The Science of Discworld II

rules of the game and a trajectory is a legal sequence of moves by the players.
    The formal setting and terminology for phase spaces is not as important, for us, as the viewpoint that they encourage. For example, you might wonder why the surface of a pool of water, in the absence of wind or other disturbances, is flat. It just sits there, flat; it isn’t even doing anything. But you start to make progress immediately if you ask the question ‘what would happen if it wasn’t flat?’ For instance, why can’t the water be piled up into a hump in the middle of the pond? Well, imagine that it was. Imagine that you can control the position of every molecule of water, and that you pile it up in this way, miraculously keeping every molecule just where you’ve placed it. Then, you‘let go’. What would happen? The heap of water would collapse, and waves would slosh across the pool until everything settled down to that nice, flat surface that we’ve learned to expect. Again, suppose you arranged the water so that there was a big dip in the middle. Then as soon as you let go, water would move in from the sides to fill the dip.
    Mathematically, this idea can be formalised in terms of the space of all possible shapes for the water’s surface. ‘Possible’ here doesn’t mean physically possible: the only shape you’ll ever see in the real world, barring disturbances, is a flat surface. ‘Possible’ means ‘conceptually possible’. So we can set up this space of all possible shapes for the surface as a simple mathematical construct, and this is the phase space for the problem. Each ‘point’ – location – in phase space represents a conceivable shape for the surface. Just one of those points, one state, represents ‘flat’.
    Having defined the appropriate phase space, the next step is to understand the dynamic: the way that the natural flow of water under gravity affects the possible surfaces of the pool. In this case, there is a simple principle that solves the whole problem: the idea that water flows so as to make its total energy as small as possible. If you put the water into some particular state, like that piled-up hump, and then let go, the surface will follow the ‘energy gradient’ downhill, until it finds the lowest possible energy. Then (after some sloshing around which slowly subsides because of friction) it will remain at rest in this lowest-energy state.
    The energy in this problem is ‘potential energy’, determined by gravity. The potential energy of a mass of water is equal to its height above some arbitrary reference level, multiplied by the mass concerned. Suppose that the water is not flat. Then some parts are higher up than others. So we can transfer some water from the high level to the lower one, by flattening a hump and filling a dip. When we do that, the water involved moves downwards, so the total energy decreases. Conclusion: if the surface is not flat, then the energy is not as small as possible. Or, to put it the other way round: the minimum energy configuration occurs when the surface is flat.
    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 shapewith 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