Some science fiction writers have imagined that if you could learn enough about four dimensions you could move in an extra dimension. In that case, you could get into bank vaults, since you could get into and out of a sealed cubical box as easily as an ordinary person can step into a square drawn on the pavement.

If you learn how to do this by watching this picture, please choose a bank not too close to the University, as most of us bank nearby, and we don't want our money stolen by four dimensional bank robbers.

More seriously, four and higher dimensions occur in science and engineering all the time: speech is traditionally sampled in 12 or 16 filter bank coeficients which change in time- so speech can be viewed as a trajectory in a 12 or sixteen dimensional space. Trying to write a program to recognise speech requires higher dimensional geometry.

In first and second year linear algebra, you study the geometry of n-space, starting off with dimensions 2 and 3. These last are important if you want to be able to write computer graphics programs.

We move on to infinite dimensional spaces in second year. These are of very practical value in understanding how to solve differential equations.

The geometry of n-space matters because when a physicist, chemist or engineer measures the state of a system, it takes some number of real numbers (decimals) to tell us what state it is in. When it changes in time, we get a path in the space. We are often interested in systems where we know the laws telling us how much tendency to change there is at every possible state of the system, so we have a vector field on n-space. Finding the time development, or predicting the future, is done by finding the flow, or in old fashioned language, in solving a system of ordinary differential equations. This is also done in second and third years.

So although the moving image is merely a thought provoking little gizmo, it is closely related to very practical matters indeed.

The Klein Bottle

This object is a klein bottle. You are seeing it in four dimensions, because it cannot live in three dimensions without being squashed, rather as a knotted circle cannot live in two dimensions.

You get a klein bottle as follows: first take a strip of paper and glue the ends together, but first do a twist so you get something like this:

Now make a duplicate of exactly the same size.

If you run your finger around the boundary, you get back to where you started, so the boundary of the mobius strip is a circle, just as the boundary of a disk is a circle.

Since you have two of them, you could sew the boundaries together, and when you have finished, the resulting surface will have no boundary at all, it will all be used up.

Try it! Get you mother to do it on a sewing machine.

You (or your long suffering mother) will discover that it is impossible, some bits of the thing get in the way, and it cannot be made- in three dimensions.

If your mum could sew in four dimensions she could make it. It would look rather weird. It has no inside, or its inside is joined smoothly to its outside.

The Real Projective Plane

These spaces are not just curiosities; there is a method of finding lines in images called the Hough Transform which accumulates evidence for a line in the space of lines in the plane. This space is actually real projective space with a hole in it. Understanding that some desirable things are actually impossible (and saving the time of computer scientists in looking for them) can be done by understanding that this space is different from a disk in a very fundamental (topological) way.

Here is a smaller version:

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