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Author Topic: E8 and other Lie groups  (Read 1666 times)
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matsoljare
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« on: February 25, 2009, 06:08:57 PM »

<a href="http://www.youtube.com/v/-xHw9zcCvRQ&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/-xHw9zcCvRQ&rel=1&fs=1&hd=1</a>

These algebraic constructs have been known for over a century, and just recently have gotten more attention due to their importance in physicist Garret Lisi's attempt at a unified theory of particle interactions, which is still incomplete and not yet proven or disproven.

They seem somewhat similar to fractals. Can anyone explain in simpler terms how exactly they are constructed, and what the original "use" for them were?



http://en.wikipedia.org/wiki/E8_(mathematics)
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cKleinhuis
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« Reply #1 on: February 25, 2009, 09:03:48 PM »

quote frome the wikipedia article:


http://en.wikipedia.org/wiki/E8_(mathematics)
Basic description
"E8 has rank 8 and dimension 248 (as a manifold). The vectors of the root system are in eight dimensions and are specified later in this article. The Weyl group of E8, which acts as a symmetry group of the maximal torus by means of the conjugation operation from the whole group, is of order 696729600."

the only i can understand in this explanation is that it is a damn big mathematical group, dimension 246 huihuihui ( am i right when i say the base for this algebraic body is , this is a lot ..

i can adopt it to linear algebra, where dimension is defined as the number of lineary independent values of the (vector) group, e.g. x,y,z as 3d coordinates forms an algebraic
group (...even a body ) and you have 3 lineary independent base vektors ( (1,0,0) ( 0,1,0) and ( 0,0,1) with those three vectors, you can create every other member
of vectors belonging to that group, e.g.
1*(1,0,0)+2*(0,0,1)+3*(0,0,1) wouild form the vector ( 1,2,3 )

but i go bancrupt when thinking about 248 of those cheesy

no, i can not help you out on this ... can anyone else ?
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divide and conquer - iterate and rule - chaos is No random!
Steve
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« Reply #2 on: July 19, 2010, 04:46:52 PM »

I think the picture you are showing is probably a 2-D projection of the 8-D Cayley graph of the E8 group. The video seems to be rotations of the Cayley graph about different axes of the Cayley space, but I'm only guessing. Wikipedia has detailed info on Cayley graphs.

The following is a Cayley graph of a group that can be represented in 2 dimensions. It comes from http://en.wikipedia.org/wiki/Group_theory. That sure looks like the beginnings of a fractal.  Sort of like the one given by Thunderwave at http://www.fractalforums.com/meet-and-greet/hello-a-simple-fractal-first-post/


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Thunderwave
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« Reply #3 on: July 20, 2010, 01:54:30 AM »

Wow, so intricate!  Thanks for linking to my hand drawn fractal, Steve.  I find this very fascinating but confusing.  I would love to research it more when I have time.
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Steve
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« Reply #4 on: July 20, 2010, 06:00:36 PM »

The interesting thing about the above graph is that it is a true fractal only if you define it as a recursion. No doubt the group that generated this has a specific set of parent groups that can be indexed by their order. The fractal itself would be described as "the limit of the Cayley graph as the order of this class of groups goes to infinity". This is an obscure way of defining it, compared to the words "bisect each segment etc...."
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Calcyman
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« Reply #5 on: July 21, 2010, 09:23:44 PM »

E8 can refer to a number of related concepts. As well as being a Lie algebra of dimension 248, it has a more comprehensible geometric aspect:

The E8 root system comprises 240 vectors in eight dimensions, in a beautifully symmetric arrangement. These can be considered to be 240 octonions, or 240 vertices of a 8-polytope.

More impressively, these 240 vectors span a lattice in eight dimensions, imaginatively called the E8 lattice. This is the optimal sphere packing in eight dimensions, analogous to the face-centred cubic lattice in three dimensions.

Three orthogonal copies of E8 lattice can be used to generate an even more awesome lattice, the 24-dimensional Leech lattice. It appears to be the optimal solution to a number of problems in 24 dimensions, including the sphere packing problem, covering problem, kissing number problem and quantizing problem. Not to mention its connection with 20 of the 26 Sporadic Groups.
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