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WHAT IS THE FOURTH DIMENSION?

When we say space has three dimensions we mean that a position in space is determined by three measurements: for example, so many miles north, east, and up from a definite place, such as the Daily Worker Office.

If we like, two of the measurements may be angles; for example, 37 degrees east of north, and 5 degrees above the horizontal, but one must be a distance or involve a distance.

Three measurements are always enough, and a fourth eithertells you nothing more or makes nonsense. For example ifwe say that an aeroplane is 12 miles north of our rotary press,four miles east, and three miles up, we say nothing fresh byadding that it is 13 miles away, and we talk nonsense if wesay it is 15 miles away.

It is a hard fact that space has three and only three di­mensions, as it is a hard fact that I have two and only two eyes. Nevertheless, mathematicians lecture and write about the geometry of four, five or many-dimensional manifolds or spaces. I even refer to it myself in one course of lectures.

The main justification is this. We use ordinary space for representing mathematical relations by means of graphs. For example, you can follow the changes in employment or real wages, or the population of Britain, much more easily from a graph than from a table of figures.

Clearly, if we are merely considering the relation between two numbers we only need two dimensions, and can make our graph on a sheet of paper. For example, if we want to show the frequencies of different heights in the population, we could draw a graph showing that 20 per cent of the men in _.a group had heights between 69 and 70 inches, 12 per cent between 68 and 69, 10 per cent between 67 and 68, and so on falling off on each side of the maximum.

If we want to do the same thing for three numbers we need solid model. For example, we can put up a wire on each squ­are of a piece of squared paper to represent the frequencies of married couples in which, the husband measures 69--70 inches and the wife 62--64, and go on. We can then make a surface through the tops of these wires, and get a solid model. When we look at it, we see at once there is a definite, but not very strong, tendency for tall men to marry tall women, and so on. There are plenty of exceptions, but no doubt about the tendency.

Now in science we constantly want to do calculations in­volving relations between four numbers, for example, the frequencies of different height combinations of father, mother and child. We can't make a graph or model of such a relation, but we can argue as if we had made one. We can say that our four-dimensional model would or would not be symmetrical, that it would have one or several "peaks" or maxima, and so on.

And just as various sections of a three dimensional model are two dimensional graphs, for example, contour lines, so various "sections" of the four-dimensional imaginary model can be represented by solids.

Again, if we have two variable numbers, say, the pressure of steam in a cylinder and the position of the piston, it is most use­ful to the engineer to graph one against the other. In the course of a stroke the point moves round a certain area, which measu­res the work done.

We often have to deal with such relations between several numbers, and we need to be able to calculate quantities cor­responding to volumes, solid angles, and so on, in many di­mensions.

In fact the geometry of many dimensions is of practical importance. It is also quite amusing.

No doubt all the practical results can be reached by ot­her methods. But it is worth noting that the pioneer in using "many-dimensional "models" in mechanics was Hertz. And Hertz was also the first man to produce and pick up radio waves; so it is no good dismissing him as an unpractical theo­rist.

If we want to represent the intensity and direction of gravity, and of electric and magnetic forces at any point in space, we can do so by using extra dimensions. There is nothing my­sterious about this. We are just attaching, say, nine more numbers to each point in space besides the three which tell us where it is. The others tell us about the forces on a body there. These numbers form a 12-dimensional manifold. But that does not mean that space has 12 dimensions.

Similarly we can think about a four dimensional mani­fold of space and time. We are using four numbers when we say that an aeroplane was 12 miles north, four miles east, and three miles up at 4,23 p.m.

This way of looking at events is very useful, because two observers in relative motion, for example, one in a moving train arid one on a platform, do not quite agree as to what events take place at the same time, any more than they do as to what objects are at rest, but one can fit all their private views of the universe into one public space-time, which is the same for everyone.

So much for the uses of many dimensional geometry. Now for its abuses. There are people who explain ghosts and other marvels by a fourth dimension of space. They say things come into our space from it, and so on. If you answer that no one can see it or feel it, they reply that one can't see or feel radio waves, but one can pick them up with a receiver. Now theories about invisible and intangible objects are only worth anything if they enable one to predict.

I believe in radio transmission because this theory lets me predict that by making the correct adjustments to a cer­tain instrument I shall hear a certain programme. The pre­dictions are not always right, but often enough.

Some people certainly see ghosts, that is to say, they see people who aren't there. I have been a ghost myself. I appeared to my grandmother while I was sitting on a lawn 20 miles away. If I had been dying or dead the vision would have got more publicity.

I cannot see how the existence of a fourth dimension of space would help us in the least to understand how a man's wife can see him when he is dying 1.000 miles away, or with the help of a medium after he is dead, supposing either of these things to be true. If there is a fourth dimension, we know enough about its geometry to say that any path thro­ugh it between points in our space is longer than the short­est distance in three dimensions.

And above all no one has explained why objects from the fourth dimension appear in our space so very rarely. So I shall go on lecturing about the geometry of many dimensions when I want to, but I shall no more believe in the real exi­stence of a fourth dimension of space than I believe that an economic depression is a real hole, or that the peak of gas consumption on Sunday mornings is a real peak which I could climb on a holiday.

1948

Література

John B.C. Haldane. Reader of Popular Scientific Essays. - Изд-во«Наука», М., 1993. - 235 с.