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Why We Live in Three Dimensions

The possibility of universes with different dimensions of both space and time has been explored by a number of scientists. If we assume that the laws of Nature keep the same form but allow the numbers of dimensions of space and time to range freely over all possibilities we can explore what happens to those laws. The fact that the laws of Nature are strongly linked to the dimension of the space in which they act was first noticed by the famous German philosopher, Immanuel Kant (1724-1804). Kant had noticed a very profound thing: that Newton’s famous inverse-square law of gravity was intimately connected with the fact that space has three dimensions.

If space had four dimensions then gravity would vary as the inverse-cube of distance, if it had 100 dimensions then as an inverse 99th power of distance. In general, an N-dimensional world exhibits a force law for gravity which falls off as the (N - 1)st power of distance.

Kant used this observation to ‘prove’ to himself that space must have three dimensions because of the existence of Newton’s inverse-square law of gravitational force He suggested that if God had chosen an inverse cube rather than an inverse square law of gravitational force with distance then a universe of different dimension - four - would have resulted. Today, we would regard this as getting the punch-line back to front: it is the three-dimensionality of space that explains why we see inverse-square force laws in Nature, not vice versa. The same idea can be used to explore the ways in which changing the number of dimensions of space and time changes other laws governing atomic structure.

Start by imagining that there could be any number of dimensions of space and of time. The chequer-board of all possibilities can be whittled down dramatically by the imposition of a small number of reasonable requirements that seem likely to be necessary for information processing and ‘life’ to exist. If we want the future to be determined by the present then we eliminate all those regions of the board marked ‘unpredictable’.

If we want stable atoms to exist along with stable orbits of bodies (planets) around stars then we cut out the strips marked ‘unstable’. Cutting out worlds in which there is only faster-than-light signalling we are left with our own world of 3+1 dimensions of space plus time along with very simple worlds that have 2+1, 1+1, and 1+2 dimensions of space plus time. Such worlds are usually thought to be too simple to contain living things. For example, in 2+1 worlds there are no gravitational forces between masses and there is an imposed simplicity of designs that challenges any attempt to evolve complexity. Notwithstanding these limitations, there has been much speculation about how working devices could be constructed in two-dimensional worlds.

Networks are extremely limited because paths cannot cross without intersecting. Worlds with more than one time are hard to imagine and appear to offer many more possibilities. Alas, they seem to offer so many possibilities that the elementary particles of matter are far less stable than in worlds with a single time dimension.

Protons can decay easily into neutrons, positrons and neutrinos and electrons can decay into neutrons, antiprotons and neutrinos. The overall effect of extra time dimensions is to make complex structures highly unstable unless they existed in conditions of extremely low temperature. Worlds with more than one time dimension do no allow the future to be predicted from the present. In this sense they are rather like worlds with no time dimension. A complex organised system, like that needed for life, would not be able to use the information gleaned from its environment to inform its future behaviour. It would remain simple, too simple to evolve.

If the number of dimensions of space or time had been chosen at random and all numbers were possible then we would expect the number to be a very large one. It is very improbable that a small number is chosen. However, the constraints imposed by the need to have ‘observers’ to talk about the problem mean that all possibilities are not available and a three-dimensional space is forced upon us.

All the alternatives will be barren of life. In recent years, theoretical physicists have found that their attempts to unify the different forces of Nature only work if the world has many more dimensions of space than three. This is reconciled with the fact that we observe the universe to possess only three large dimensions by assuming that the 'other' dimensions are very small ('compactified'), perhaps as small as 10-33 cm in extent, and are therefore imperceptible to us. So far it is not known how such a splitting of space came about during the very early history of the universe.

In summary, three-dimensional worlds with a single arrow of time are peculiar. The alternatives are too simple, too unstable and too unpredictable for complex observers to evolve and persist within them. As a result we should not be surprised to find ourselves living in three large dimensions subject to the ravages of a single time. There is no alternative.

Further Reading: John D. Barrow 'The Constants of Nature', Jonathan Cape (2002), Chapter 10.

Animations by Nick Mee from the CD_ROM 'POLYTOPIA II: Honeycombs and Polytopes' Published by Virtual Image