The shape of a soap bubble is another example. Why is it round? The way to answer that question is to compare the actual round shape with a hypothetical non-round shape. What's different? Yes, the alternative isn't round, but is there some less obvious difference? According to Greek legend, Dido was offered as much land (in northern Africa) as she could enclose with a bull's hide. She cut it into a very long thin strip and enclosed a circle. There she founded the city of Carthage. Why did she choose a circle? Because the circle is the shape with greatest area, for a given perimeter. In the same way, a sphere is the shape with greatest volume, for a given surface area; or, to put it another way, it is the shape with the smallest surface area that contains a given volume. A soap bubble contains a fixed volume of air and its surface area gives the energy of the soap film due to surface tension. In the space of all possible shapes for bubbles, the one with the least energy is a sphere. All other shapes have larger energy, and are therefore ruled out.

You may not feel that bubbles are important. But the same principle explains why Roundworld

(the planet not the universe, but maybe that, too) is round. When it was molten rock, it settled into a spherical shape, because that had the least energy. For the same reason, the heavy materials like iron sank into the core, and the lighter ones, like continents and air, floated up to the top. Actually, Roundworld isn't exactly a sphere, because it rotates, so centrifugal forces cause it to bulge at the equator. But the amount of bulge is only one-third of one per cent. And that bulging shape is the minimum-energy configuration for a mass of liquid spinning at the same speed as the Earth's rotation when it was just starting to solidify.

The physics here isn't important for the message of this book. What is important is the 'Worlds of If' point of view involved in the application of phase spaces. When we discussed the shape of water in a pond, we pretty much ignored the flat surface, the thing we were trying to explain. The entire argument hinged upon non-flat surfaces, humps and dips, and hypothetical transfers of water from one to the other. Almost all of the explanation involved thinking about things that don't actually happen. Only at the end, having ruled out all non-flat surfaces, did we observe that the only possibility left was therefore what the water would actually do. The same goes for the bubble.

At first sight, this might seem to be a very oblique way of doing physics. It takes the stance that the way to understand the real world is to ignore it, and focus instead on all the possible alternative unreal worlds. Then we find some principle (in this case, minimum energy) to rule out nearly all of the unreal worlds, and see what's left. Wouldn't it be easier to start with the real world, and focus solely on that? No, it wouldn't. As we've just seen, the real world alone is too limited to offer a convincing explanation. What you get from the real world alone is 'the world is like it is, and there's nothing more to be said'. However, if you take the imaginative leap of considering unreal worlds, too, you can compare the real world with all of those unreal worlds, and maybe find a principle that picks out the real one from all the others. Then you have answered the question 'Why is the world the way it is, rather than something else?'

An excellent way to approach 'why' questions is to consider alternatives and rule them out. 'Why did you park the car round the corner down a side-street?' 'Because if I'd parked outside the front door on the double yellow lines, a traffic warden would have given me a parking ticket.' This particular 'why' question is a story, a piece of fiction: a hypothetical discussion of the likely consequences of an action that never occurred. Humans invented their own brand of narrativium as an aid to the exploration of I-space, the space of 'insteads'. Narrative provides I-space with a geography: if I did this instead of that, then what would happen would be ...

On Discworld, phase spaces are real. The fictitious alternatives to the one actual state exist, too, and you can get inside the phase space and roam over its landscape -provided you know the right spells, secret entrances and other magical paraphernalia. L-space is a case in point. On Roundworld, we can pretend that phase space exists, and we can imagine exploring its geography. This pretence has turned out to be extraordinarily insightful.

Associated with any physical system, then, is a phase space, a space of the possible. If you're studying the solar system, then the phase space comprises all possible ways to arrange one star, nine planets, a considerable number of moons and a gigantic number of asteroids in space. If you're studying a sand-pile, then the phase space comprises the number of possible ways to arrange several million grains of sand. If you're studying thermodynamics, then the phase space comprises all possible positions and velocities for a large number of gas molecules. Indeed, for each molecule there are three position coordinates and three velocity coordinates, because the molecule lives in three-dimensional space. So with N molecules there are 6 N coordinates altogether. If you're looking at games of chess, then the phase space consists of all possible positions of the pieces on the board. If you're thinking about all possible books, then the phase space is L-space. And if you're thinking about all possible universes, you're contemplating U- space. Each 'point' of U-space is an entire universe (and you have to invent the multiverse to hold them all ... )

When cosmologists think about varying the natural constants, as we described in Chapter 2 in connection with the carbon resonance in stars, they are thinking about one tiny and rather obvious piece of U-space, the part that can be derived from our universe by changing the fundamental constants but otherwise keeping the laws the same. There are infinitely many other ways to set up an alternative universe: they range from having 101 dimensions and totally different laws to being identical with our universe except for six atoms of dysprosium in the core of the star Procyon that change into iodine on Thursdays.

As this example suggests, the first thing to appreciate about phase spaces is that they are generally rather big. What the universe actually does is a tiny proportion of all the things it could have done instead. For instance, suppose that a car park has one hundred parking slots, and that cars are either red, blue, green, white, or black. When the car park is full, how many different patterns of colour are there? Ignore the make of car, ignore how well or badly it is parked; focus solely on the pattern of colours.

Mathematicians call this kind of question 'combinatorics', and they have devised all sorts of clever ways to find answers. Roughly speaking, combinatorics is the art of counting things without actually counting them. Many years ago a mathematical acquaintance of ours came across a university administrator counting light bulbs in the roof of a lecture hall. The lights were arranged in a perfect rectangular grid, 10 by 20. The administrator was staring at the ceiling, going '49, 50, 51 -'

'Two hundred,' said the mathematician.

'How do you know that?'

'Well, it's a 10 by 20 grid, and 10 times 20 is 200.' 'No, no,' replied the administrator. 'I want the exact number.'13 Back to those cars. There are five colours, and each slot can be filled by just one of them. So there are five ways to fill the first slot, five ways to fill the second, and so on. Any way to fill the first slot can be combined with any way to fill the second, so those two slots can be filled in 5 X 5 = 25 ways. Each of those can be combined with any of the five ways to fill the third slot, so now we have 25 x 5 = 125 possibilities. By the same reasoning, the total number of ways to fill the whole car park is 5 x 5 x 5 ... X 5, with a hundred fives. This is 5100, which is rather big. To be precise, it is