Help me relate to E8
November 16, 2007 7:50 PM Subscribe
E8: what's in it for me? (inspired by this post)
Do we ever encounter E8 in daily life? I'd like to know more about E8 and I tried reading some of the links suggested in that great FPP and discussion, but unfortunately most of it went way over my head. I'm wondering if E8 is the new fractals: it seems like fractals only came to popular attention fairly recently, but now they're everywhere, e.g. in biology (self-similar structures like fern leaves), engineering (noise in electronics), and software (modeling landscapes). Does E8 show up in or lurk behind anything mundane?
Do we ever encounter E8 in daily life? I'd like to know more about E8 and I tried reading some of the links suggested in that great FPP and discussion, but unfortunately most of it went way over my head. I'm wondering if E8 is the new fractals: it seems like fractals only came to popular attention fairly recently, but now they're everywhere, e.g. in biology (self-similar structures like fern leaves), engineering (noise in electronics), and software (modeling landscapes). Does E8 show up in or lurk behind anything mundane?
E8 is a massive, complex structure and I am not sure you will find it recognizably in any human-scale natural structures. However it is not so baffllingly abstruse that you can't get the basic idea by reading through some of the better descriptions. You don't need to understand the detail, just the basic concept.
Symmetry groups are fairly abstract, but if you can understand the symmetry group for a triangle (rotations through 0, 120, 240 degreees) it is not hard to imagine the symmetry groups for constructs in higher dimensions (eg a cube, a tetrahedron, a hypercube, and so on).
The most compelling link to reality is that E8 does seem to have some kind of isomorphism to quantum electrodynamics and gravitation. I looked through the E8 paper and I'm absolutely not qualified to make a judgement on the validity of the model, but basically it seems to provide a structure which links QED and gravitation in a fairly simple geometric whole.
If true, it's an astounding breakthrough. Responses I've seen have ranged from jubilation through cautious optimism to outright rejection. Myself, I figure that enough of it seems to be true that that the boundaries will probably be mopped up over the next few years.
It is a much more elegant concept than string theory, just as Copernican astronomy was a more elegant construct than epicycles, even though both systems could potentially describe reality to arbitrary degrees of accuracy.
posted by unSane at 8:54 PM on November 16, 2007
Symmetry groups are fairly abstract, but if you can understand the symmetry group for a triangle (rotations through 0, 120, 240 degreees) it is not hard to imagine the symmetry groups for constructs in higher dimensions (eg a cube, a tetrahedron, a hypercube, and so on).
The most compelling link to reality is that E8 does seem to have some kind of isomorphism to quantum electrodynamics and gravitation. I looked through the E8 paper and I'm absolutely not qualified to make a judgement on the validity of the model, but basically it seems to provide a structure which links QED and gravitation in a fairly simple geometric whole.
If true, it's an astounding breakthrough. Responses I've seen have ranged from jubilation through cautious optimism to outright rejection. Myself, I figure that enough of it seems to be true that that the boundaries will probably be mopped up over the next few years.
It is a much more elegant concept than string theory, just as Copernican astronomy was a more elegant construct than epicycles, even though both systems could potentially describe reality to arbitrary degrees of accuracy.
posted by unSane at 8:54 PM on November 16, 2007
Best answer: Quietgal, the shortest answer, to the best of my experience, is that, although E8 is a very "fundamental" object, it's not a ubiquitous phenomenon among familiar objects the way self-similarity (or fractal geometry) is. That in mind, I'll try to sketch out a little of why it can still be considered an important, elementary object to study. I don't know what your background is, so pardon me if I tell some things you already know. Maybe the most interesting link for answering your question is at the very end.
E8 is a Lie group, and fully answering the question "What is a Lie group?" is too deep to attempt here. It's possible to give a short answer that's not really helpful for understanding unless you already have a lot of background. But for starters, a Lie group is a group, and one of the nicest elementary, expository articles on the concept of a group is The Power of Groups at Plus magazine. Again, you can take an abstract, axiomatic approach to defining a group; the article avoids that tack and concentrates on thinking of a group as a sort of collection of symmetries of a geometric object - which is a legitimate view, although not always the most suitable one to choose in any particular problem. There's a partner article, Symmetry Rules which should give you some ideas about physicists' feelings on symmetries being at the heart of natural order, in some sense or other.
Mathematicians like to classify things - and for groups, that means they'd like to have, in some systematic way, a descriptive catalog of "every group there is." That's too broad a problem to have any reasonable answer. Less ambitiously, but still not exactly a Saturday morning pencil-and-paper exercise, there's the possibility of classifying all finite groups. And one approach to that is looking at ways of factoring groups into smaller, less complicated groups - very, very analogous to the way you can factor an integer (76466) into smaller integers (346x221). Now, with integers, as you probably did all the way back in grade school, you can keep on factoring the factors (2x13x17x173). The ultimate building blocks for integers, from this viewpoint, are the prime numbers, which can't be factored any further.
Finite groups can also be factored, and broken down to bits that don't factor any more, and the ultimate unfactorable bits are called simple groups. So, now, instead of a vague problem of "classify all finite groups", you have a two-stage program: (1) classify all finite simple groups, and (2), which I'll only be vague about, understand how factors get put together to make products - it's a bit more complicated than multiplication of integers; there are more ways to do it. Anyway, (2) is not so awfully hard.
(1), on the other hand, is both awful and hard - but it has been done. You can get a sense of it in Wikipedia's List of Finite Simple Groups article - never mind the detail; look at the structure of the answer in the box up at the top. Every finite simple group either belongs to one of the inifinite families of finite simple groups enumerated under point 1 (each family has infinitely many groups in it, but they're all structurally similar, they just come in smaller and larger sizes), or... it's a black sheep. It's one of the "exceptional" or "sporadic" groups listed under point 2.
Now, it's important to get that finite simple groups are a platonic, really real thing - they are what they are, we don't invent them. The naming or cataloging system, however, is artificial - not without rhyme or reason, but no more intrinsically meaningful than the relationship between Dewey Decimal numbers and subjects of books. So it's possible there's another system, or point of view for classifying these, that doesn't make such an awkward picture, with these special, sporadic "exceptional" groups that have to be looked at each as its own little world, rather than fitting nicely into a family. In a way, I kind of doubt that such a better perspective exists, but that's just gut feeling. From the current point of view, of course, you can see that these black sheep might naturally attract interest and curiosity; you can construct them rather abstractly without any thought of geometry or symmetry, but then again they have to be symmetries of something... can we figure out what? Is it something interesting? Well, generally yes; not too much detail here, but for example the Mathieu groups are symmetries of quite interesting things, and if you want one further bit of reading that really captures the interest and thrill of discovering connections to these groups, the book Sphere Packings, Lattices, and Groups is a good and accessible read that works at several levels.
The E8 that's currently in the spotlight is the Lie group E8, however. (There's a little amibguity there. E8 could also refer to a Lie algebra but to the extent we care about it here, classifying Lie algebras and classifying Lie groups is essentially the same problem.) Lie groups are not finite groups; for the simplest interesting example, think of the rotational symmetries of a circle. You can turn it through any angle, that is, any number between 0 and 2π, infinitely many choices, as opposed to the symmetries of a square or triangle, which have finite symmetry groups. As with finite groups, an program to classify Lie groups quickly becomes a program to classify simple Lie groups that can't be broken down into simpler building blocks. The result (Wikipedia: imple Lie Groups) is really much nicer, and far less difficult than the finite group picture. Look at the box. You see just four infinite families, and 5 exceptional or "sporadic" groups, the largest of which is E8. So, again, it should be clear that there's a certain intrinsic curiosity about this object just due to its unique position in Lie theory.
Unlike classification of finite simple groups, which depends on the incomprehensibly vast Feit-Thompson theorem, the tidy Lie group result has correspondingly tidy, pretty, geometric ideas underlying it. One of these is the idea of a Dynkin diagram. Now, to get back to the point about where these things actually manifest in nature: the best recommendation I have is this, from John Baez on, in his words, "how Dynkin diagrams infest so much of mathematics." It should give you a certain sense of the ubiquity, or underlying fundamental nature of these objects, and if you follow the back-links to earlier articles you can fill in quite a lot of the picture.
You'll notice Baez describes E8 as related to the "octooctonionic projective plane." A final recommendation for reading - back at a quite accessible level again - is the Plus magazine articles on quaternions and octonions which again should do a bit suggest why they can be regarded as a fundamental and pervasive presence in math - even if not terribly visible in familiar objects the way your example of fractal geometry is. Any hope that E8 represents the ultimate organizing agent for particle physics, though - I think at this stage that's just hope and maybe a little mysticism engendered by its unique position as the "largest, weirdest thing that we still have to regard as simple because it doesn't break down any further."
Woof! That's a lot, and not all directly relevant, but I hope it helps - or at least stimulates some more curiosity.
posted by Wolfdog at 7:33 AM on November 17, 2007 [13 favorites]
E8 is a Lie group, and fully answering the question "What is a Lie group?" is too deep to attempt here. It's possible to give a short answer that's not really helpful for understanding unless you already have a lot of background. But for starters, a Lie group is a group, and one of the nicest elementary, expository articles on the concept of a group is The Power of Groups at Plus magazine. Again, you can take an abstract, axiomatic approach to defining a group; the article avoids that tack and concentrates on thinking of a group as a sort of collection of symmetries of a geometric object - which is a legitimate view, although not always the most suitable one to choose in any particular problem. There's a partner article, Symmetry Rules which should give you some ideas about physicists' feelings on symmetries being at the heart of natural order, in some sense or other.
Mathematicians like to classify things - and for groups, that means they'd like to have, in some systematic way, a descriptive catalog of "every group there is." That's too broad a problem to have any reasonable answer. Less ambitiously, but still not exactly a Saturday morning pencil-and-paper exercise, there's the possibility of classifying all finite groups. And one approach to that is looking at ways of factoring groups into smaller, less complicated groups - very, very analogous to the way you can factor an integer (76466) into smaller integers (346x221). Now, with integers, as you probably did all the way back in grade school, you can keep on factoring the factors (2x13x17x173). The ultimate building blocks for integers, from this viewpoint, are the prime numbers, which can't be factored any further.
Finite groups can also be factored, and broken down to bits that don't factor any more, and the ultimate unfactorable bits are called simple groups. So, now, instead of a vague problem of "classify all finite groups", you have a two-stage program: (1) classify all finite simple groups, and (2), which I'll only be vague about, understand how factors get put together to make products - it's a bit more complicated than multiplication of integers; there are more ways to do it. Anyway, (2) is not so awfully hard.
(1), on the other hand, is both awful and hard - but it has been done. You can get a sense of it in Wikipedia's List of Finite Simple Groups article - never mind the detail; look at the structure of the answer in the box up at the top. Every finite simple group either belongs to one of the inifinite families of finite simple groups enumerated under point 1 (each family has infinitely many groups in it, but they're all structurally similar, they just come in smaller and larger sizes), or... it's a black sheep. It's one of the "exceptional" or "sporadic" groups listed under point 2.
Now, it's important to get that finite simple groups are a platonic, really real thing - they are what they are, we don't invent them. The naming or cataloging system, however, is artificial - not without rhyme or reason, but no more intrinsically meaningful than the relationship between Dewey Decimal numbers and subjects of books. So it's possible there's another system, or point of view for classifying these, that doesn't make such an awkward picture, with these special, sporadic "exceptional" groups that have to be looked at each as its own little world, rather than fitting nicely into a family. In a way, I kind of doubt that such a better perspective exists, but that's just gut feeling. From the current point of view, of course, you can see that these black sheep might naturally attract interest and curiosity; you can construct them rather abstractly without any thought of geometry or symmetry, but then again they have to be symmetries of something... can we figure out what? Is it something interesting? Well, generally yes; not too much detail here, but for example the Mathieu groups are symmetries of quite interesting things, and if you want one further bit of reading that really captures the interest and thrill of discovering connections to these groups, the book Sphere Packings, Lattices, and Groups is a good and accessible read that works at several levels.
The E8 that's currently in the spotlight is the Lie group E8, however. (There's a little amibguity there. E8 could also refer to a Lie algebra but to the extent we care about it here, classifying Lie algebras and classifying Lie groups is essentially the same problem.) Lie groups are not finite groups; for the simplest interesting example, think of the rotational symmetries of a circle. You can turn it through any angle, that is, any number between 0 and 2π, infinitely many choices, as opposed to the symmetries of a square or triangle, which have finite symmetry groups. As with finite groups, an program to classify Lie groups quickly becomes a program to classify simple Lie groups that can't be broken down into simpler building blocks. The result (Wikipedia: imple Lie Groups) is really much nicer, and far less difficult than the finite group picture. Look at the box. You see just four infinite families, and 5 exceptional or "sporadic" groups, the largest of which is E8. So, again, it should be clear that there's a certain intrinsic curiosity about this object just due to its unique position in Lie theory.
Unlike classification of finite simple groups, which depends on the incomprehensibly vast Feit-Thompson theorem, the tidy Lie group result has correspondingly tidy, pretty, geometric ideas underlying it. One of these is the idea of a Dynkin diagram. Now, to get back to the point about where these things actually manifest in nature: the best recommendation I have is this, from John Baez on, in his words, "how Dynkin diagrams infest so much of mathematics." It should give you a certain sense of the ubiquity, or underlying fundamental nature of these objects, and if you follow the back-links to earlier articles you can fill in quite a lot of the picture.
You'll notice Baez describes E8 as related to the "octooctonionic projective plane." A final recommendation for reading - back at a quite accessible level again - is the Plus magazine articles on quaternions and octonions which again should do a bit suggest why they can be regarded as a fundamental and pervasive presence in math - even if not terribly visible in familiar objects the way your example of fractal geometry is. Any hope that E8 represents the ultimate organizing agent for particle physics, though - I think at this stage that's just hope and maybe a little mysticism engendered by its unique position as the "largest, weirdest thing that we still have to regard as simple because it doesn't break down any further."
Woof! That's a lot, and not all directly relevant, but I hope it helps - or at least stimulates some more curiosity.
posted by Wolfdog at 7:33 AM on November 17, 2007 [13 favorites]
Since you asked whether E8 is the "new fractals", it might be worth emphasizing that both mathematicians and physicists have been interested in it for a long time. It plays an important role in string theory, for example. The media has picked up on two stories both related to E8 recently, but this isn't indicative of a trend in the scientific community.
(In particular, contrary to what I've seen written in a few places, the two stories are unrelated except inasmuch as they both involve E8 -- Lisi's proposal has nothing directly to do with the computation of Kazhdan-Lusztig polynomials.)
posted by em at 11:12 AM on November 17, 2007
(In particular, contrary to what I've seen written in a few places, the two stories are unrelated except inasmuch as they both involve E8 -- Lisi's proposal has nothing directly to do with the computation of Kazhdan-Lusztig polynomials.)
posted by em at 11:12 AM on November 17, 2007
Trying to answer this question myself, the most concrete occurrence I could find of E8 was that it's related to the symmetry group of the icosahedron. It isn't the same as the symmetry group of the icosahedron, but it's the "root system" of that group. Attempting to understand what a root system of a group is has been beyond me so far.
posted by hattifattener at 12:56 PM on November 17, 2007
posted by hattifattener at 12:56 PM on November 17, 2007
Response by poster: Thanks, everyone, for the answers so far. Wolfdog, you've given me plenty of reading for a cold winter night - I just hope I can understand some of it. I have a PhD and I earn my living as a scientist, but sometimes MeFi makes me feel really dumb.
And unSane, you're a screenwriter according to your profile - what are you doing in this thread? Are you some kind of deeply closeted math geek? People like you make me feel really dumb.
Anyway, keep the replies coming! I'm a glutton for feeling really dumb.
posted by Quietgal at 1:10 PM on November 17, 2007
And unSane, you're a screenwriter according to your profile - what are you doing in this thread? Are you some kind of deeply closeted math geek? People like you make me feel really dumb.
Anyway, keep the replies coming! I'm a glutton for feeling really dumb.
posted by Quietgal at 1:10 PM on November 17, 2007
This thread is closed to new comments.
Be sure to check out this previous E8 thread.
posted by ericb at 8:53 PM on November 16, 2007