics)/Archive 1

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Archive 1

organization

Hey, just wondering if someone can "lamenise" this article a bit... It's linked from a variety of different articles related to string theory, few of which make any effort to explain it. Obviously it's a bad idea to remove any of the information that is there, but if there's someone who can understand the article, they should add a few lines explaining what E8 x E8 means with minimal jargon and maths.

     18 mathematicians? or 19? the bbc article says 19... http://news.bbc.co.uk/2/hi/science/nature/6466129.stm — Preceding unsigned comment added by 61.68.184.249 (talk) 06:38, 24 December 2006 (UTC)

And a third

There is one compact one (which is usually the one meant if no other information is given), one split one, and a third one.

So, the distinguishing feature of one of these forms is that it's not one of the other two. :) --Starwed 13:56, 19 March 2007 (UTC)

very very recent news as per 2day

http://web.mit.edu/newsoffice/2007/e8.html - - - 84.226.45.163 07:33, 19 March 2007 (UTC)

http://www.timesonline.co.uk/tol/news/uk/science/article1533648.ece follow up article in the Times about the solving of the E8 problem. Philbentley 10:45, 19 March 2007 (UTC)

http://atlas.math.umd.edu/ David Vogan is giving a lecture on the computation at MIT on Monday, March 19, at 2 PM in Building 1, Room 190. See an introduction to the calculation at AIM, and read the Press Release.

http://www.aimath.org/E8/ Mathematicians Map E8

http://www.aimath.org/E8/E8release.txt this is a press release link about "Mathematicians solve E8 structure..."

Actually, David Vogan has already talked about it on January 8, 2007, the day it actually happened, at the Joint Mathematics Meetings (annual biggest American Mathematical Society conference)! But with all the brouhaha, let us not forget that computer tables are not a replacement for understanding. Mark Goresky made first tables of Kazhdan-Lusztig polynomials more than 20 years ago, but as of 2007 we still struggle to complete understand the conceptual side. Arcfrk 03:38, 21 March 2007 (UTC)

Witchcraft!

Well, not really, but I have degrees in both math and physics and have no idea what this article is talking about. It really needs some clarity, or at least better internal linkage. --NEMT 04:27, 21 March 2007 (UTC)

All that buzz

The discovery may be all over the news (see the links above), but after reading a few descriptions of it in the popular press, I get the firm impression that all without exception writers didn't have a faintest clue about what had actually been discovered, proved, solved, or computed (nor the distinction between the four). I suspected from first following the link to David Vogan's personal account of the story that "the solution to a 120-year problem', as one source put it, was referring to computation of Kazhdan-Lusztig polynomials of $E_{8},$ but besides a cursory reference in the AIM press release, it's essentially impossible to tell. Which is made all the worse by the timing, because the computation itself was actually finished on January 8, 2007 and first reported by David Vogan only a few hours later in a seminar talk at the Joint Mathematics Meetings in New Orleans. As for the majority of people who crave for understanding what is it all about ("c'mon, it's on the news, so it couldn't be all that difficult to explain"), here is my metaphor: they would have had a better chance of reproducing the full score of a Mozart Piano Concerto than understanding Kazhdan-Lusztig polynomials, or even what the group (actually, Lie algebra or even root system) $E_{8}$ constitutes, with or without the help of Wikipedia. Which brings me to my point: while some consider buzz about mathematics among general public to be beneficial, we have to keep in mind that it has very limited impact in the long term, not in the least because the attention span is so short and the willingness to immerse oneself in systematic learning of a scientific theory is so low. So I would caution against trying to follow the buzz and not the substance, and rework a fundamental article, if about a fairly obscure topic, to satisfy the minute curiosity of the crowds: it may (and almost certainly, will) fail to make them appreciate the mathematical beauty of it "here and now". On the other hand, it would inevitably introduce an element of exigency into the article, which we may bitterly regret later, when the moods change. (Although the curiousity may be there, and should be encouraged in some way.) We are, after all, writing an Encyclopaedia, and not What's New in Mathematical Sciences. Incidentally, isn't there a Wiki Project deading specifically with current events? Arcfrk 04:39, 21 March 2007 (UTC)

Well said. While what you say about avoiding "exigency" in our writing is spot on, I do think that taking opportunities that come up like this to improve an article are not misspent. As long as our desire is not simply to please the masses, we can view this as something of a spontaneous "article improvement drive". After all, even if most people--even people with math majors--cannot hope to understand the complexity here, I should be able to understand this as a graduate student with a reasonable amount of training in geometry and topology. The article is in pretty good shape by this standard--even if it's a rather arbitrary one. :) I suspect I'm saying more that you agree with than not. VectorPosse 05:30, 21 March 2007 (UTC)

I made some relevant comments earlier on the page. Also, with a bit of digging I found some helpful online background which I added to the Kazhdan–Lusztig polynomial article. Here's what the press responds to in this story: big. Specifically, the fact that the computation required substantial time on a powerful machine, and especially the fact that the output was almost too big to write out, both excite the imagination. Also titillating is the vague idea that this may lead to fundamental breakthroughs in the foundations of theoretical physics, such as supersymmetric string theory. (Which is equally mysterious.) Remember the coverage of the Fields Medal for resolving the Poincaré conjecture? That conjecture (theorem) is easier than this to state for a layman, but that was hardly relevant; the story was "reclusive genius rejects top prize".

This, too, shall pass. Let us use this focus of attention to further our perpetual goal: provide a solid article for those with the interest and stamina to learn, and provide a stimulating overview that may draw in others. For, not only must we recruit and train the next generation, we also must retain the appreciation and support of fans and funders. Mathematicians are dirt cheap compared to, say, engineers; but mathematics still needs support. (Here's a Wikipedia example: How long have we been waiting for the developers to incorporate blahtex?) --KSmrq^T 10:05, 21 March 2007 (UTC)

Rather than focus too much on this article too much I feel it would be good to take an overview articles on simple lie groups to the same level as Manifold. There are scattered examples of some accessable writing, circle group, but nothing which really brings this all together, in a coherent accessable whole. --Salix alba (talk) 10:49, 21 March 2007 (UTC)

From what I know, and I happen to know a lot about this particular development, what the [Atlas] group has done up to date, and that includes the so-called mapping (what a horrible choice of the term) of $E_{8}$ , it is nowhere near the importance of establishing Thurston's geometrization or proving the Poincare conjecture. If they classify the unitary dual for all groups, it will be big progress, but still not as impressive as classifying all finite simple groups (and they explicitly mention the Atlas of finite groups as their inspiration). As far as computations go, personally I am infinitely more thrilled by factoring large (1000 digits and more) integers. It also required serious breakthroughs in algorithms light years beyond the idea of using the Chinese remainder theorem. Arcfrk 01:04, 22 March 2007 (UTC)

Background for non-specialists

I added a section at the top entitled "Background" that I hope will provide some motivation and explanation of what this is all about. It's all kind of "stream of consciousness", so take a look at it and see how you think it might be improved. Greg Woodhouse 00:46, 21 March 2007 (UTC)

I've tweaked it a bit. The main structural change was to describe (er, handwave about) simplicity in terms of the groups/algebras themselves rather than via the representation theory. One thing I'm a bit concerned about (exactly as much after my tweaks as before) is that it's more an introduction to Lie groups and algebras generally than to E8. Gareth McCaughan 03:07, 21 March 2007 (UTC)

But why is it here? Introductory material for Lie groups needs to be at Lie group. Otherwise it risks being in nine or so places (for each sort of Dynkin diagram). Charles Matthews 19:08, 21 March 2007 (UTC)

That's a reasonable question, and perhaps we should add a pointer to the main article. But the point was not to provide a general introduction to Lie groups and Lie algebras, but to allow the non-specialist to get his or her bearings long enough to get some idea of what this is all about. That's why I tried to focus on representations (many readers will have some experience with matrices and be able to understand linear and affine symmetries, even if the more abstract notions of Lie groups and Lie algebras are unfamiliar. It was a first stab at making the article (or, at least the introductory portions of it) more accessible. Greg Woodhouse 19:26, 21 March 2007 (UTC)

I'm all for introductory material. We have to respect the encyclopedia structure, also. WP is the first encyclopedia to be written as hypertext, so that we do assume people click on the Lie group link if they need to. That way we may be less annoying to people who do not need to. Charles Matthews 20:04, 21 March 2007 (UTC)

Well put, Charles! I tried to say the same thing below but not as concisely. Arcfrk 01:07, 22 March 2007 (UTC)

Move to "E8 (mathematics)"?

I know really nothing beond basic information about this so I'm sorry if I'm way off base here but should this page be under E8 (mathematics) insted of it be a redirect to E₈ (mathematics)? The pages listed with this one on Lie group are all listed in that same format: G2 (mathematics), F4 (mathematics), E6 (mathematics),and E7 (mathematics).

Again, sorry if I'm way off. Scaper8 15:27, 21 March 2007 (UTC)

Well it was at E8 (mathematics) until a few days ago. I have no problem with either name, but we should be consistent. If we are going to keep this here then we should move G2, F4, E6, and E7 accordingly. Incidentally, does the name E₈ display properly for everyone (being unicode and all)? -- Fropuff 16:05, 21 March 2007 (UTC)

The manual of style[1] is not particularly clear on the subject, and [2] suggests there may be some technical problems in some browsers. Even in mozilla the title appears messed up in the title bar. I would suport moving back to E8. --Salix alba (talk) 16:09, 21 March 2007 (UTC)

I concur for the reasons above. I cannot read it in my IE6.X browser. Also, either er change this one back, or change ALL the other ones to match, creating even more illegible article titles. - CobaltBlueTony 16:10, 21 March 2007 (UTC)

I agree that the E8 name is preferrable to E₈. Oleg Alexandrov (talk) 02:54, 22 March 2007 (UTC)

Same for E₈ manifold. Oleg Alexandrov (talk) 02:55, 22 March 2007 (UTC)

To answer your question, for me the title is listed as E₈ but the article itself uses the underscore Scaper8 16:38, 21 March 2007 (UTC)

a vote for move back to E8 (mathematics)?

Yes

Tom Ruen 08:35, 22 March 2007 (UTC) For consistency with other group articles, and compatability with all browsers.
I'm not sure we need a formal vote as consensus already appears to be forming. (If I must vote, though, my vote is yes.) VectorPosse 09:29, 22 March 2007 (UTC)
Also E₈ manifold. --Salix alba (talk) 09:38, 22 March 2007 (UTC)
CobaltBlueTony 12:17, 22 March 2007 (UTC) In agreement with all above.
The subscript title displays on this IE computer, but is barely legible. Septentrionalis PMAnderson 14:30, 22 March 2007 (UTC)

No

I went ahead and moved it back, as their seems to be no disagreement. I'll do the same for E8 manifold. -- Fropuff 17:58, 22 March 2007 (UTC)

Typographical issues

The E₈ in the title of this article looks fine on my Mac (at home), but, at least on this XP box it just displays as a box. Should the article be renamed? (Oh, and I do not want to start a Mac vs. PC discussion here. I just mean to call attention to a practical problem that can make the article title unreadable by a number of users. Greg Woodhouse 19:31, 21 March 2007 (UTC)

Yup title subscript is a box for me too in Internet Explorer. Tom Ruen 20:12, 21 March 2007 (UTC)

It is a box in Firefox as wellJason Smith 05:02, 22 March 2007 (UTC)

I think the title should be "E8" and not "E₈". I encounter the same font problems. KyuzoGator 18:37, 22 March 2007 (UTC)

Mac vs. PC? Literate adults use linux. Michael Hardy 20:07, 22 March 2007 (UTC)

Variety is the spice of chaos

Can we please decide on and consistently use either E₈ (with italics) or E₈ (without italics) inline, or offer some explanation for using both. And do not use $E_{8}$ . Because I went to the trouble of making the article consistent (it had all three!), only to see later editors blithely ignore that. I don't much care if different articles adopt different conventions; the literature varies, too. But please, can we stick with one in this article?! Thanks.

Oh, and never, ever, ever use E₈, as if '' were somehow magically equivalent to TeX $.

Also, we don't see the names of the Lie algebras much, but (except in the lead) I chose to go with lower-case bold instead of lower-case Fraktur inline, because it's clear enough to those who need to know, and it typesets much cleaner. Thus so(n) instead of ${\mathfrak {so}}(n)$ is my preference within Wikipedia's current limitations. --KSmrq^T 23:26, 22 March 2007 (UTC)

I promise I will never use

E_{8}

. --NEMT 00:37, 23 March 2007 (UTC)

This is one of things we'll never see a consensus on. But yes an article should be internally consistent. I prefer E₈ to E₈ but I also think colour is spelled wrong. Of course

E_{8}

is completely valid too, and decrees to not use it will surely meet with opposition. -shrug- Life goes on. -- Fropuff 00:58, 23 March 2007 (UTC)

Clear as so much mud

The 248-dimensional adjoint representation of E₈ transforms under SU(2)×E₇ as:

$(3,1)+(1,133)+(2,56)\,\!$

It would be nice to define this notation before using it. Septentrionalis PMAnderson 13:47, 22 March 2007 (UTC)

It is not obvious wny (1,-1,0,0,0,0,0,0} is not a simple root. Septentrionalis PMAnderson 13:47, 22 March 2007 (UTC)

As a non-specialist, what I really want is some indication of why the exceptional symmetries are exceptional, why E8 is the biggest. I think that's the most puzzling thing for the non-specialist. --tcamps42

Also for the specialist, it seems. Charles Matthews 20:15, 31 March 2007 (UTC)

give it more focus!

This all over the news now, before today i didn't have any idea about E8 and Lie groups, i think i need lots of reading to understand this. can any one help explain this article in more simple terms and give it more focus , as it seem that E8 is big thing !!. --Zayani 17:36, 19 March 2007 (UTC)

Just another confused reader :) - Zephyris _Talk 18:41, 19 March 2007 (UTC)

One word: wow. Or perhaps I should say "hear hear" ? (Epgui 02:15, 16 November 2007 (UTC))

theory of everything

would it be too soon to add this as a link in the article somewhere? --24.214.236.85 21:43, 15 November 2007 (UTC) http://arxiv.org/pdf/0711.0770

nvm. i see it has already been done. --24.214.236.85 21:46, 15 November 2007 (UTC)

An encyclopedia should provide relevant and verified content, notice that Garret Lisi's work has been hyped by the media but it hasn't even been published in a peer-reviewed journal.

I added: In 2007, Garrett Lisi proposed a controversial theory of everything based on E₈. But maybe one should also add some of the initially rather enthusiastic comments of Lee Smolin and the less encouraging ones of Jacques Distler? Discrepancy (talk) 19:14, 24 March 2008 (UTC)

Nice title

I like what has happened with the title. Can the same be done with E6, E7, F4 and G2? Nilradical (talk) 20:42, 14 August 2008 (UTC)

Explanation for spam declaration

Could we get some rationale for a proclimation from R.E.B. based on critical evaluation of the linked material he declares "spam". This of course presumes having used Mathematica to run through various 2D/3D projections of translational/rotational paths of E8. Without this due diligence, I suspect the spam label should be assigned to its source.

Reference the example pics and determine if the beautiful E8 symmetry (not available elsewhere from what I have seen) is spam. The first is a 2D projection much like the ubiquitous pic on the main page with edges norm'd to sqrt(2) and edge colors based on projected edge length (not angle) using a color spectrum determined from a specific Mathematica color gradient (BrightBands)). The second is another projection in 3D with edges norm'd to sqrt(6).

There are 6720 edges and 240 vertices. The vertices have various 3D shapes/sizes/colors/shades based on a theoretical mapping to particle physics. This of course is not part of E8, but can be easily ignored when viewing these beautiful structures.

Image:E8.png

Image:E8a.JPG —Preceding unsigned comment added by Jgmoxness (talk • contribs) 03:53, 3 July 2008 (UTC)

Jgmoxness (talk) 04:07, 3 July 2008 (UTC)

This challenge for due diligence also applies to "Giftlite" —Preceding unsigned comment added by Jgmoxness (talk • contribs) 01:06, 4 July 2008 (UTC)

R.E.B continues to remove w/o discussion - latest declaration "strange". Please provide an analysis of the tool from first hand experience and explain what is "strange". There is no other more flexible 2D/3D visual E8 representation capability on the net. If there is - it would be great to link here. Jgmoxness (talk) 14:34, 22 August 2008 (UTC)

E8 construction

I have added a {{fact}} ("citation needed") tag at the construction section of E8. Can we point this (and maybe also other) sections a bit more closely to the references at the bottom? 68.20.132.105 (talk) 01:38, 23 January 2009 (UTC)

Did you have a specific issue with this claim? In my sources this is the definition of the group of Lie type, and I think it is just a general fact about Lie groups and Lie algebras (the "fundamental theorem" more or less). JackSchmidt (talk) 03:55, 23 January 2009 (UTC)

Jargon

As is the case with most mathematics and physics articles on Wikipedia, this page is impenetrable to the non-expert. I'm not the only one who finds this; several other editors have expressed this view on the talk page. Please take this concern seriously, as readers should not need a post-graduate understanding of maths to get the gist of Wikipedia articles. I expect to get a feel for a subject from Wikipedia, and after reading this article I am none the wiser on the importance of the E8 pattern. Also note that, as it also far too common in physics and maths pages, there are no in-line citations, so the whole thing could be original research as far as any of us lesser mortals would know. Fences and windows (talk) 20:49, 30 March 2009 (UTC)

There were a lot of news stories in 2007, some of those could be useful in helping explain this to the (educated) layman: http://news.google.co.uk/archivesearch?q=e8+maths+physics&btnG=Search&um=1&ned=uk&hl=en. I'm surprised that all mention of the connection of the Lie Group to the symmetry of particles has been excised; Lisi's TOE might not be peer-reviewed, but it is notable, as it appeared all over the press, and has been cited quite a few times. Even a Wikilink to An Exceptionally Simple Theory of Everything would be a start. Fences and windows (talk) 21:12, 30 March 2009 (UTC)

Is there any reason why the omission of in-line citations in mathematics related articles has become a tradition (e.g., Baker-Campbell-Hausdorff formula)? Or is it just because math people think that math is clear by itself and there is not need for citations. --Marozols (talk) 00:48, 31 March 2009 (UTC)

Have a look at Wikipedia:Scientific citation guidelines --Salix (talk): 06:41, 31 March 2009 (UTC)

Zome Model

While I like the Zome model pic, it seems to be not an E8 representation, but two 600 Cells (of 120 vertices - the dual of the 120 Cell (one embedded at a ratio of the Golden Ratio)). This is, per Richter, isomorphic to E8 (at least in 2D but not E8). We might want to clarify this in the main page. Jgmoxness (talk) 02:06, 5 September 2009 (UTC)

Lisi theory -> "E8 theory"

At Talk:An_Exceptionally_Simple_Theory_of_Everything#Requested_move some editors apparently not acquainted with E₈ in other contexts are proposing to move that article to "E8 theory", which I feel would be ambiguous and giving Garrett Lisi's theory excessive weight. Please come and help discuss this. --JWB (talk) 03:55, 26 November 2009 (UTC)

e8 in Lab

7 Jan 10

New Scientist : Gosset 421 polytope (e8) appears in lab for first time

What about including this information in this article?

Cheers , Mateus Zica (talk) 09:52, 8 January 2010 (UTC)

A new Petrie projection - opinions please

I would like to get opinions of those watching this (and related) pages. Do you find the new E8 Petrie projection more attractive? Is is distracting?

Old	New

Please look at the description of how it is created to understand more completely the parameters of its creation. Of course, there are many color schemes to choose from that could be applied. Jgmoxness (talk) 14:01, 4 January 2010 (UTC)

I think including it twice on the same page is distracting. The older version with more of a solid dull green would be better for at least one of the images if both must appear. You might want to address the "correctness" complaint more directly. To me it appears that you have intentionally omitted part of the drawing, perhaps because the drawing is too dense to see. Assuming that is the case, wouldn't it be better to use a drawing that is easier to understand (say, for the navbox at the top of the article)? JackSchmidt (talk) 14:52, 4 January 2010 (UTC)

I agree that twice is distracting. Niout created/added the template - a nice touch for all related topics, but for this page only it creates a redundancy with the second representation. All representations shown so far omit edges. I will post one in commons and link here in Talk with all 6720 for comparison (and even a version with all edges (not just the sqrt(2) filter). Maybe even a short .gif animation going through all edge lengths. As for correctness, if desired I will share more of the code than what is already explained/provided in the image description and in the freely downloadable tool itself.

(Jgmoxness)

I emailed the author of the original (left) drawing, who duplicated the graph made by Peter McMullian. I wasn't aware that it was filtering edges. Can we see a hires (SVG) graph with no edges filtered? Tom Ruen (talk) 23:11, 4 January 2010 (UTC)

ok, looks like I we both made some assumptions. After generating full 6720 sqrt(2) edges in hi-res .svg E8 Petrie diagrams with super fine (.001) edge widths, the diagram does still look ok(not "TOO blotchy" as the lower res diagrams show, but maybe not as clear as the one in my original post). This version is much more like the Stembridge/Vogan representations with a seemingly similar edge color assignment.

See also an interestingly beautiful image with all 28,680 edges from all 4 edge sets of sqrt(2)*sqrt(1,2,3,4) with respective edge counts of (6720, 15120, 6720, 120) which sum to a total of the Binomial[240,2] (basically a combined single hi-res version of those in the .gif animations:

At this point - I don't have a problem with any of these diagrams being "the one shown". Let's discuss preferences.Jgmoxness (talk) 03:24, 5 January 2010 (UTC)

I'm a bit confused, but sounds like the orginal image from McMullian is "complete"! I figured each vertex should have 56 edges (6720*2)/240, while the original diagram (copied from McMullian) appear to show 28 edges/vertex (at least radiating from the outer vertices). SO if must have overlapping edges?! But not all edges have overlaps, SO some must have a least triple overlap? Can you draw the "full edgeset", colored them by overlap counts?

p.s. The narrow lines are good. I don't like the black central point - should be removed. Also with bright edges, it might be more clear to have black vertices? Also it should be made clear that the the 28680 edge set (complete graph), really isn't the E8 polytope! Tom Ruen (talk) 03:52, 5 January 2010 (UTC)

Okay, I confirmed this fact from by [3] "So in the picture, there should be 56 lines emanating from each of the black dots. But some of the lines cast shadows on top of each other, so in fact you see fewer. For example, there are only 28 visible edges emanating from each of the points on the outermost ring." Tom Ruen (talk) 04:00, 5 January 2010 (UTC)

Yes - I agree with most aspects of this last assesment. I will remove the center dot (it is a holdover from some of my code simplification(e.g. a s/w hack)). You suggest (as Vogan did in his representation)that black vertices are better - ok can do.

Now for a mathematically non-rigorous group theory/poltytope geometry discussion based on HOW I created these diagrams.
I generate the 240 8D vectors (the vertices of split real even E8 Lie group, the 128 half integers, 112 integers) from their base permutations.
I calculate all possible edge lines (grouped by norm'd 8D lengths). That total is your "complete graph".
I then take selected 2 (or 3) 8D projection vectors and dot product each vertex in order to project them into 2 (or 3) dimensions.
Each projected edge line from the 8D edge groups (from every possible pair of projected vertices) is given a color and sorted based on norm'd 2D (or 3D) projected length. Color gradients are selected from ~50 pre-defined or user defined patterns.
Based on the work of Richter, I have deduced the exact projection vectors the Petrie projection (as noted in my original (60% sqrt(2) edge length) graph description). This may not be "new", but I haven't seen anyone else derive them. I have seen them manually created by drawing two sets of 4 complex roots (one set of 4 is the other multiplied by the Golden Ratio) that are then rotating in 30 increments. You get the same 2D projection - but from a completely different method. Here is a sqrt(6) 6720 graph.

I suspect the combination of the two will look more like the "complete graph" than the McMullen, Stembridge/Vogan, Rocchini/Ruen versions.

Interesting discussion - need to work the graphs... Jgmoxness (talk)

WELL, I decided it was time I made my own as well, so I computed the 112+128 vertices given on the wikipage, selected edges by vertex pairs distance sqrt(2), and used the projection basis given by Jgmoxness. Then I computed overlapping edges: first split edges on middle-vertices, then counted duplications of all copies split subedges, and drew edges by lines with color&width of the overlap counts: (black=1, red=2, orange=3, yellow=4, lightgreen=5, green=6), drawing higher order counts last. I could use SVG resolution to better show it, but here it is. It looks pretty good. (I also tried looking at 1..6 counts individually, and found some nonsymmetry artifacts, maybe due to accuracy of basis vectors?!) I think something like this might be useful! Tom Ruen (talk) 07:14, 5 January 2010 (UTC)

E8 polytope edges colored by overlap order	edges to a single vertex at 8 radii

The 56 edges/vertex on the outer vertices come as a sum of "order" 1,2,3 edges, as shown in (10 black)/(8 red)/(10 orange): 10+2*8+3*10=56. Tom Ruen (talk) 07:25, 5 January 2010 (UTC)

I added a second image showing edges radiating from 8 radii of vertices in the projection. Again, counting edges times overlap order should sum to 56 (or maybe more for interior vertices if unrelated overlaps computed...). Tom Ruen (talk) 07:37, 5 January 2010 (UTC)

Great work - and fast too! BTW - I calculated the basis projection vectors to over 10^-12 accuracy from perfect rotational symmetry of specific complex roots. The asymmetric artifacts are most likely from pixelation errors in low res images (the .gif animation gives a perfect example of this). Therein lies the value of vector graphics :-)Jgmoxness (talk)

Thanks! Oh, on the issue of "different edge sets", any edges besides the 6720 set will be "interior", but can represent other polytopes with the same vertex arrangement, like the Kepler-Poinsot polyhedron are nonconvex (star) polyhedra, and Schläfli-Hess 4-polytopes are star 4-polytopes. Tom Ruen (talk) 20:30, 5 January 2010 (UTC)

Some notes on my draw (the old one): all overlapping edges are deleted, not only, bu also all edges that are "covered" by someone else. The SVG renders are strange beasts: color, thickness, transparency and over all, edge's drawing order, are carefully designed to force a good antialiasing on the SVG renders. My compliments for your good works and new improvements. Rocchini (talk) 08:55, 7 January 2010 (UTC)

Thanks for your input Rocchini, AND your new polytope graph - Gosset 2 41 polytope, and recognizing overlapping vertices by colors! [4] Tom Ruen (talk) 00:23, 8 January 2010 (UTC)

These are .svg maps generated using e8Flyer.nb. It confirms some (not all) of Ruen's numbers of overlapping edge counts (looks best at 2000px resolution - no asymmetry
The edge counts obtained are:
Ring 1:
InE8={{overlap,count},...,Total}
{{1,10},{2,16},{3,30}},56}
InView={{overlap,count},...,Total}
{{1,10},{2,8},{3,10}},28}

Ring 2:
InE8={{overlap,count},...,Total}
{{1,16},{2,28},{3,12}},56}
InView={{overlap,count},...,Total}
{{1,16},{2,14},{3,4}},34}

Ring 3:
InE8={overlap,count},...,Total}
{{1,22},{2,28},{3,6}},56}
InView={{overlap,count},...,Total}
{{1,22},{2,14},{3,2}},38}

Ring 4:
InE8={{overlap,count},...,Total}
{{1,24},{2,32}},56}
InView={{overlap,count},...,Total}
{{1,24},{2,16}},40}

Ring 5:
InE8={{overlap,count},...,Total}
{{1,28},{2,28}},56}
InView={{overlap,count},...,Total}
{{1,28},{2,14}},42}

Ring 6:
InE8={{overlap,count},...,Total}
{{1,32},{2,24}},56}
InView={{overlap,count},...,Total}
{{1,32},{2,12}},44}

Ring 7:
InE8={{overlap,count},...,Total}
{{1,32},{2,24}},56}
InView={{overlap,count},...,Total}
{{1,32},{2,12}},44}

Ring 8:
InE8={{overlap,count},...,Total}
{{1,44},{2,12}},56}
InView={{overlap,count},...,Total}
{{1,44},{2,6}},50}

The 3 smaller displays are obtained by clicking on vertices to generate corresponding nearest sqrt(2) edges from each. The large graph is obtained by selecting all 240 vertices.

Of course, in 3D the edges don't overlap - as shown below by adding a third projection vector of:
Z={0.338261212718, 0, 0, -0.338261212718, 0.672816364803, 0.171502564281, 0, -0.171502564281}
With output analysis of vertices 43, 16, 145:
As opposed to the full set of 6720 sqrt(2) edges in 3D:

Jgmoxness (talk) 05:00, 12 January 2010 (UTC)

Gosset

Anyone care to explain Gosset lattice, the early days of the Weyl group, whether Elie Cartan was the first to call this E8, and other bits of history? Charles Matthews 10:34, 20 March 2007 (UTC)

Thorald Gosset enumerated the "semi-regular" polytopes, some of which become infinite-radius (ie tilings in N-1 dimensions). One of these have at each margin (N-2 element), a simplex, and two cross-polytopes. One starts in three dimensions, with the triangular prism. In higher dimensions, one uses the previous member as a vertex figure, to get polytopes of (from 3d), 6, 10, 16, 27, 56, 240, and infinity, the last being a tiling.

The polytopes of 27, 56, 240 vertices are not connected to regular figures of that dimension, but represent an entirely new group. The attributions of 2_21, 3_21, 4_21 are due to Coxeter (see, eg "Wythoff Construction" in "Twelve Essays" (1968, reprint Dover 1998). The Coxeter graphs resemble the Lie graphs for E_6, E_7, E_8, names apparently used by Catalan.

The connexion to higher dimensions is not understood by me, but apparently, there are connections between (vertex+dimension) for various figures, eg E_8 = 240+8, E7 = 126+7 etc.

--Wendy.krieger (talk) 10:26, 30 January 2010 (UTC)

E8 subgroups

Does anyone have a reference for the subgroups of E8 shown in the picture?

Why is it called "Simple subalgebra tree of E₈" instead of "Simple subgroup tree of E₈"? Marozols (talk) 19:56, 19 February 2009 (UTC)

One of the other images uploaded by this user had (and still has) errors, but he took it more or less verbatim from a reliable source. I think the subalgebra/subgroup thing is not a significant change. Subgroups of the connected component correspond to subalgebras. I wish diagram uploaders took more time to describe how the image was created and checked. At any rate, I too would be interested in a reference. Google is too polluted with finite subgroups of E8(C) to find the Lie subgroup inclusions. JackSchmidt (talk) 20:38, 19 February 2009 (UTC)

The E8 subALGEBRA tree is wrong. The subgroup tree is wrong too

The graph listed as subalgebra tree is wrong.

The paper of Dynkin "Semisimple Lie algebras of Simple Lie algebras" gives a list of the regular semisimple Lie algebras, which includes, for example, Lie algebra type A8=sl(9).

[Edit:] The Lie subalgebra A8(=sl(9)) integrates to a Lie subgroup via the adjoint action, i.e. Ad(sl(9)) is a subgroup of the Adjoint group Ad(E8). Therefore this subgroup tree is wrong since it does not include sl(9).

At any rate, such a subgroup tree MUST not be listed without precise citations. This is a very tricky subject, and it is very easy to say/write something wrong!!! Tition1 (talk) 16:56, 19 May 2010 (UTC) Tition1 (talk) 16:45, 19 May 2010 (UTC)

A very serious problem with clarity

The first sentences of this article are:

"In mathematics, E8 is the name given to a family of closely related structures. In particular, it is the name of some exceptional simple Lie algebras as well as that of the associated simple Lie groups. It is also the name given to the corresponding root system, root lattice, and Weyl/Coxeter group, and to some finite simple Chevalley groups."

But right after we learn that E8 may refer to a number of different things, the very next paragraph, "Basic description", begins as follows:

"E8 has rank 8 (the maximum number of mutually commutative degrees of freedom) and dimension 248 (as a manifold)."

This creates a serious problem with clarity, in that this so-called "Basic description" does not say which of the many meanings of E8 it is a basic description of.

The answer can be inferred by reading further, especially if you are already familiar with the concepts. But that is assuredly not how a clear description should proceed.Daqu (talk) 23:02, 16 December 2007 (UTC)

Removed by author as ultimately unfruitful and perpetually distracting --85.156.216.218 (talk) 15:19, 20 May 2010 (UTC)

not so unique?

In the Basic Description, the author writes, ``The compact group E8 is unique among simple compact Lie groups n that its non-trivial representation of smallest dimension is the adjoint representation ... However this is also true of SO(3) (and SU(2) if one counts the natural rep on C² as 4-dimensional real). Simplifix (talk) 09:23, 4 August 2010 (UTC)

Zome Model

The latest changes to the description of the Zome model needs to be verified. I have replicated it mathematically (as shown on the page) by using 2 concentric 120 vertex 4D 600 cells (at the golden ratio) projected into 3-space using x-y-z unit orthogonal vectors. At most (per Richter), it would only be isomorphic to E8 (not a true projection of the 8D E8 root system or the 4_21 polytope). The edge counts and lengths are different and the roots are not derivable directly from a Cartan matrix and simple roots (AFAIK). It should also be noted, that not all 3360 edges of 4D length √2(√5-1) are able to be included in the physical Zome. While this model is still interesting, especially since it may hint at the connection between folding the E8 Dynkin to H4 (600 cell), it needs to be referenced accurately.Jgmoxness (talk) 13:53, 30 March 2011 (UTC)

Huh?

All of this mathematical Jargon undoubtedly helps explain this structure to Math professors and graduate students, but the rest of us have no idea what you're talking about. Could somebody add a short description in simpler language, please? Thanks! Ahudson 15:24, 19 March 2007 (UTC)

Is it a good idea to dumb down these concepts though? Maybe a simple english article (linked by the main?) would be a better idea. — Preceding unsigned comment added by 78.32.171.170 (talk) 14:29, 1 December 2011 (UTC)

Huh^2. I'm a math/s graduate and I haven't the faintest clue what this article is about.84.9.128.80 16:07, 19 March 2007 (UTC)

Huh^3. I have an undergrad degree in mathematics and this is all Greek to me. I read the news story at BBC online and came here for more information. The article is completely inaccessible to the layperson.BAW 17:21, 19 March 2007 (UTC)

This is a tricky topic in any regards. I don't really know the ins and aouts of it myself so regard everything below with a pince of salt. Heres a few pointers, Coxeter groups describe here describe the symmetry of polyhedra in n-dimensions. For example A2 is related to the symmetry of a triangle. The lie groups are a bit more complicated than the coxeter groups, but intermatly related. Probably the simplest Lie Group is the Circle group any you might get some idea of whats involved from there. Theres another group which describes the symmetry of the sphere there you can rotate around three axis, and each triple of angles will give one element in the group, so there are an infinite number of elements related is quite a complex fashion. Some of the lie groups are sub-groups of the symmetry group of the sphere. There things are also related to n by n matricies and also solutions of polynomial equatation x^n+a x^(n-1)+.... E8 itself is rather hugh related to symetries of an object in 128-dimensional space. --Salix alba (talk) 19:11, 19 March 2007 (UTC)

Gentlemen: It's called "higher" math for a reason. Lie groups are typically brushed over if you are getting your PhD in mathematics unless your major calls for more. This is an incredibly arcane subject. Don't expect a proper education on a wiki page. If you really want to learn this from wiki, you need to go back to Lie groups & Lie Algebras. If you are lost there, go back further. I majored in algebraic topology, and even I got little more than the brush-over.

And if you don't know what the "simple" means, then you probably need to find a good (undergraduate) program to enter.

NathanZook 22:59, 19 March 2007 (UTC)

I don't think Lie groups are that arcane. We begin with the idea of a group of things we can do, with the stipulation that

if we can do one thing after another we can do the combination,
we can do nothing, and
anything we can do we can "undo" (so that the combination amounts to doing nothing).

We can make this tangible with concrete examples.

Example 1. On a parade ground a member of a marching band can perform an "about face". Done twice, this amounts to doing nothing.
Example 2. We can also include "left face" and "right face", which undo each other; and each done twice is an about face.
Example 3. If we loosen up and allow any amount of turning, however small, our group becomes a Lie group.

In fact, the rotation groups, SO(n), are classic examples of Lie groups.

I will not pretend that a "simple Lie group" is quite so simple as its name suggests, nor will I undertake to explain the classification of Lie groups in detail; but the basic ideas are still accessible.

Some groups can be described as a combination of pieces; the ones that cannot, earn the name "simple".
Therefore we wish to catalog and understand the simple groups, and in doing so we discover a fascinating phenomenon.
Most of them fall into one of a handful of systematic families; the rotation groups are a good example of this. However, a very few do not; these are the "exceptional groups". (Perhaps it may help some readers to think of irregular verbs.)
The most complicated of these is our subject today, E₈, and it is far more complicated than the others. Think of it, if you will, as the Mount Everest of groups. It appears it has been conquered.

It would be foolish to think we can convey all that is required in training, sacrifice, and perseverance to conquer either goal. It would be foolish to think we can convey the view from the top and the personal impact of getting there. And it would be foolish to think the story ends here. But I suspect, in the long run, that this conquest may have far-reaching implications, especially for physics. --KSmrq^T 15:22, 20 March 2007 (UTC)

Sincere thanks for an eloquent reply and clear overview.Rich 20:16, 21 March 2007 (UTC)

Huh^{999,999,999,999,999,999 . . . . . .}????? I'm confident about getting into Stuyvesant High School, and I read the recent Scientific American article about "A Geometric Theory of Everything", but I still don't understand the basics of Lie algebra, let alone E₈ theory. —The Doctahedron, Ph. D. (not!), 68.173.113.106 (talk) 04:04, 25 November 2011 (UTC)