Thursday, August 16, 2007

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.

3 comments:

toomanytribbles said...

i've always liked this depiction of a tesseract. apod's featured it twice:

http://antwrp.gsfc.nasa.gov/apod/ap070318.html

Jonathan Shock said...

Hi there,

Yes, I'd seen that one before. There are many much more impressive videos out there showing projections of higher dimensional objects, but there didn't seem to be anything that might show why they are the way they are.

I'll improve mine when I have time but for the moment I hope that it's instructive.

J

toomanytribbles said...

look what i found!!
http://www.flatlandthemovie.com/