Friday, April 03, 2015

All is geometry

Here's a little puzzle. Suppose you pick a number randomly between 0 and 1: call it x. Now pick a second one, also between 0 and 1: call it y. What's the average distance between x and y?

Hands up those who called out zero! Well, you clever, but you bad! I mean the distance you could measure with a ruler, no negative distances.

Not so trivial now, is it.

So here's a picture of us choosing x and y (figure 1). Let's agree that the order makes no difference to the gap between them, so we'll just arrange things so that x is the bigger one (don't care, if x = y).

Figure 1
We'll let z be the distance measured by our mathematical ruler between the two numbers - we want to know the average value of z over many trials. We can plot x and y as separate choices in the X-Y plane (figure 2).

Figure 2

The space of possible choices for x and y is the shaded triangle above. The area of the triangle is one half - we'll need this later.

Now, we add the Z-axis where we can plot the distance between them, z = x-y.

Figure 3

The volume z = x-y is a right-angled triangular pyramid. As you can see, when y is zero, z is just equal to x so z slopes upwards on the X-axis. When y = x, z is zero - that's where the pyramid meets the X-Y plane. The volume of the pyramid is a third the base-area times the height so that's (1/3) * (1/2) * 1 = 1/6; we'll need that in a moment.

I hope you'll agree that this three-dimensional volume showing the variation in distance between two randomly chosen points in the [0, 1] interval is not exactly obvious. To work out the average value of the distance z, we consider a shape which "sits" on the shaded area of figure 2 above, which has the same volume as figure 3 but is of uniform z-height.

Here's the picture (figure 4).

Figure 4

As you can see, z here is a constant 1/3, because (1/3) * (1/2) = (1/6), the same volume as before.

So that's the answer: the average distance between two randomly-chosen points is one third. Did you guess that correctly?

[Note: you might be surprised by how hard this problem is in general].


Here's another puzzle. You throw an ordinary six-faced die twice (example: getting 5 and 2). Call the difference between the two scores z (example: z = 5-2 = 3).

What's the average value of z? Fancy a guess?

Suppose the faces of the die were labelled (1/6), (2/6), ..., (6/6). Then these six points are uniformly distributed along the interval [0, 1] so we could use the result we already showed above and estimate that the average distance between them should be one third.

Of course the actual values on the die are six times bigger, so we would expect the value of z here also to be six times bigger, so 6 * (1/3) = 2.

Rough, obviously. We're talking a continuous approximation to a discrete distribution. We can get the right answer by simply listing cases.

6-1 = 5, 6-2 = 4, 6-3 = 3, 6-4 = 2, 6-5=1, 6-6 = 0
5-1 = 4, 5-2 = 3, 5-3 = 2, 5-4 = 1, 5-5 = 0
4-1 = 3, 4-2 = 2, 4-3 = 1, 4-4 = 0
3-1 = 2, 3-2 = 1, 3-3 = 0
2-1 = 1, 2-2 = 0
1-1 = 0

Total of differences = 35; total number of cases = 21 so average difference = 35/21 = 5/3 = 1.67 .. or, as we say, two to the nearest whole number :-).

(Do you see how you could draw pictures of these cases paralleling figures 2-4 above? You'd get a lumpy three-dimensional histogram rather than the smooth pyramid.)


There is a point to this amateur theorising. Mathematics sometimes seems very abstract and textual: axioms and long strings of deduction. This is to take a very lexical-syntactic view; to reduce ourselves to automated theorem-provers. The language of mathematics is about something, and that something is structure, geometry. The entities of mathematics are structural objects, often of high-complexity, of infinite size and inhabiting many dimensions. Our axioms and theorems describe properties of these structures; our proofs are like a blind man feeling his way around an elephant, encountering its aspects deduction by deduction.

When you truly understand a mathematical object in its full "shape", its nature is obvious. But it's hard for mortals to encompass the infinite - mere glimpses are sometimes as good as it gets.

Since physics is finding mathematical structures which can be brought into correspondence with measured reality, it is equally true - in some sense - that physics is geometry. There is a programme in physics, Geometrodynamics, which takes this rather literally.

You might also like to take a look at the hottest of new physics ideas: ER = EPR (wormholes in general relativity = quantum entanglement).
"I’ve mentioned before that John Wheeler was one of my heros during my formative years. Back in the 1950s, Wheeler held a passionate belief that “everything is geometry,” and one particularly intriguing idea he called “charge without charge.” There are no pointlike electric charges, Wheeler proclaimed; rather, electric field lines can thread the mouth of a wormhole. What looks to you like an electron is actually a tiny wormhole mouth. If you were small enough, you could dive inside the electron and emerge from a positron far away."

"Where did all this come from?" you ask wonderingly. Amazingly, I was working this out in bed this morning at 6.30 am. I have no idea why.

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