They have gone to plaid!
Posted on 03/19/2007 8:34:38 PM PDT by LibWhacker
Roger Highfield describes a heroic mathematical enterprise that could lay bare the fundamentals of the cosmos
Mathematicians have successfully scaled their equivalent of Mount Everest. Today they unveil the answer to a problem that, if written out in tiny print, would cover an area the size of Manhattan.
At the most basic level, the calculation is an arcane investigation of symmetry – in this case of an object that is 57 dimensional, rather than the usual three dimensional ones that we are familiar with. Although this object was first discovered in the 19th century. there is evidence that it could contain the structure of the cosmos.
Mathematicians are known for their solitary style of working, but the combined assault on what is described as "one of the largest and most complicated structures in mathematics" required the effort of 18 mathematicians from America and Europe for an intensive four-year collaboration.
The feat may baffle most people but could have unforeseen implications in mathematics and physics, which won’t be evident for years to come, said the American Institute of Mathematics.
"The group of symmetries of this strange geometry called E8 is one of the most intriguing structures that Nature has left for the mathematician to play with," commened Prof Marcus du Sautoy of Oxford University, currently in Auckland. "Most of the time mathematical objects fit into nice patterns that we can order and classify. But this one just sits there like a huge Everest." advertisement
What makes this group of symmetries so exciting is that Nature also seems to have embedded it at the heart of many bits of physics. One interpretation of why we have such a quirky list of fundamental particles is because they all result from different facets of the strange symmetries of E8. I find it rather extraordinary that of all the symmetries that mathematician’s have discovered, it is this exotic exceptional object that Nature has used to build the fabric of the universe. The symmetries are so intricate and complex that today’s announcement of the complete mapping of E8 is a significant moment in our exploration of symmetry."
For the feat, the team used a mix of theoretical mathematics and intricate computer programming to successfully map E8, (pronounced "E eight") which is an example of a Lie (pronounced "Lee") group. Lie groups were invented by the 19th century Norwegian mathematician Sophus Lie to study symmetry.
Underlying any symmetrical object, such as a sphere, is a Lie group. Balls, cylinders or cones are familiar examples of symmetric three-dimensional objects. Today’s feat rests on the drive by mathematicians to study symmetries in higher dimensions. E8 is the symmetries of a geometric object that is 57-dimensional. E8 itself is 248-dimensional.
"E8 was discovered over a century ago, in 1887, and until now, no one thought the structure could ever be understood," said Prof Jeffrey Adams, Project Leader, at the University of Maryland. "This groundbreaking achievement is significant both as an advance in basic knowledge, as well as a major advance in the use of large scale computing to solve complicated mathematical problems."
"This is an exciting breakthrough," said Prof Peter Sarnak at Princeton University. "Understanding and classifying the representations of E8 and Lie groups has been critical to understanding phenomena in many different areas of mathematics and science including algebra, geometry, number theory, physics and chemistry. This project will be invaluable for future mathematicians and scientists."
The ways that E8 manifests itself as a symmetry group are called representations. The goal is to describe all the possible representations of E8. These representations are extremely complicated, but mathematicians describe them in terms of basic building blocks. The new result is a complete list of these building blocks for the representations of E8, and a precise description of the relations between them, all encoded in a matrix, or grid, with 453,060 rows and columns. There are 205,263,363,600 entries in all, each a mathematical expression called a polynomial. If each entry was written in a one inch square, then the entire matrix would measure more than seven miles on each side.
The result of the E8 calculation, which contains all the information about E8 and its representations, is 60 gigabytes in size. This is enough to store 45 days of continuous music in MP3-format. If written out on paper, the answer would cover an area the size of Manhattan. The computation required sophisticated new mathematical techniques and computing power not available even a few years ago.
"This is an impressive achievement," said Hermann Nicolai, Director of the Albert Einstein Institute in Potsdam, Germany. "While mathematicians have known for a long time about the beauty and the uniqueness of E8, we physicists have come to appreciate its exceptional role only more recently - yet, in our attempts to unify gravity with the other fundamental forces into a consistent theory of quantum gravity, we now encounter it at almost every corner," he said, referring to efforts to combine the theory of the very big (general relativity) with the very small (quantum mechanics). "Thus, understanding the inner workings of E8 is not only a great advance for pure mathematics, but may also help physicists in their quest for a unified theory."
So I'm to believe in a 57-dimensional model, and creationism is a fairy tale?
No thanks - I'll stick with Genesis.
what time is McGuyver on ?
Great, just great. I went to public school. So that means I have 57 dimensions I am clueless about.
Something tells me we're a bit short on being ready for that next level, dude...
They have gone to plaid!
Where does Duncan Hunter stand on this? /s
The foolishness of God is greater than the wisdom of man.
E-8 is so gay.
E-9 is where it's at.
"Belief" doesn't apply to mathematics and physics -- those sciences don't rely on faith. Consider:
God's Love is infinite and beyond our comprehension, yet He granted us human Love as a model, to help us understand. A child can love, and in that act becomes one with God.
God's Universe includes complexities (perhaps 57-D ones) that are beyond our comprehension, yet He granted us the mathematics to describe the Universe with models, to help us understand. Mathematicians and physicists can appreciate the beauty of those models, and in doing so, appreciate the Creation itself (whether they admit it out loud or not).
I see no contradiction in believing that God created a Universe of infinite majesty and complexity, and yet inspired the author of Genesis to phrase it in a simpler way that would help all people grasp the Creation.
Nothing in this fancy 57-D mathematical model requires that in order to appreciate it, you have to give up your religious beliefs. They are not in opposition.
That said, of course you're welcome to reject the mathematics -- it's a free country.
Not at all what is invited in this calculation. Don't be obtuse. Seeing some of the mathematical facets of God's plan is not blasphemy.
Now, now... If you were paying attention in school, you should have a pretty good handle on four of them. So actually, you're only clueless about 53 of them! ;-)
Fair criticism. What is so intriguing about E8 is that it pops out of physicists equations in a lot of different areas. There's likely something fundamental about it. But you're right that the definition of how it is going to change things is a little vague. It seems to me one of the biggest barriers is the complexity of the description required. Sampling a single configuratin of the E8 out of 60 gig of numbers and formulae is going to be very comptationally intensive. Will probably limit it's usefulness until computers get faster.
But kudos to the folks who did this. It's a massive accomplishment that may be very important.
Personally, I think the fabric of the universe is based on Palindromes. How can it be that all numbers can result in a palindrome? It's utterly baffling to me. If we understood why palindromes exist, I think we would find that answer to be the key to much of the universes puzzles.
So I'm to believe in a 57-dimensional model"
For the feat, the team used a mix of theoretical mathematics"
This is imaginative science.
I guess I missed the part in the article where they honored God. Sorry.
"Belief" doesn't apply to mathematics and physics -- those sciences don't rely on faith."
"the team used a mix of theoretical mathematics"
That relies on faith.
Cheers!
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