Professor Stewart's Hoard of Mathematical Treasures (48 page)

A century and a half ago, Victorian England faced a similar problem, the equally baffling concept of the Fourth Dimension. Mathematicians had encountered it when looking for something else: William Rowan Hamilton had spent decades searching for a natural algebra of 3D space, much as complex numbers are a natural algebra of 2D space, and had been forced instead to settle for a natural algebra of 4D space, which he called quaternions. Scientists were finding that 4D thinking helped them sort out a
lot of basic physics. Spiritualists, claiming to put people in contact with the dead, realised that the Fourth Dimension was a great place to locate the Spirit World, because we can’t go there to check whether spirits exist, but they can see us from their superior vantage point if they do. The same went for ghosts, another Victorian obsession; ghosts could appear from, and disappear into, that extra dimension. And what we now refer to as ‘hyperspace theologians’ were quick to see the advantages of locating God in the Fourth Dimension. From there, He could watch over every part of His creation, while remaining outside it - just as we can see an entire printed page at a glance, without being embedded in the paper.

All this interest was somewhat tempered by a contrary view: the Fourth Dimension does not exist, indeed cannot exist: it is inconceivable. The debate managed to muddle two distinct questions: the structure of physical space, and the possibility of logically consistent mathematical spaces that differ from the orthodox 3D model. Philosophers got in on the act, mostly arguing for the orthodox 3D model - which was a bit surprising given their propensity to claim that nothing really exists and everything we perceive is an illusion.

Into this intellectual hornet’s nest dropped a polite clergyman and headmaster of a prominent boys’ school, Edwin Abbott Abbott. Yes, two Abbotts, to distinguish him from his father, Edwin Abbott. In 1884 Abbott published one of the most curious and original books ever written, a mathematical fantasy called Flatland.

Abbott hit on a cunning way to get his fellow Victorians to accept the possibility of a Fourth Dimension, by quietly pursuing an analogy in which 2D creatures living in a 2D world found the very idea of 3D space inconceivable, indeed heretical - and only then springing 4D on his readers, when he had softened them up. His hero, the modestly named A. Square,
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lives a humdrum life with his linear wife and polygonal children in a pentagonal house in the 2D universe of Flatland, which is a Euclidean plane. Abbott wove in some biting satire about the suppression of women and the poor in Victorian society, as well. And a few quotations from Shakespeare, and some allusions to Aristotle.

Anyway, A. Square lives in a 2D universe and can conceive of no other. That is how things are, that is how things have always been, and that is how things always will be. And in any case, the Third Dimension is religious heresy and the priesthood of Circles will make short shrift of anyone who dares to mention it. And so A. Square continues his humdrum life, until one day he undergoes an epiphany, and becomes totally converted to the notion of a 3D world. This change of heart is triggered by a visitation from ... the Sphere.

A. Square meets the Sphere.

Now, the limitations of his 2D nature prevent him from seeing the Sphere as a single object. Instead, he sees the circles
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in which it meets his flat world. A dot materialises into thin air in an empty room. Then it grows to a circle, expands to a larger
circle, then shrinks back to a dot and vanishes. (You see why Victorian ghostbusters liked the idea of 4D.) He thinks the Sphere is some sort of priest, but one capable of changing its size. We Spacelanders have the luxury of visualising the geometry: a Sphere of fixed size passes through the Flatland plane, and the intersection changes as it does so.

Now, says Abbott - though not in these words - a Victorian Spacelander, trying to contemplate the Fourth Dimension, is in the same position as A. Square trying to contemplate the Third. Protestations about the natural order, or the alleged impossibility of extra dimensions, carry no more weight in Spaceland than they do in Flatland. Abbott confines his mathematical discussion to enumerating the edges and vertices of a cube, and a 4D hypercube, but the point comes over quite strongly.

By analogy, if we Spacelanders met a Hypersphere from the Fourth Dimension, we would perceive only the sequence of spheres that arises when it meets our 3D space. Like A. Square, we see a dot, materialising into thin air in an empty room. Then it grows to a sphere, expands to a larger sphere, then shrinks back to a dot and vanishes. (You see even more clearly why Victorian ghostbusters liked the idea of 4D.)

We can turn this loose geometric analogy into solid algebra, using coordinates. We are used to expressing a point in the plane using two numbers (x, y). Similarly, points in space can be expressed as triples (x, y, z). Here our familiar 3D space runs out of new directions, but mathematically we can explore the behaviour of quadruples (x, y, z, w), and this is what mathematicians mean by a 4D space. Such a space comprises all possible quadruples, not just one. And it has a natural ‘geometry’, because we can define distances using an extension of Pythagoras’s theorem, and once we have distances, we also have angles, and circles, and most of the other stuff that we associate with geometry. Now we can say what we mean by hyperspheres, hypercubes, and all sorts of cute geometrical objects. It all fits together beautifully, and once we’ve become
accustomed to the language, these new types of space start to feel just as real as the one we live in.

Around 1900, physicists and mathematicians suddenly realised the advantages of thinking of time as a (not ‘the’) Fourth Dimension. Soon everyone was very happy talking about 4D spacetime. Today, designers of video games talk of 4D graphics, meaning 3D graphics that move. If we view A. Square’s encounter with the Sphere as a movie, we are in effect using time as a surrogate for a third spatial dimension. Our own encounter with a hypersphere can be visualised using time as a surrogate for a fourth spatial dimension.

However, that’s a surrogate, not the reality. The Sphere from the Third Dimension existed, unchanged, as time passed. Only his intersection with Flatland changed in time. Moreover, time is not the only surrogate for an extra dimension of space. We could employ colour, or temperature, or an entirely new physical quantity, instead.

For instance, suppose the dimension of ‘colour’ varies from yellow to blue through intermediate shades of green, and the universe is a plane on which coloured figures move about. By a trick of perception, they interact, and perceive each other, only when they have the same colour. Now the green creatures would mimic Flatland. So would the yellow ones, and the blue ones. But these three ‘parallel worlds’ really are parallel - in the sense of not meeting. They are separated along the ‘colour dimension’. Now a Sphere could be represented as a yellow dot surrounded by circles that become greener as they expand, and then shrink back to a blue dot. From our 3D perspective we could pull them apart ‘along the colour dimension’ and see the whole thing as a conventional geometric sphere, with shades of colour parallel to its equator. But we don’t have to do that: the colour image is entirely adequate.

When we call the set of quadruples of numbers a space, we are emphasising the 4D analogues of traditional 3D geometry. However, the numbers that appear in the quadruple (x, y, z, w) do
not have to be spatial measurements in the usual sense. They might, for example, be coordinates

(price, colour, weight, temperature) in the space of all woolly pullovers, with colour ranging on a numerical scale from yellow (0) to blue (1). A specific woolly pullover, with coordinates
would have

So, although a woolly pullover is a 3D object, we are representing a few of its key features in a 4D mathematical space. In short, woolly-pullover-space is 4D.

Economists use this approach to represent the state of the nation’s economy, but now they work in, say, a space with a million dimensions, whose coordinates show the prices of a million goods. Astronomers represent the locations and velocities of the eight planets of the solar system
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using six numbers for each planet - three for location and three more for velocity. So the state of the planets, at any given time, defines a point in a 48-dimensional space.

Just as A. Square discovered - to his initial incredulity - that his 2D world was really just part of a higher-dimensional universe, so physicists are beginning to wonder whether the same applies to our 3D world. According to string theory - well, one popular version of many different string theories - space may actually be 10D. The number 10 is not an arbitrary choice, but because this kind of ToE works only in 10 dimensions.

Of course, string theory may not correspond to reality. But
science has taught us many times that the world is more complicated than the one we perceive. If relativity and quantum theory are ever unified, then our view of our world will have to change, just as it did when those two theories were first proposed.

All very well, but: Why don’t we notice those missing dimensions?

There are at least three possible answers.