Visualizing 4+ dimensions…

Jon Shock of Jon’s Travel Adventures and our guest blogger for the month has kindly written a clear and elegant post on how to visualize more than 3 dimensions. He’s even created a movie for it!

Enjoy -L

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You mentioned emergence as an alternative for string theory. In fact it turns out that emergence is one of the most exciting and active areas of string theory itself. This comes from the study of black holes and, more recently a discovery of a holographic correspondence in string theory where it was found that there were dual descriptions of a set of physical theories.

The two descriptions of each theory are extremely different. One of them is of a four dimensional gauge theory.A gauge theory describes the interactions of the force carrying particles – the photon (mediating the electromagnetic force), the gluons (mediating the strong force), the weak bosons (mediating the weak force) and the graviton (mediating gravity). This particular gauge theory is somewhat like the strong force, which we know to be described by QCD (quantum chromodynamics). The dual description of this gauge theory is a ten dimensional theory of gravity, which is a string theory in a specific space-time.

The idea is that in a limit of the gauge theory, higher dimensions may be emergent from the dynamics of the lower dimensional theory. That is, in a limit of the gauge theory the appropriate description is one of a higher dimensional theory of gravity.

This emergence is termed holography because we find that a high dimensional theory can be described by a lower dimensional theory – just as a 2d hologram encodes a 3d image.

OK, so that’s the about the quickest description of emergence in string theory that I can come up with. I’d be happy to explain it in more detail some time. We first learnt about holography along these lines from studying black holes. There is a superb video by Bousso on this subject which can be found here: http://www.uctv.tv/search-details.asp?showID=11140

OK, onto trying to visualise extra dimensions.

In fact it’s pretty easy to understand higher dimensions from some simple visualisations. The video you posted 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 desciptions 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 just animated this in Mathematica but can’t seem to make an avi from it. If you have Mathematica I can send you the file.

[Jon has put this up on YouTube here: http://youtube.com/watch?v=J5QlUdNMHWs -ed]

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.http://www.geom.uiuc.edu/~banchoff/Flatland/ Chapter 16 is particularly relevant.

Anyway, in terms of lectures for the next level, I would suggest the lectures by Clifford Johnson which you have already put up on the site. He’s an excellent speaker and his introductory lectures are very good. Also the lectures of Barton Zwiebach are very good.

There’s another series of lectures by Lerche here: http://indico.cern.ch/conferenceDisplay.py?confId=a032483

I have only been able to read the transparencies, which look broad ranging and reasonably detailed. In the cafe I’m in at the moment I can’t download the video. I would imagine it’s good, but couldn’t guarantee anything.

I’m afraid that I haven’t found many videos that you could embed, without ripping and putting onto youtube or the like. I don’t know what the copywrite situation is with this.

There is a nice colloquium by Shamit Kachru, who is an excellent speaker, talking about string theory and cosmology here: http://vmsstreamer1.fnal.gov/VMS_Site_03/Lectures/Colloquium/050302Kachru/index.htm

If you go to this page: http://streamer.perimeterinstitute.ca/mediasite/viewer/FrontEnd/Front.aspx?&shouldResize=False

and look on the left for ” Summer School: Strings, Gravity & Cosmology” you will find a host of great videos. Unfortunately it’s really difficult to try and view these independent from their special player (you need IE, too!). It would be great if you could persuade such institutes to let you have such material. The lectures on perturbative string theory, again by Clifford Johnson, are probably excellent and just the right level for first year grad students. I just wish I could rip them and watch them at my own comfort and time. I’ve watched a few, but I just don’t like this format of streaming.

All the best,

J

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