The Science of Discworld II
is an entire universe (and you have to invent the multiverse to hold them all â¦)
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 âcombinatoricsâ, 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 roofof a lecture hall. The lights were arranged in a perfect rectangular grid, 10 by 20. The administrator was staring at the ceiling, going â49, 50, 51 â¦â
âTwo hundred,â said the mathematician.
âHow do you know that?â
âWell, itâs a 10 by 20 grid, and 10 times 20 is 200.â
âNo, no,â replied the administrator. âI want the exact number.â 2
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 à 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 à 5 = 125 possibilities. By the same reasoning, the total number of ways to fill the whole car park is 5 à 5 à 5 ⦠à 5, with a hundred fives. This is 5 100 , which is rather big. To be precise, it is
78886090522101180541172856528278622
96732064351090230047702789306640625
(weâve broken the number in two so that it fits the page width) which has 70 digits. It took a computer algebra system about five seconds to work that out, by the way, and about 4.999 of those seconds were taken up with giving it the instructions. And most of the rest was used up printing the result to the screen. Anyway, you now see why combinatorics is the art of counting without actually counting ; if you listed all the possibilities and counted them â1, 2, 3, 4 â¦â youâd never finish. So itâs a good job that the university administrator wasnât in charge of car parking.
How big is L-space? The Librarian said it is infinite, which is true if you used infinity to mean âa much larger number than I can envisageâ or if you donât place an upper limit on how big a book can be, 3 or ifyou allow all possible alphabets, syllabaries, and pictograms. If we stick to âordinary-sizedâ English books, we can reduce the estimate.
A typical book is 100,000 words long, or about 600,000 characters (letters and spaces, weâll ignore punctuation marks). There are 26 letters in the English alphabet, plus a space, making 27 characters that can go into each of the 600,000 possible positions. The counting principle that we used to solve the car-parking problem now implies that the maximum number of books of this length is 27 600,000 , which is roughly 10 860,000 (that is, an 860,000-digit number). Of course, most of those âbooksâ make very little sense, because weâve not yet insisted that the letters make sensible words. If we assume that the words are drawn from a list of 10,000 standard ones, and calculate the number of ways to arrange 100,000 words in order, then the figure changes to 10,000 100,000 , equal to 10 400,000 ,
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