Ian Stewart

by Professor Stewart's Hoard of Mathematical Treasures

  However, it’s not that simple. Mathematicians have invented logically consistent geometries with 4, 5, 6, or even infinitely many, dimensions. Whichever number you like, including 10. So, on the face of it, there is nothing sacred about 3D space. It might be a historical accident, which could have been different in another run of the universe. It might be sacred after all, the only possibility for reasons we don’t yet appreciate. Or it might not actually be 3D, despite appearances. And even if it is, there’s no reason to expect it to be the neat, tidy 3D space of Euclid. In fact, thanks to Einstein’s general theory of relativity, we think space is curved, in ways that Euclid never dreamed of, and sort of mixed together with bits of time.

  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.

  Edwin Abbott Abbott . . .

  ... and his book.

  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,47 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 circles48 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

  (27.43, 0.62,1.37, 22.61)

  would have

  price = £27.43

  colour = bluish-green

  weight = 1.37 kilograms

  temperature = 22.61°C

  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 system49 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.

  • They don’t exist, and string theory is wrong.

  • They do exist, but they are curled up so small that we can’t see them. From a distance, a hosepipe looks 1D, but close up it has a circular cross-section, adding two more dimensions. If that cross-section were really really small - much smaller than the diameter of an electron, say - the hosepipe would be convincingly 1D unless you developed very delicate experimental techniques to probe those hidden dimensions. Now replace the hosepipe by our apparently 3D space, and the circular cross-section by an equally tiny 7D sphere, and you’ll get the idea.

  • Our space really is 3D, but it is embedded in a surrounding 10D space - and we don’t perceive the bigger space because we can’t look, or move, in those directions. Just as A. Square was confined to the plane of Flatland, we may be confined to a 3D slice of that 10D space. Mathematically, this kind of behaviour is entirely natural: dynamical systems often have ‘invariant subspaces’, and anything that lives in those subspaces can’t escape from them. Try moving yourself into the past to appreciate what I mean. Physicists have taken to calling such subspaces ‘branes’, a term derived from ‘membrane’ via ‘m-brane’, an m-dimensional subspace.

  A hosepipe seems to be 1D, but closer up we see it has two more dimensions. We can draw this schematically as a line with circles attached to each point.

  Extra dimensions of space (here the 2D plane) shown schematically as spheres. In string theory, the spheres have more dimensions than we can draw. The spheres support quantum vibrations, which endow particles with properties like spin and charge.

  All this talk of ‘hidden dimensions’ may be needlessly mystical. Physics presented us with very similar things long ago, but no one started babbling about increasing the dimension of space. An electromagnetic field - which we use to send radio, TV and phone calls - has six extra coordinates for each point in space: three for the magnetic field’s strength and direction, and three more for the electric field’s strength and direction. Maxwell’s equations for electromagnetism are naturally defined on a 9D space.

  So the extra 7 dimensions required for string theory need not actually be spatial in any meaningful sense. They might be - in fact, are - new physical quantities, like colour or temperature, that enter into the string theory equations. So talking of them as hidden dimensions of space makes string theory seem more mysterious than it really is.

  Slade’s Braid

  In the 1880s, the American medium Henry Slade used to convince people that he had access to the Fourth Dimension - the Spirit World - using a strip of leather with two cuts along it. He would get someone to make a mark on the leather, to prevent substitution. Then he would hold it under the table for a few moments, and produce it again - braided!

  Start here . . .

  . . . end here.

  In 4D space, strips can be passed over each other and woven together by pushing one temporarily into the Fourth Dimension, moving it into the right position, and then pulling it back into ordinary 3D space. That is what Slade pointed out, and what he claimed proved he had the ability to access the Fourth Dimension.

  How did he do it?

  Answer on page 334

  Avoiding the Neighbours

  Place each of the digits 1-8 in the eight circles, so that neighbouring digits (that is, those that differ by 1) do not lie in neighbouring circles (connected directly by a line).

  Answer on page 335

  Keep the neighbours apart.

  Career Move

  A mathematician who had spent his entire research career in pure mathematics - starting with topological algebra, then a bit of algebraic geometry, then some geometric topology, thinking of moving into algebraic topology or maybe geometric algebra - began to wonder if perhaps it was time he did something more obviously practical. He knew that those subjects did have applications, but he had never worked on such things, preferring the intellectual challenges of abstract thought.

  He had never been against applied mathematics, you understand - just hadn’t done any himself.

  Maybe, he thought, it’s time for a change.

  Weeks went by, and still he had not translated his thoughts into deeds. The prospect of engaging with the real world made him very nervous. He’d never done it before. But he found the idea appealing, nonetheless. The problem was to pluck up enough courage to take the plunge.

  One day, walking along the corridor of the Mathematics Department, he saw a sign on a door. ‘Seminar on gears - today at 2.00.’ He looked at his watch: 1.56. Dare he? Could he actually ... go in? It was a big step. In an agony of indecision, he stood outside the door, shifting from one foot to the other, listening to the sounds of the lecturer preparing to start the talk. Finally, at 1.59, he plucked up his courage, opened the door, and slid into a vacant seat. Now he would begin his career move to practical applications of mathematics!

  The speaker picked up his notes, cleared his throat, and began. ‘The theory of gears with an integer number of teeth is well known—’

  A Rolling Wheel Gathers No Speed

  A wheel of radius 1 metre rolls along a flat horizontal road at a constant speed of 10 metres per second, without slipping and without bouncing off the road. At a fixed instant of time, is any point on the wheel stationary? If so, which?

  Assume that the wheel is a circular disc, the road is a straight line, and the wheel lies in a vertical plane. ‘Stationary’ means that the instantaneous velocity is zero.

  Answer on page 335

  Point Placement Problem

  You have a line of unit length, whose two endpoints at 0 and 1 are missing, and an unbounded supply of points - as one does. You are required to place the points successively on the line, so that:

  • The second point and the first point lie in different halves of the line. (To avoid ambiguity, the midpoint at is excluded: neither point is allowed to lie in that exact position. So one ‘half’ runs from 0 to , excluding both, and the other runs from to 1, excluding both.)

  • The third point and the first and second points all lie in different thirds of the line. (To avoid ambiguity, the points at and are now excluded.)

  • The fourth point and the first, second an
d third points all lie in different quarters of the line. (The points at and are now excluded - remember, we have already excluded .)

  Now keep going, obeying, for increasing n, this rule:

  • The nth point and the previous n - 1 points all lie in different ths of the line. (All points , for m = 0, 1, 2, . . . , n, are excluded.)

  Got that? Here’s the question: how long can you keep this process going?

  At first sight, the answer seems to be: as long as you wish. After all, you can divide the line into indefinitely ever finer pieces, and choose points in whichever of those are appropriate.

  I really don’t expect you to get the correct answer here, but I don’t want to give it away immediately, so you’ll find it on page 336. It’s amusing to try placing the first five or six points. Even then, it’s not as easy as it sounds.