E8

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.

-cvj

(Thanks Tameem.)

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