Talk:E6 (mathematics)

Diagram
Someone contacted us mentioning that File:E6Hasse.svg is missing the #11. Just thought I'd drop this here where it's more likely to be seen. § FreeRangeFrog croak 04:27, 1 May 2013 (UTC)


 * I can confirm that the Hasse diagram is missing a root (I already started to clarify the root poset definition and that the Hasse diagram of this poset is shown). Actually, I don't see why the roots are at all labelled by integers; we discussed that at the University of Minnesota some days ago and agreed that such a labelling is not natural at all. These are roots in the root system and can as such naturally be expressed in simple roots. I will propose new versions of the Hasse diagrams reflecting their nature as roots. The tree stump (talk) 08:41, 1 May 2013 (UTC)


 * I provided a new graphic - but I don't see why it is not getting updated on the wiki page to its newest version... The tree stump (talk) 08:08, 6 May 2013 (UTC)


 * Ok, I checked the code and diagrams. The "missing nodes" are not really missing, they are hidden behind a node in front. This is an artifact from the code that determines the horizontal position of the node (which is based on the root vector for that node and a dot product with a small table of the same rank. I just need to add a randomizer). This creates a small potential for overlap. I will take care of it. The node numbering is based on output from SuperLie Mathematica code for a given Cartan Matrix input (generated directly from a Dynkin diagram as graphical user interface (GUI) input. I will add the root vector text on each edge line in the diagram or some other way to add the detail (it may be hard to fit it in these larger groups).


 * I am also considering reworking the color coding of the edges and nodes. I have created an interactive demonstration that provides much more than the Hasse diagram. See below for an example:

Jgmoxness (talk) 15:36, 31 May 2013 (UTC)


 * To be honest, I certainly prefer the simple latex output that I provided in type E6. It is as simple as possible, though still reflecting all information about the root poset. In particular, all information that you show in your new graphics on the left is reflected in this simple figure -- except for which simple root corresponds to which node in the Dynkin diagram, this information can simply be added by rechecking that your provided Dynkin diagram has the vertices labelled consistently with the simple roots in the root poset. What do you think? The tree stump (talk) 08:32, 2 June 2013 (UTC)
 * I have no problem leaving your (manually created in LaTex?) .png output on the E6 page. Of course, as you point out, the size of E6 is about the limit for doing this by hand (as well as with your style for labeling nodes).
 * I updated my algorithm based generation of the E6,F4, E7 and E8 Hasse diagrams to include the nodes and edge labels for indicating which simple roots are added to get to the next higher grade node. The color of the edge is assigned based on the color of the Dynkin node with the same number as the simple root added. The Cartan Matrix row is the same as the weight of the simple root vector number. The assignment of the Dynkin node number here is based on the off diagonal row and column where the Cartan Matrix entry is changed from 0.

Jgmoxness (talk) 17:41, 2 June 2013 (UTC)


 * Can you output in .svg?
 * I wasn't yet able to convert the output to svg. The Sage output is tikz, which I can run pdflatex on to get a pdf. If I try to get an svg from it, it becomes huge and not very nice - any better ideas?
 * The output is actually Sage output (I only had to adjust some parameters in order to get the output showing the complete diagram), I also uploaded png's in types E7 and E8. If that's okay with you, I'd add these (and also create images for types F and G):




 * As I wrote above, I'd prefer to use my png's (or svg's if I get it converted properly). Would that be okay with you? The tree stump (talk) 11:27, 3 June 2013 (UTC)
 * No, I will try to account for your desire for simplicity, while trying to keep desirable aspects of mine. Jgmoxness (talk) 13:47, 3 June 2013 (UTC)


 * Hmm.. I prefer User:The tree stump's version due to its simplicity and clarity. Mark M (talk) 14:17, 3 June 2013 (UTC)
 * Agreed, but I will try to fix it.Jgmoxness (talk) 23:47, 3 June 2013 (UTC)
 * I am also in favor of User:The tree stump's Sage graphic (even over the graphic I originally proposed, which you may find at http://www.math.umn.edu/~will3089/docs/E6RootPoset.svg). Although User:Jgmoxness's software package seems very nice, I am not fond at all of the styling of the root poset currently displayed on the E6 page.Nathan.f.williams (talk) 06:27, 4 June 2013 (UTC)
 * Nathan's diagram lacks both info and artistic style. WP is not just about the logic - most mathematicians need a bit of help on that right brain part of the equation.
 * Let's split the difference. Put Tree Stump's vertically mirrored mono-color plain text .png without added simple root indicators on F4 and E8. We can leave my stylized color .svg with added simple root indicators on E6 and E7.


 * Jgmoxness (talk) 13:17, 4 June 2013 (UTC)

In the end, I don't care enough as long as the shown figs are not wrong; I can provide other root posets or can try to get svg's, but I certainly don't need to. Nevertheless, here are some further comments on your images: The tree stump (talk) 13:56, 4 June 2013 (UTC)
 * The edge labels ("simple root indicator") are very redundant in the sense that if I want to know the edge label, I just check in which coordinate the the two roots differ, same for the colors. Also see that some of your edge labels are sitting on top of each other...
 * The text "Hasse diagram of E7 root poset with root vector node references and edge labels identifying added root position" is weird, since the vertices of the root poset are positive roots, thus "root vector node references" is misleading.
 * The name "E7HassePoset.svg" should maybe be "E7RootPoset.svg" ? At least HassePoset doesn't make any sense since posets are usually identified with their Hasse diagrams.
 * Yes, one can calculate the edge label, but we can save a bit of neural energy waste by giving it in the diagram.
 * I can change the text under the diagram (unless you want to do it - it is a wiki).
 * I didn't want to step on your file name for the diagram (in case you create an .svg version). What's in a file-name, it indicates the Lie Group, that it is a Hasse diagram and based on root "posets". Lighten up, it's just a file name.
 * You do what you think you need to do - and so will I.
 * Jgmoxness (talk) 00:12, 5 June 2013 (UTC)

Concentric Hulls of E6
I removed this material from Jgmoxness since it doesn't make sense to me and has no documentation. The projection vectors are useless unless you give explicit 6D roots, while the roots give in the article are in 9D or Alternative one in 6D, and neither work. The convex hull of the projective vertices seem to have D_5d symmetry, from an incomplete icosidodecahedron. It looks like just one of infinitely many projections that serve no clear purpose to anyone. Tom Ruen (talk) 20:32, 24 October 2017 (UTC)


 * Tom Ruen is using deliberately pejorative language as an unfortunate logical fallacy (ad hominem attack). It doesn't make sense because he doesn't understand the root system reference to taking perpendicular vertices from E8 (mathematics) to E7 (mathematics), and then E7 to E6. This was being clarified as he removed the content during my edits. The purpose is clear once you understand E6 in context to E7 and E8 and their relationship to the 3D projection used to create H4 600-cell and H3 as well as rhombic triacontahedron from the 6-cube and D6 to H3.


 * His entire argument relies on the lack of a description for the particular E8 derived roots of E6 I am using. Indeeed, the article is lacking in its description of such a derivation. So instead of excluding my content, we should enhance the article to include a more relevant E6 root system based on perpendicularity of E8. That is, in addition to only those derived from 9D and the "alternate" roots he seems to understand.


 * From the root system article:
 * The root system E7 is the set of vectors in E8 that are perpendicular to a fixed root in E8. The root system E7 has 126 roots.
 * The root system E6 is not the set of vectors in E7 that are perpendicular to a fixed root in E7, indeed, one obtains D6 that way. However, E6 is the subsystem of E8 perpendicular to two suitably chosen roots of E8. The root system E6 has 72 roots.


 * Since perpendicularity to α1 means that the first two coordinates are equal, E7 is then the subset of E8 where the first two coordinates are equal, and similarly E6 is the subset of E8 where the first three coordinates are equal. This facilitates explicit definitions of E7 and E6 as:


 * E7 = {α ∈ Z7 ∪ (Z+½)7 :  ∑αi2 + α12 = 2, ∑αi + α1 ∈ 2Z},
 * E6 = {α ∈ Z6 ∪ (Z+½)6 :  ∑αi2 + 2α12 = 2, ∑αi + 2α1 ∈ 2Z}


 * Jgmoxness

A big graphic of coordinates seems particularly ugly, i.e. File:E6-roots.png. Also, why not multiply by 2, giving 1 and 2 values instead of 1/2 and 1? It looks like the 72 roots are: 40 of 60 permutations of (±1,±1,0,0,0,0), and 32 of 64 (±1,±1,±1,±1,±1,±1). So I see it is even signs for the second one. What's the pattern with the 110000? Tom Ruen (talk) 01:45, 25 October 2017 (UTC)


 * Worse, actually these roots are not the same length: sqrt(2), sqrt(3/2). That's clearly wrong. I removed the new section for now. Tom Ruen (talk) 01:49, 25 October 2017 (UTC)


 * I see your point, but the norm length for the roots from E8 are based on the 8D E8. As such they are 8D norm of Sqrt[2]. I changed the picture to include the 8D roots with description modified. The argument you are making suggests the root system article I am referencing is flawed.


 * Your questions regarding multiplying by 2 to make it look nicer is missing the ENTIRE point that these are a selected subset of split real even E8 roots (as shown in the E8 article). Please stay on focus rather than trying to fit everything into a too narrow view of the topic.


 * E6-roots-of-E8.svg


 * I repeat the root system article:
 * E6 is the subset of E8 where the last three coordinates are equal.


 * I am going to correct that and revert your undo - please don't undo that w/o getting a third person to weigh in. PLEASE.
 * Jgmoxness


 * I see your 8-dimensional coordinates are the same as the alternative description, where the last 3 are folded into 1 dimension. Tom Ruen (talk) 03:56, 25 October 2017 (UTC)
 * I see your "E6 to H3" projection of the E6 roots, 1_22 polytope, the 12 vertices of the dodecahedron and 20 vertices of the icosahedron ALL come from the 32 vertices of a 6-demicube inside the 1_22 polytope, (±1/2,±1/2,±1/2,±1/2,±1/2,±[1/2,1/2,1/2]). The other 40 vertices project (of your irregular 20-vertex polyhedra) project into a dihedral symmetry. So the H3 connection really has nothing to do with E6. It's 100% D6 to H3 folding. Tom Ruen (talk) 09:41, 25 October 2017 (UTC)
 * You are again focusing on a limited view of what is relevant to the readers. Yes, the icosahedron and dodecahedron are the +/-1/2 integer elements of E8 (in this case the even 6-demicube (as described on the regular dodecahedron article)). This is by the way NOT related to D6 at all, except we use the same basis vectors to fold it to H3. The 2 partial concentric icosadodecahedrons (20 of 30 vertices scaled at the golden ratio) come from the 40 of 60 vertices of D6. Thus, it is pedagogical to show in 3D how E6 is relates to both the 6-demicube and partial D6 (along with E7 and E8). It is really a beautiful picture! In fact -I find it interesting you fight against it so desperately. What's up with that? The WP needs more 3D visualizations of E8 based geometry. Please get other experts to deny this is valuable before disallowing it.
 * Jgmoxness


 * I think I have to agree with Tom Ruen that it's not clear to me what about E_6 is being illustrated by this figure or why it's significant. The traditional thing to ask at this point is, do you have a source that supports this representation?  (I will certainly not deny that it is a beautiful picture.)  --JBL (talk) 15:26, 25 October 2017 (UTC)


 * I wouldn't think that identifying polyhedron shapes from vertex coordinates would be something that needs a source, kind of like 2+2=4. All I am doing is (as stated in the text of the image) projecting E6 vertices and identifying the concentric hulls using a basis common to multiple related articles. The fact is the image is correct and the statements are true. They have pedagogical value in the ability to identify elements of other polytopes (6-demicube, D6, which I have clarified in the description). If you agree they look beautiful and agree that they are accurate, are you denying value in seeing the link to other polytopes and Lie Group symmetries (e.g. E7 and E8)? Really?


 * FWIW - Dechant and Koca have done similar work on this in terms of visualizing the vertex projection symmetries / shapes. The only difference is they don't pull it together in a single ordered opacity 3D image.
 * Jgmoxness


 * The use of sources in this context is (1) to prevent people from doing their own research on whatever interests them and dropping it into Wikipedia, where it doesn't belong, and (2) for determining the appropriate weight to assign to something (space, text, figures, etc.). For example, it's a perennial problem on articles about things from basic number theory that people want to add all sorts of (true!) statements about what digits a perfect square can end with in base 12 (or whatever), and these kinds of additions distort an encyclopedic treatment away from the significant portions of the material.  Evidence that other people have done this before (in the form of sources) provides verifiability, complies with the policy WP:OR, and helps determine how much weight to give a given statement, figure, or whatever.  There are lots of pretty pictures that one could, in principle, draw.  --JBL (talk) 22:01, 25 October 2017 (UTC)
 * I have been around WP long enough to know the game. Sometimes the OR argument is valid, other times it is thrown into the mix along with colluding with cliques of authors who gang up to prevent fresh content. Not sure if one or both of these are in play here, but I would like to get a few more opinions before we settle this. If you want to use OR on interesting graphics that may not be well documented in papers, I suggest we take a deep dive into the volume of stuff that Tom Ruen has done. While we worked on extending the 2D projections (much of which I GAVE HIM from my work), he knew most of it could be considered OR. So it seems we have the pot calling the kettle black. So Tom, get started verifying all your projections with sources. That means papers that show basis vectors and defined vertex groups as you tried to suggest earlier was suspect in this image. Or you could simply stop being irritated about our dialogue on FB and stop trying to bully me by preventing interesting info on E6. Maybe?
 * Jgmoxness
 * I agree that Tom is also often guilty of the kind of thing I'm talking about, and indeed I have from time to time removed his additions from various articles. Unfortunately for you, that other people have sinned does not exempt you from following WP policies.  --JBL (talk) 00:16, 26 October 2017 (UTC)
 * "stop trying to bully me by preventing interesting info on E6" This is a preposterous losing argument. You are presenting what looks to me as nonsense as facts with no sources, and if calling nonsense nonsense is bullying, it is up to you to defend your material in a way that at least one person can understand. You've already admitted to me you have no interest in standard terminology, mixing and matching as your intuition leads you, whether anyone else can understand what you're doing being irrelevant. My challenges have already helped correct some of your errors that could have remained here for years if no one else looks at it, and just assumes you know what you're talking about. It's not my job to agree with you. Tom Ruen (talk) 00:18, 26 October 2017 (UTC)

Joel - I have given you sources for this - check out P. Dechant's latest papers. Also check out Koca. Oh, and check out Baez's article on D6 to H3 (he uses my work as a basis for that - also referenced in his AMS blog). In fact, I think we are remiss in these related WP pages not having an article for Dn or D6 that includes the D6 to H3 stuff of Baez, et. al.

Tom - I am not making the case for keeping it in based on bullying. I am suggesting your frustrated by your inability to understand the importance in the pedagogical aspect of showing E6 in 3D (something lacking on WP). Yes, it is easy to find someone else who lacks the insight to agree with you. But combine the frustration in not understanding value of 3D models as they relate to others in larger Lie groups with your dislike of me from social media is causing you to not be fair and impartial WRT this article. If you were honest and ethical (which I have evidence in dealing with you in the past - you have some growth needs in that area), you would admit this to yourself.

Yes, collaboration does help get the content right. That is the value in WP. In fact, it seems your initial rejection of the picture forced me to highlight the fact that the E6 article was lacking certain important content. I fixed that and all I got was more crap from you on that not liking the look of it (yet, the content now seems correct). See, you can learn something new from me !!! So go one step further and try to understand how 3D concentric hulls with no unused vertices within can be insightful on the structure of Lie groups! You have been hell bent on only being concerned with exterior hulls. WHY?? I know you want to get into 3D after running the course on 2D (from my work). Now you probably want to pull another stunt and do work based on my stuff w/o citation and call it your own. That's why you re-engaged with me on FB to determine the substructure of the 6-cube to rhombic 30-hedron.

If you keep trying to block this valid pic on E6, your going to need to start putting in sources for all your work. I am going to start calling you out on all your OR across the hundreds of articles with your stuff.

So instead of doing that - let me put that image back and see what others think.

Jgmoxness


 * I thought you were going to communicate with Baez on it and see what he thinks of this discovery of yours? I'm glad to await his response. I'd be much happier if JBL wanted to play "bad cop" in this game, and then I could offer sympathy for your plight. Tom Ruen (talk) 16:16, 26 October 2017 (UTC)
 * The only significance I can see to this projection is that the result has 5-fold symmetry, or D_5d symmetry to be more explicit. Your statement of irregular 20-vertex polyhedra actually diminishes whatever significance is here. I see the A4 Coxeter plane of the 1_22 polytope has 5-fold symmetry. I wonder if your 3D projection will match the A4 Coxeter plane projection when viewed along a symmetry axis? In 2D the 5-fold symmetry actually doubles to 10-fold, probably becayse of the projected D_5d symmetry.Tom Ruen (talk) 07:03, 27 October 2017 (UTC)
 * I was right, the 5-fold symmetry axis of the 3D projection, projected into 2D matches the A4 Coxeter plane vertices, for whatever this is worth. So you could say there's a projection from E6 to A4, and then a folding to H2, but the 3D basis is taking a detour into something else, that still allows the H2 projection. The advantage of the 3D model is all 72 vertices are unique (so a reverse projection is possible back into 6D) while the 2D projection has overlapping vertices. Tom Ruen (talk) 11:15, 27 October 2017 (UTC)
 * 1 22 polytope A4 Coxeter plane.pngUp 1 22 t0 A4.svg


 * Very good analysis. Maybe that needs to go into the article along with mine. I find it interesting that you are willing to take the 3D model generated from my 2010 discovered basis vectors and rotate it in 3 space (with π φ/n based rotation matrices which I also gave you) to show that the 3D model projects to well known 2D symmetries. Cool (as we discussed on FB messenger before you started deleting my WP stuff). Yes, we also show the 8D, 6D and 4D connections as well. Cool.
 * Yes, you could skip over 3D just because all concentric hulls within are not perfect, but.... that isn't the only purpose. Why is it you feel the need to erase the significance of the 3D in enabling all this insight (along with unique 3D vertex positions and minds eye perspective) - all because the generation of E6 from perpendicular E8 removes 20 of the 60 D6 roots used to form φ concentric regular icosadodecahedrons. Yes, those resulting two concentric hulls are not regular (for good reason), but they exist and inform the user to these n=1 to 8 dimensional symmetries within E8. Why leave out n=3? It's all good.
 * You are perfectly willing to splatter hundreds of every 2D projection symmetry image across all the Lie group and polytope pages and find it difficult to admit ONE, yes ONE, 3D concentric hull image. It isn't for consumption of valuable space. It isn't for lack of specific utility in the image. Indeed, this very image helped you determine the A4 connection to E6. Yay!! If you still insist it is for lack of utility, you would need to answer the why ALL of the hundreds of 2D stuff is important enough to keep, other than they are beautiful and help understand the landscape of this stuff - which is the same as this 3D image you don't seem to like). Jgmoxness (talk) 15:11, 27 October 2017 (UTC)


 * BTW - you should highlight the overlap count differences in the 2D showing how E6 is different from D6 and/or A4 (and E8 and E7 too). I can do it for you if you like. Maybe that might be OR, not sure (despite it being basic hyper-dimensional geometry).Jgmoxness (talk) 15:16, 27 October 2017 (UTC)
 * The significance is still questionable. The connection of E6 to A4 was always clear, E6=[32,2,1], A4=[32,1]. Myself, I don't know what visualizations are valuable for root systems, but at least 2D you can clearly see what's there. The E8 roots as rings in H3 also seems questionable, but at least it is a complete folding projection from H4, unlike this 3D projection which fails to maintain full icosahedral symmmetry. Tom Ruen (talk) 15:26, 27 October 2017 (UTC)


 * So why do you get to decide for the world whether icosahedral symmetry of two of 4 concentric 3D hulls must determine value? That sounds a bit arbitrary. All that does is prevent insight into relationships that go beyond the limited view of geometry you seem to want to keep in a box (pun intended). Jgmoxness (talk) 15:30, 27 October 2017 (UTC)


 * Tom Ruen> I don't know what visualizations are valuable for root systems
 * Well, I guess that answers my question - you don't know enough to make a case for value (or lack of value) for root systems. Therefore, the case is closed for your POV WRT my image. Now let this image stay and go about making more images for these pages. Jgmoxness (talk) 15:38, 27 October 2017 (UTC)


 * If you draw a polytope projected into a Coxeter plane and the polytope projection has a lower symmetry than the coxeter plane's symmetry, then it is NOT a member of that Coxeter plane. If 2 of 4 rings of vertices have 8-fold symmetry of a Coxeter plane and the rest have 2-fold symmetry, then it simply doesn't belong to that group. It's something different. If you can find papers that talk about incomplete foldings and describe their significance, it might justify something being on Wikipedia. Like I said, ask Baez to look at it and tell you if its worth anything at all, and if anyone is doing any such things that can be referenced. Tom Ruen (talk) 15:42, 27 October 2017 (UTC)


 * This is a 2D projection using the 3D D6 to H3 basis that shows E6 and you can still see the 5 and 10 fold symmetry. Just for fun of course.


 * E6 vertices projected to the 3D H3 basis vectors:
 * u = {1, &phi;, 0, -1, &phi;, 0}
 * v = {&phi;, 0, 1, &phi;, 0, -1}
 * w = {0, 1, &phi;, 0, -1, &phi;}
 * Which are then rotated by π φ/5 around v and π/5 around w - and shown in 2D with the resulting {u,v} basis.



Jgmoxness (talk) 16:07, 27 October 2017 (UTC)

BTW - This 2D projection shows that E6 is a partially alternated form of E8. Projecting E8 using the same basis gives the full decagonal Coxeter plane, but it is fundamentally different in that it is a 6D basis (with {0,0} appended rather than the simplistic 5D with {0,0,0} appended of A4. It works fiine for a 5D object like A4. Yet in a sense, by avoiding the 3D progression and over-simplifying the 2D using your A4 basis is a naive way to get to a decagonal Coxeter planes for higher dimensional objects like E6 (or E8 plus folded to H4 using same), D6, and 6-Cube for that matter.





Jgmoxness (talk) 16:54, 27 October 2017 (UTC)
 * I often disagree with Tom on the value of certain illustrations, but here I agree that these ones seem arbitrary, uninformative, and to the extent that they are not those things, original research. —David Eppstein (talk) 01:42, 28 October 2017 (UTC)


 * I don't see any citations in any of the above, only an editors own work which requires quite a bit of explanation. Wikipedia isn't the place for original research and I think we better see something like this explained in an outside source to show it is noteworthy and worth putting in before putting it into Wikipedia. Could you please provide full citations for where you say 'FWIW - Dechant and Koca have done similar work on this in terms of visualizing the vertex projection symmetries / shapes. The only difference is they don't pull it together in a single ordered opacity 3D image'. Dmcq (talk) 11:08, 28 October 2017 (UTC)


 * Could I also point out that presenting matrices as images is against the image use policy in WP:IUP. Dmcq (talk) 11:27, 28 October 2017 (UTC)
 * I stopped supporting WP for several years because of the likes of responses like this. I am truly sorry I reengaged after Tom asked for help on understanding the internal structure of the 6-cube to rhombic 30-hedron projections I did based on the E8 to H4 folding matrix that has been so successful for many articles. Yet, it looks like nothing has changed. So never mind, I'll let WP stay ignorant of the insight I could provide, you're not worth the time. Fix the matrices yourself if you really care about the matrix as image rule -that is VERY important! 14:52, 28 October 2017 (UTC)
 * Yes I'm afraid that is what domain experts sometimes say. I'm sorry but that's the way Wikipedia is. It isn't ArXiv, it is an encyclopaedia and it has to follow rather than lead and it aims to be easy to edit so people can fix problems and improve articles - it doesn't depend on single expert contributors editing articles. If you wish people to stop being ignorant about something you should write a paper and have it peer reviewed or at least have some other people reference it. I'll have a look at fixing the format of those matrices. Dmcq (talk) 17:02, 28 October 2017 (UTC)
 * Well this is them translated into LaTeX, however I believe it is a list of 72 entries rather than the matrix format meaning anything, it should be easy enough now to reformat as desired. I'll leave it to others to decide if all this should be in the article. Dmcq (talk) 18:23, 28 October 2017 (UTC)

$$ \left[ \begin{smallmatrix} -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} \\ \tfrac{1}{2} & \tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} \\ \tfrac{1}{2} & -\tfrac{1}{2} & \tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} \\ \tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & \tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} \\ \tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & \tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} \\ -\tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} \\ -\tfrac{1}{2} & \tfrac{1}{2} & -\tfrac{1}{2} & \tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} \\ -\tfrac{1}{2} & \tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & \tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} \\ -\tfrac{1}{2} & -\tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} \\ -\tfrac{1}{2} & -\tfrac{1}{2} & \tfrac{1}{2} & -\tfrac{1}{2} & \tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} \\ -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} \\ -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ -1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ -1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \end{smallmatrix} \right]

\left[ \begin{smallmatrix} -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 \\ \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} \\ \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & -\tfrac{1}{2} & \tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} \\ \tfrac{1}{2} & \tfrac{1}{2} & -\tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} \\ \tfrac{1}{2} & -\tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} \\ \tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} \end{smallmatrix} \right]

\left[ \begin{smallmatrix} -\tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} \\ -\tfrac{1}{2} & \tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} \\ -\tfrac{1}{2} & -\tfrac{1}{2} & \tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} \\ -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & \tfrac{1}{2} & -\tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} \\ -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} \\ 0 & 0 & 0 & 1 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \end{smallmatrix} \right]

\left[ \begin{smallmatrix} 1 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} \\ \tfrac{1}{2} & \tfrac{1}{2} & -\tfrac{1}{2} & \tfrac{1}{2} & -\tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} \\ \tfrac{1}{2} & \tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} \\ \tfrac{1}{2} & -\tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & -\tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} \\ \tfrac{1}{2} & -\tfrac{1}{2} & \tfrac{1}{2} & -\tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} \\ \tfrac{1}{2} & -\tfrac{1}{2} & -\tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} \\ -\tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & -\tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} \\ -\tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & -\tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} \\ -\tfrac{1}{2} & \tfrac{1}{2} & -\tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} \\ -\tfrac{1}{2} & -\tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} \\ \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} & \tfrac{1}{2} \end{smallmatrix} \right] $$
 * Dmcq (talk) 18:23, 28 October 2017 (UTC)


 * If this pic is OR (a notion which I dispute), then I strongly suggest you take a harder look at everything Tom Ruen is doing with Coxeter plane projections. Ask for sources for his basis vectors and vertex coordinates. There's much OR in there (if you define simply projecting hyper-dimensional geometry for visual perspective OR). For some reason he gets a pass on this. It must be nice to get inside the cabal to assert authority with bias attached. Well, that is WP I guess.Jgmoxness (talk) 18:31, 28 October 2017 (UTC)
 * Please see WP:OTHER. It can be a valid argument but only really when connected with some positive argument. It is not reasonable to require people to take a hard look at everything another editor has done before doing anything here. If you want to point at specific problems on another article about some obscure polytope there may be a problem with too few editors watching it in which case I would advise putting a note at WT:MATH. Dmcq (talk) 19:54, 28 October 2017 (UTC)
 * Dmcq, thanks for the example formatting. Interestingly this shows the advantage of Jgmoxness's image, which can be scaled down on the article and clicked for a more readable scale. 72 8D vector is just huge whatever we do, but the Latex formatting as-is is bulky, and no better since you can't copy&paste from it. Tom Ruen (talk) 21:11, 28 October 2017 (UTC)
 * On the other hand, Jgmoxness's image is incredibly ugly and unreadable. What the two together show is that it's a dumb idea to list 72 vectors that can be described by two natural-language sentences.  --JBL (talk) 21:17, 28 October 2017 (UTC)
 * Agreed. The following section E6_(mathematics) does exactly that, same basis with the last 3 coordinates projected into one. For fun, I also pasted in my own spreadsheet coordinates (more nicely sorted) into the ugly graphic description File:E6-roots-of-E8.svg#Tab-delimited_listing. Tom Ruen (talk) 19:49, 29 October 2017 (UTC)
 * I've changed the LaTeX version to make it almost equally small and nasty to show that these things can be done, but yes there isn't much point if two lines can describe the whole thing. Dmcq (talk) 21:24, 29 October 2017 (UTC)
 * Ha! At least you showed how it's done, in case Jgmoxness is interested in adding a LaTex export for other smaller uses. Tom Ruen (talk) 21:45, 29 October 2017 (UTC)