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Archive 1 |
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)
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)
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..."
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)
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 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) 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)
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 , 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)
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)
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 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)
Yes
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)
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)
Mac vs. PC? Literate adults use linux. Michael Hardy 20:07, 22 March 2007 (UTC)
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 . 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 is my preference within Wikipedia's current limitations. -- KSmrq T 23:26, 22 March 2007 (UTC)
The 248-dimensional adjoint representation of E8 transforms under SU(2)×E7 as:
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
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)
One word: wow. Or perhaps I should say "hear hear" ? ( Epgui 02:15, 16 November 2007 (UTC))
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
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)
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)
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: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)
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)
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)
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)
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)
7 Jan 10
What about including this information in this article?
Cheers , Mateus Zica ( talk) 09:52, 8 January 2010 (UTC)
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)
(
Jgmoxness)
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)
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)
![]() E8 polytope edges colored by overlap order |
![]() edges to a single vertex at 8 radii |
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)
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)
-- Wendy.krieger ( talk) 10:26, 30 January 2010 (UTC)
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)
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)
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)
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)
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)
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)
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)
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)
Huh999,999,999,999,999,999 . . . . . . ????? 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)
![]() | This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 |
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)
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)
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..."
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)
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 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) 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)
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 , 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)
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)
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 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)
Yes
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)
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)
Mac vs. PC? Literate adults use linux. Michael Hardy 20:07, 22 March 2007 (UTC)
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 . 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 is my preference within Wikipedia's current limitations. -- KSmrq T 23:26, 22 March 2007 (UTC)
The 248-dimensional adjoint representation of E8 transforms under SU(2)×E7 as:
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
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)
One word: wow. Or perhaps I should say "hear hear" ? ( Epgui 02:15, 16 November 2007 (UTC))
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
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)
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)
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: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)
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)
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)
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)
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)
7 Jan 10
What about including this information in this article?
Cheers , Mateus Zica ( talk) 09:52, 8 January 2010 (UTC)
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)
(
Jgmoxness)
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)
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)
![]() E8 polytope edges colored by overlap order |
![]() edges to a single vertex at 8 radii |
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)
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)
-- Wendy.krieger ( talk) 10:26, 30 January 2010 (UTC)
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)
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)
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)
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)
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)
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)
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)
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)
Huh999,999,999,999,999,999 . . . . . . ????? 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)