E8 and the Gosset polytope 421

No, not another flower from my garden. This is a two dimensional projection (originally hand drawn in the 1960s by Peter McMullen, of a polytope that lives in eight dimensions, known as the Gossett polytope 421. Click here to be taken over to the American Institute for Mathematics (AIM) site for more information about it. (This image was computer generated by John Stembridge, and you can get higher resolution there for use on your T-shirts and so forth.)

What does this all pertain to? A new result from a team of mathematicians. They’ve done what some are calling the mathematician’s equivalent of mapping the genome of a Lie group, the one called E8. Groups pertain to symmetries. Symmetries are operations that don’t change something, for example rotating an equilateral (all sides the same) triangle in the plane by 60 degrees is such an operation, as is by 120 degrees, or you could leave it alone. Those three operations for the discrete “group” of operations called Z3. Symmetries are of vital importance in physics and mathematics, and one of the things that is important to do is understand some of the more complicated groups that appear. our mathematician friends devote a great deal of their lives to classifying groups, describing them, and determining important properties.

The group E8 is one of the “exceptional Lie groups” that is of considerable interest in various areas of mathematics and physics. It is, from my perspective, a fascinating object that shows up in several places in string theory, for example, for reasons that are not at all clear. I first fell in love with when I first heard about it as an undergraduate in one of my other favourite contexts – Penrose tilings. In three dimensions, the nicest Penrose tilings have the symmetry of a regular icosahedron or dodecahedron, and the discrete symmetries of that shape are associated to an exceptional “subgroup” of SO(3) (the name for the group of symmetries corresponding to rotations in three dimensions) which can be shown to map to E8.

Don’t worry too much about the terminology that I’m using here. You’ll get more of the gist of it as you go along, and I’ll give some links at the end. What did the mathematicians do? They essentially mapped out the structure of E8, in a sense, which is (I think) equivalent to classifying and explicitly exhibiting the building blocks of all its representations. E8 is a 248 dimensional group, which means that if I were to try, using its most natural representation, to describe all its elements and the relations between them using points in a space, I would need a 248 dimensional space (recall that we live in just the three spatial dimensions), and the fact that it is a Lie group means that the resulting set of points forms a smooth mathematical shape known as a “manifold”. Given this, you might imagine that this is a rather large amount of information. Here’s one visual way the BBC put it:

What came out was a matrix of linked numbers, which together describe the structure of E8. It contains more than 60 times as much data as the human genome sequence.

Each of the 205,263,363,600 entries on the matrix is far more complicated than a straightforward number; some are complex equations.

The team calculated that if all the numbers were written out in small type, they would cover an area the size of Manhattan.

Let’s not get carried away though. Having more data does not mean that you worked harder to get it. Mapping the human genome project involves a much harder task, but the analogy is still a good one, if not taken too far.

There’s a very good description of their work on the AIM site here, and before you ask what it is good for, let me jump right in with one of their quotes:

According to project leader Jeffrey Adams, “This is basic research which will have many implications, most of which we don’t understand yet. Just as the human genome does not instantly give you a new miracle drug, our results are a basic tool which people will use to advance research in other areas.” This could have unforeseen implications in mathematics and physics which do not appear for years.

e8 dynkin diagramThe picture above is a projection of the “root system” of the group, for those who know what that means. There are 240 roots, which are vectors in eight dimensions. The eight dimensions are the dimensions of the “Cartan subalgebra” – this is the “rank” of the group, giving part of its name. (Recall from your studies of SU(2) or SU(3) that you can locate the roots of the group as vectors in space formed by the Cartan subalgebra? This generalizes all that. Above right is the Dynkin diagram of E8, but I’ll leave it to you to find out more about that elsewhere, in view of time – I should be writing two lectures.)

You can find out a lot more by going to the AIM site I linked above (they’ve lots of links from there to further explanations – and I love the unexpected mention at the end of Fry’s electronics!), and also Mark Chu-Carroll’s post on Good Math, Bad Math, or for a more technical discussion (and more links) to John Baez’ post over at the n-category cafe, where you can learn (from the latter) for example that the picture above tells you about the densest lattice packing of balls in eight dimensions.


(Thanks Tameem.)

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21 Responses to E8

  1. Plato says:

    Is Coxeter related to these issues?

    Geometry is a branch of mathematics that deals with points, lines, angles, surfaces and solids. One of Coxeter’s major contributions to geometry was in the area of dimensional analogy, the process of stretching geometrical shapes into higher dimensions. He is also famous for “Coxeter groups,” the inversive distance between two disjoint circles (or spheres).

    See here.

    I find the image above very pleasing. I know you said they weren’t flowers from your garden but I wonder sometimes at the complexity of such flowers if given to handling in a computer sequence. I have my other reasons of course, but that is beside your point.

    Polytopes in regards to Coxeter as well were worth looking in too?

    The Man Who Saved Geometry

    Coxeter lectured about “the beauteous properties of triangles,” about circles and spheres, and about the Platonic solids: the tetrahedron, cube, octahedron, icosahedron, and dodecahedron. According to a recent cosmological hypothesis (and a similar theory put forth by Plato) the dodecahedron is a potential model for the shape of the universe-bound by 12 walls, each the shape of a pentagon.

    See here for a media report

    When you mentioned Dynkin diagrams I was trying to understand the way in which mapping is done in dimensional spaces. I first learn of the Dynkin diagrams from PP cook here

    Unfortunately I had not gone much further with Dynkin Diagrams until you mention your post today. Although, I was still looking at Sylvester surfaces and Cayley. Of course the genus figures.

    It just a different way of thinking in the non-euclidean geometries?

  2. Mary Cole says:

    What a hugely satisfying image. Why is symmetry so pleasing? I was in a rather old fashioned toy shop yesterday and discovered a ‘spirograph’ set. I was really pleased to see these are still being made, as I loved them as a child!

  3. Blake Stacey says:

    Clifford Johnson wrote, emphasis added:

    E8 is a 248 dimensional group, which means that if I were to try, using its most natural representation, to describe all its elements and the relations between them using points in a space, I would need a 248 dimensional space (recall that we live in just the three spatial dimensions), and the fact that it is a Lie group means that the resulting set of points forms a smooth mathematical shape known as a “manifold”.

    . . . and some people complain about ten dimensions. :-/

  4. donna says:

    Weird. I see that pattern all the time when I’m going to sleep at night….

  5. Thomas Larsson says:

    I would need a 248 dimensional space

    . . . and some people complain about ten dimensions. :-/

    More relevant here is that some people complain about 496 = 2*248 gauge bosons, since experiments only see 12.

  6. Plato says:

    More relevant here is that some people complain about 496 = 2*248 gauge bosons, since experiments only see 12.

    Maybe you could elaborate some more and give the experiment you are referring? If one was to see this diagram and it’s complexity as structure above, does this experiment not appeal to further possibilities ?

  7. Clifford says:

    He’s talking about the fact that if you were to use E8xE8 as a four dimensional gauge group you would have 496 gauge bosons, whereas the Standard Model of particle physics has gauge group SU(3)xSU(2)xU(1), which gives the 8 gluons of the strong interactions and the four gauge bosons associated the weak interactions and electromagnetism. 8+4=12.

    This brilliant observation has nothing whatsoever to do with the discussion of the structure and beauty of E8, of course.


  8. Antonio says:

    The pattern is strikingly similar to those religous circles that is very familiar here in Asia. I must hav last seen them in Hindu, with elephants around its border or a variation thereof in Buddhism with the great Boddhisatva seated in enlightened bliss in the center. I congratulate the West, in finally establishing their rational foundation which this side of humanity has timelessly accepted by faith.

  9. Arun M Thalapillil says:

    Interesting post. Thanks for pointing this out. Is it to be understood that a similar feat has already been achieved for E6 ?

  10. Plato says:

    G -> H -> … -> SU(3) x SU(2) x U(1) -> SU(3) x U(1)

    Here, each arrow represents a symmetry breaking phase transition where matter changes form and the groups – G, H, SU(3), etc. – represent the different types of matter, specifically the symmetries that the matter exhibits and they are associated with the different fundamental forces of nature

    Layman thinking.

    Okay I am now thinking in terms of the phase changes?

    While someone may of said they had trouble with ten dimensions, let alone 248, that number “248” is supposed to be held to the context of the “eight dimension?” Any of those points of the 248, “within” the context of the 8th dimension as well?

    While it is “complex in it’s arrangement” given the image above, it is held to an aspect of how we view the symmetry in that case?

    It is a quantum “directional view of the complexity of information” at that level?

  11. khanna salil says:

    greetings !!!!
    the ancient scriptures mention an 8 sided flat with a 0 face and a 9 face. each of these has 8 opposites out of which one is a polar & another is a reciprocal opposite. sometimes the polar & reciprocal opposites are the same. the final balance is achieved in its rotation of individuality. once that stops it collapses into anhillation or completeness depending on its velocity. its velocity may be much higher than the c-square of light. then suddenly all the illusionary dimensions created by the rotation disappear instantaneously back into its root. i believe modern maths will be able to reach that level of understanding in our lifetime.

  12. stefan says:

    the Maharishi mahesh yogi and quantum physicist John Hagelin ( now professor at mum.edu ) relates the 8 dimensions to the prakritis in vedic literature …. 8 fold divided nature of Prakriti
    the ninth is Paraprakriti, undivided or unified prakriti
    unmanifest prakriti the tenth is Purusha

    john hagelin claims when he spoke of the gossett polytrope 421 as a picture of the unified field that Transcendental meditation ( TM ) as taught by maharishi allows the mind to experience sequentially the unified field
    which the maharishi calls ATMA

  13. Elliot says:

    Anyone have an opinion (positive or negative) on this recent paper by Garrett Lisi using E8 as a unification framework?



  14. Blake Stacey says:

    And “Vedic physics” literature says that when the Vedas speak about wild and domesticated animals, they’re really talking about fermions and bosons. Oh, and annihilating the native peoples of India — the civilization which built cities like Mohenjo-Daro — really refers to the annihilation of antimatter.

    No joke. You can look it up in Meera Nanda’s Prophets Facing Backward, if you’re so inclined.

  15. Clifford says:

    Hi Elliot,

    I’ve nothing intelligent to say that has not already been said in (a small subset of) the comment stream of Bee’s post over on Backreaction. See Moshe and Aaron’s comments, for example. They seem to get at the heart of the matter, and quite clearly present the things that also puzzle me.

    It looks like a pretty pattern, – not dissimilar to other work in the literature on using E8 to play with the particle spectrum – but that’s it so far. You need more than pretty patterns to do physics, as I’m sure Garrett Lisi knows. Perhaps further work might uncover something concrete. I wish Garret Lisi all the best, and encourage him to keep trying (perhaps guided partly by addressing the questions people are asking) and see if he can beat things into shape, just as we’re all doing at various stages in our respective research programs.



  16. Elliot says:


    Thanks for your thoughts.


  17. Mary Cole says:

    So – on the subject of E8: is this all bunkum?


    Jason Cole (From Mary’s account!)

  18. Clifford says:

    Please read the comments directly above.


  19. Mary Cole says:

    I apologise on Jason’s behalf for not reading the above comments with sufficient care. He just enjoys the opportunity to use the word ‘bunkum’!

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