How to work with extra dimensions

I'm posting an e-mail that I sent to Lee at Scitalks which was added as a blogpost on the scitalks blog. Sounds a little circuitous, but having written it I wanted to add it here, with the little video I made.

Having mentioned that the video explanation of extra dimensions was in no way accurate, I wanted to add my two cents about extra dimensions and so sent the following message:

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In fact it’s pretty easy to understand higher dimensions from some simple visualisations. The video on Scitalks was fine, up to the fourth dimension, and then things started to get a bit cryptic.

One thing I should mention first is that trying to visualise objects in higher dimensions is good to feel comfortable with what you’re studying. However, sometimes if we try and visualise the things that we are calculating, we may stifle, or bias our calculations. In my opinion it is usually best to become familiar with the mathematics, the purest descriptions of our theory and try to ‘visualise’ in this language rather than trying to use common sense from what we see around us. A good example is quantum mechanics. If we try and visualise what is going on on very small length scales we will quickly tie ourselves in knots and not be able to progress as far as we can by exploring the mathematics of our system. Of course it’s important to be able to translate the mathematics back into what you will actually see in your experiment.

So, given that caveat I will explain how we can build ourselves a hypercube (or at least the frame of a hypercube), the higher dimensional generalisation of a cube.

The way we will do this is to start with less than three dimensions and see what rules we have to follow to go up in dimensions. We will see that we can extrapolate these rules to however many dimensions we want.

Start with a point, a zero dimensional object.

Take this point and turn it into two points. Now pull one of the points apart from the other one, say a distance L away, and join the two points with a piece of elastic. Now you have a line, a one dimensional object.

Now do a similar thing, but this time turn the elastic into two pieces of elastic (on top of each other - now you have four points given by the two ends of the pieces of elastic). Pull the pieces apart in the direction perpendicular to their length. While you’re pulling them apart keep the two ends joined by more elastic which will grow to length L. Now you have a square, something with four edges (pieces of elastic) and four points, or vertices. This object is two dimensional.

Now repeat the process. Take your square and replicate it with another square, on top of the first one. Join the vertices of the two squares with four pieces of elastic which will be stretched in the direction perpendicular to the face of the square. Pull the squares apart to a distance L. Now you have a cube. This object lives in three dimensions. It has six faces, 12 edges and eight vertices.

So, what rule have we developed? We have taken our previous object, replicated it, joined the vertices of the two objects together and pulled them apart in a direction perpendicular to the directions they lie, until the two copies are a distance L apart.

Let’s do that again.

Take your cube and replicate it. Join the vertices of the twin cubes to each other, again by elastic, and pull them apart to a distance L in the direction perpendicular to the direction they are living. There seems to be a problem though, in the last example, the square could live on a piece of paper and you could pull the two squares apart vertically to create the cube, we seem to have run out of directions. We need to pull the two cubes apart in the fourth dimension to a distance L.

Though we can’t really picture this realistically (at least I can’t) we can draw the projection of this onto 2-dimensions, just as easily as we can draw the projection of a cube onto a piece of paper.

When we pulled the one square from the other, we did this in the third dimension, say height, from your paper. So we don’t have another direction to go to pull the cube apart any more. This is where we have to imagine, as best we can, that we take the cube, split it into two and, joining the vertices of one cube with the other pull the two cubes apart to a distance L in the fourth direction, to create an object with 16 vertices and 40 edges. It’s almost as easy to draw the projection of this object onto a two dimensional piece of paper as it is to draw the projection of a three dimensional cube onto a piece of paper.

I've created a short animation of the above construction using Mathematica. We see the projection onto 2-dimensions. When I have time I'll let the objects rotate and slow it down a bit too.



In terms of mathematics, it’s even easier to go to higher dimensions. As an example, we might want to know the length of a line in two dimensions, going from some point (0,0) to (x,y). The length, as we know is the square root of x^2+y^2.

In three dimensions for a line going from (0,0,0) to (x,y,z) the length is the square root of x^2+y^2+z^2.

Well, let’s stop labelling directions as x,y,z etc and label them x_1, x_2, x_3,x_4, etc. (1,2,3,4 are simply labels). Now it’s easier to keep track of them.

Now a line in four dimensions stretching from (0,0,0,0) to (x_1,x_2,x_3,x_4) has a length of the square root of (x_1)^2+(x_2)^2+(x_3)^2+(x_4)^2.

Well, if you can work out the 2-dimensional example, I would suggest that it’s pretty easy to calculate the 4, 10, or any dimensional example. Imagining it isn’t easy, but as long as you have a mathematical handle on the objects that are living in your higher dimensional theory, you should be doing fine.

I would suggest having a read of Flatland, a romance in many directions - not for it’s political correctness, but for an idea of how things would be if we didn’t live in 3 dimensions. Chapter 16 is particularly relevant.

More on the KITPC

The two day opening ceremony for the KITPC is coming up in a little over a week. After the official opening at the Great Hall of the People (with talks from Fred Kavli amongst others) there will be two days of lectures at the KITPC from many highly eminent scientists. The morning of the 26th will see a public lecture given by Michael Freedman, winner of the Fields medal for his part in the solution of the Poincare conjecture. Title:

'How Topology will save Moore's law' - Some thoughts on the human condition expressed through the history of the concept of "number." This concept carries us finally into the interface of topology, quantum mechanics, and computer science.

After this there will be lectures on subjects from quantum information, the search for habitable planets, laser technology, dark matter, the origin of the universe (from Henry Tye) to the future of physics, both worldwide and in China.

The closing speech will be by David Gross on Sunday the 27th. I'll report on what I get a chance to go to. I'll try and get some photos of the building too to give those who are thinking of coming to the future meetings a bit more of an idea about the place.

Turing universality prize from Wolfram

Not quite the big bucks of the Millenium prizes, but a nice $25,000 for proving that an exceedingly simple Turing machine (an abstract symbol manipulations device) discovered by Stephan Wolfram is in fact a Universal Turing machine (A Turing machine which can simulate any other Turing machine, and therefore any computer program, which generally have much more complex rules).

The contest is to show that a particular two state, three colour Turing machine is universal.

This comes from the newly created Wolfram blog. See here for details about the competition, more about Turing machines, and more about this particular, incredibly simple algorithm.

[Thanks to Blake Stacey for pointing out the typo in the above. Indeed the Universality of the 2,5 Turing machine had been proved already, by Matthew Cook]

With seconds to spare

A few random links...

• First to a site which I mentioned, perhaps a year ago, but is always worthy of a plug. There's big news in the Beijing music scene this month as Sonic Youth are due out here. The tickets are very expensive but I would love to go and see them, so will attempt to pick some up tomorrow. Pandora is one of the best resources I've found to discover great new music. By selecting a band or song to begin with, this online radio player choses songs based on the style of music from your original choice. By telling the station whether you like a particular track or not it will evolve to give you more and more of the type of music you really like. Listening to a station today seeded with Sonic Youth I've found a host of new bands I'd never previously heard of.

• Discovery sites (Digg, Pandora, Stumbleupon, del.ici.ous Plime, and presumably many more) are becoming big business and allow the user to find web content based on what other people with similar tastes have liked in the past, rather than based on a specific web search. It's a powerful tool but also an easy way to fritter away time jumping from one interesting site to another, not taking anything of any real content in. It's definitely one of the many new branches of web technology which will be interesting to keep an eye on.

• On a side note I wonder if anyone has tried to develop a community del.ici.ous site where papers from the ArXiv can be tagged for searching more appropriately than the current search functions available online. At the very least I would like my pdf files to be tagged with the basic information about each paper and not have to type it all into an unweildy title name.

• While we're on the topic of the ArXiv I'll point to both the papers by Jacob Bourjaily (here and here )and the subsequent positive analysis by Lubos Motl. The papers are geometric engineering of F-theory (a 12 dimensional theory where the complex dilaton-axion of IIB string theory take on a geometric interpretation as the complex structure of a two-torus). The geometry of the compactified dimensions determines the symmetries and matter content of the low energy theory and so in these papers the standard model gauge groups and matter content are constructed via this process of geometric engineering, 'unfolding' from larger gauge groups.
• Jacob shows that not only is a three flavour standard model a generic feature of a particular (and perhaps natural, from the string theory perspective) orbifold singularity but that smaller gauge groups are more natural from this construction than grand unified theories with large gauge groups. He points out however that the fact that the standard model symmetries aren't completely 'unfolded' is not obvious from this model.
• Lubos gives a more thorough explanation although the general idea of the papers isn't too difficult to understand, even without knowing the background in great detail.

• There was lots of news a couple of weeks ago about a huge calculation of the structure of an immense but important symmetry group. I'll just leave this as a series of links, if you want to read the maths there's some fascinating discussion at John Baez's site and if you just want to look at the pictures, well, they're pretty too. The American Institute of Mathematics has some details here, too.

• The subsequent edition of This Week's Finds also contains a wealth of interesting material on solar flares plus more information about groupoids.

Apart from that I've been rather too busy calculating to read any new science over the last week or so.

• From the BBC comes the news that the new wing of the Tate Modern has been given the go ahead. Without a doubt it's an exciting piece of art but it has a lot of critics who are not impressed. I like the look from the artist's impressions and the claim that London is becoming riddled with different styles of architecture is a century or so too late. The architects, Herzog and de Meuron, are also the designers behind the Beijing Olympic stadium - The Bird's Nest. I go past this whenever I go to the airport and would love to get there some time to take some photographs myself.

• My parents recently came back from a trip to Paris where they took this rather fun picture:

This is an original image.

• From Kevin at the Weifang Radish came some useful hints for internet use in China, definitely good if you want to keep up with blogs and news but don't want the lag of proxies.
  

Right, the weekend beckons and I have another few hours to work before I start running around the city again. Happy Easter to all...