Day 121 - Unexpected Retrievals

There were three things I intended to blog about today - here's what I wrote down to remind myself:

Primes occuring in nature
Visualising dimensions
Trivial when expressed properly

Now let's see how they turn out when I actually write them. Behind-the-scenes blogging, that is. Straight from my brainstorm to your brainwave (why does the brain have such weather-themed imagery?).

Today I watched, with Julia (under some duress), a one-off Horizon programme called Alan and Marcus Go Forth and Multiply. Basically, the comedian Alan Davies gets taken on a whirlwind tour of higher mathematics by Oxford professor Marcus Du Sautoy. The usual things come up - Monty Hall problem, dimensions, prime numbers, and it does prevent a couple of things (such as a potted description of what Perelman achieved) in interesting and original ways, helped along by Alan's charm and Marcus' obvious enthusiasm and intelligence. I highly recommend viewing it - on with the point, as they say in the pencil factory.
While discussing the Riemann zeta function and its relation to the prime numbers, they also demonstrated an interesting parallel in nature - the frequency of the vibrations of a quartz crystal correspond closely to (I presume) the modulus of a certain section of the zeta function.
This got me thinking about why prime numbers are so important in nature, and in particular, quantum physics, etc.
It's not that surprising.

Let's suppose we have a large group of objects of some kind, held together somehow. Further assume that these bonds are relatively weak - in other words, it won't take much to break up this big group into several smaller groups. Now, also suppose that the forces that act upon this group to break it up act pretty much symmetrically over the whole group. In other words, this breaking-up force will be distributed evenly. This suggests that the parts it is broken up into will be evenly sized as well.
For example, a group with 6 objects in it, broken up in a symmetric fashion, won't break up into a group of 5 objects and a group of 1 object. It will be broken up into groups of the same size, e.g. 2 groups of 3 objects.

See where I'm going with this? The size of groups of objects will tend towards factorisation, which in turns entails that they will tend towards sizes which are prime numbers, which can't be factored in a non-trivial way.

Now, this is a very general heuristic sketch - billiard balls, for example, don't naturally group into 2s or 3s. If, however, the bonds between objects in questions were sufficiently weak, and the breaking-up forces were sufficiently symmetrical, such a tendency towards primes would occur.
The objects may not be physical - they may be waves, or frequencies of waves, or properties of frequencies of waves, and so on. In any physical situation in which have some kinds of objects grouped together, and forces acting on them symmetrically, it would naturally tend towards primes.

Moreover, such weak bonds and symmetrical forces would be more likely to occur at very small scales, where there is simply less room for forces to act non-symmetrically. So, primes would crop up frequently in quantum physics. Since the zeta function generates the primes, in some fashion, it's also expected that it would generate, in some fashion, these quantum phenomena.

Alriiiiiiight, enough heuristics.

Another problem mentioned in this program was how to visualise in more than three dimensions. This is usually seen as some in built limitation of our brains, in that we can only visualise spacially in up to three dimensions - which is only natural, since we perceive things in three dimensions.
I think, however, that it's a lot easier than people think, and we're being held back by what think a dimension should be.

Take the paradigmatic cases: for 0-dimensions a point, 1 dimensions a line, 2 dimensions a square, 3 dimensions a cube, and 4 dimensions a hypercube, and so on.
Up to 3 dimensions, we can visualise these easily. We can build them from the previous one, by extending it in a different direction, in some sense.
We stretch out a point along a length, to get a line. We stretch out a line along a width, to get a square. We stretch out a square along a height, to get a cube. Then our minds hit a brick wall - where is the fourth direction in which to stretch our cube?

Take your pick! The fourth direction does not need to be spatial. Perhaps it can't be, since all the space we know and love only has three dimensions. A dimension is just another co-ordinate; another piece of information telling us where to look for a point.

Example. A point is 0 dimensions, because no co-ordinates need to be given, we don't need any information to locate any point within the point. Because there's only one point. We know where it is, we don't need to look.
Now, for a line, we need 1 number, to know how far along the line our point is, and then we're done. A line is 1-dimensional.
In a square, we need 1 number to tell us which line the point lies on - which cross-section of the square we need. Then another number to tell us how far along that line.
Similarly the cube - the first co-ordinate tells us the square, the second the line, the third which point on that line.

So! To make a fourth dimensional cube, a hypercube, watch carefully (I have nothing up my sleeves). Take time to be the fourth dimension. Suppose a cube suddenly springs into existence at time 0, and winks out at time 1. Now, if we searching through time as well as space, we have a 4-dimensional cube. The first co-ordinate tells us at which time we're looking - which cube we're searching in. The second which square within that cube, the third which line within that square, the fourth which point within that line.

Blam. A 4-dimensional cube, plain and simple. What's wrong with that? We can't hold it in our minds eye like the others, since our eyes can only see cross-sections of time. We are still, however, perfectly aware of time, and use duration in our visualisations, so we can easily visualise this magically appearing and disappearing cube. In a way, we already do when we visualise any cube. Now we're just taking notice of the time it's there for. Nothing mysterious or esoteric about the fourth dimension.

We can do something else, however. Instead of time, why not use colour as the fourth dimension? Let 0 be the colour white, 1 the colour black, and a continuous sliding scale of grey in between. Now let's have an infinite collection of cubes, one for each possible shade. Note we need infinite cubes, just as there are infinitely many squares in a cube, and infinitely many lines in a square, we need infinitely many cubes in a hypercube.

Now, to find any point, we specify the first co-ordinate - precisely which shade of grey we're looking at, then pick up that cube. The second gives us the cube, and...you've gotten the idea by now.

In fact, any continuous variable that can be applied to cubes could be used. How much that particular cube is adored by the media, on a scale from 0 to 1, taken continuously. The probability that cube will turn into a flying pig, ideally. They're not spatial things, but we can think about them, we can visualise them. We can visualise time, and colour, and billion other non-spatial things. Use them in your search for higher dimensions.

This can be carried out higher dimensions. Want a 5-dimensional hyperhypercube? No problem! Just take your infinitely many different shades of grey cubes, and have them existing over a duration as well (not hard - most things exist over some duration). 6 dimensions? Take our temporary coloured cubes and just factor the height that the mould on them grows to. Let your mind run free!

Phew, how exciting. The third thing: "trivial when expressed properly." Here's what I meant:

I believe that any mathematical proposition, when expressed in the right way, becomes obviously and trivially true. The difficulty in mathematics lies in working out exactly how to say what you want to prove; translate it into the right language, look at it in the right way, and there's nothing to it.
A quick example, till I think of some more: factorisation of integers into primes. Look at it one way, and it's pretty obvious - primes are, by definition, things which can't be divided anymore. Take any integer, and divide it into smaller bits, then those bits again, and so on, and eventually you'll have to stop, and all you'll have left are primes. Easy.
Now, however, think about the integers like this: you start with 0. Then add 1 to make 1. Then add 1 to make 2, and so on. The integers are everything you get proceeding in this way, adding 1 each time. Now work out analysis, then complex analysis, then the theory of the Riemann zeta function, then let the primes be defined by the sequence which satisfies a certain functional equation involving the zeros of the zeta function. Now try and prove factorisation into primes. A lot harder.

Better examples to follow!