Is this the fabric of the universe?

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."

I have been reading a lot of books and web sites on string theory. It all seems very interesting, all these extra dimension and so fourth. I was just curious, it is supposed that these extra dimensions could be real, I have yet to read how the first three dimensions that we take for granted in this universe are real physical things. I know that the term dimension is used in plotting locations and trajectories of objects in space on paper, but are they real physical things that exist in the real universe?

Thank you.

Steve

> I have yet to read how the first three dimensions that we take for granted in this universe are real physical things. I know that the term dimension is used in plotting locations and trajectories of objects in space on paper, but are they real physical things that exist in the real universe?

Here's how I see it.

Consider the concept of "distance". My house is a mile down the road from my friend's house. Does that "mile" have physical reality? No, it does not. The road between the houses does, of course. But the "distance" is merely a mental model I impose on the two houses, to describe where they sit in relation to one another. I could just as well use "a half-hour's walk" to describe the relative positions; that too is just a conceptual model (the walking is real, but using it to describe relative position is an abstraction).

I view our three physical "dimensions" in a similar fashion. Objective reality is hard for us to grasp unless we impose conceptual models on it that help simplify and structure it. Regardless of whether we use cartesian (X,Y,Z) coordinates, or cylindrical or spherical or anything else, we are just imposing a mental model onto the objective reality to help with our comprehension.

So, in my opinion, the answer to your question is "No", dimensions do not have physical reality. They are a tool for helping to understand, describe, and work with, reality. Those who claim that dimensions have physical reality are, in my opinion, confusing their mathematical models with the Universe they are trying to model.

Going one level higher, again in my opinion, God is the only One who is capable of comprehending the Universe without resorting to models. But that's my personal view of God; I don't expect anyone else to necessarily buy into it. And in any case, my view of God is necessarily just a model, too, since I cannot directly comprehend God's infinite nature, at least not today...

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