Sunday, 13 March 2011

Cartesian what now?

OK guys, for today's post (jeez I need to post more often, jeez), I have a little bit of maths for you to comprehend. OK, it's not that maths-y, it's the sort of thing that I would have put up on Dimensionality back when it was alive. Let it be known before I start my multi-dimensional ramblings that this is something that I worked out a few weeks ago, but it has actually been worked out differently and 'properly' by mathematicians decades ago. Sure, maybe I'm wrong in some places because I'm not au fait with the proper notation and I'm not going into tesseracts and shit like that, here's the theory. And just the theory.

Before we begin, I need to inform you we'll be talking a lot about dimensions. You know, those planes of existence in which we live. The four dimensions we can perceive with our human eyes are 0, 1, 2, and 3. Below you can see these represented graphically. You see we can graphically show a 0-dimensional object, a point. Then we can extend that point into a new dimension, the first dimension, created a line from point A to point B. We can extend that line and thus points A and B again, creating a square, and a third point, C. That's the second dimension; like this screen. We can then pull points A, B and C out of their two dimensions and into a third, creating a cube. This offers us point D. Like the other points, D can be represented in any of the lower dimensions, but it's only when we open those dimensions up that we see the true placement of D. For example, squash that cube back down into a 2D shape and it goes right on top of one of the other points, so in a 2D plane we see it as being the same as that other point. It's only in a higher dimension that we realise just how far away certain points are.

Similarly, you could be right next to someone else in the first 3 dimensions, but maybe in the fourth or fifth dimensions, you're aeons away. It's a scary thought.

Some display schtuff: 1D coordinates (x-variables) are brown, 2D coordinates (y-variables) are blue, and 3D coordinates (z-variables) are yellow.
OK. Let's get started.

Science is based on comparing different variables, right? For every variable, one value in that variable corresponds to a different value in a different variable. This was the starting point for me. We compare two variables together and we can display them in a table; much like the times table you know so well from primary school. That grid with numbers on one side (1 to 10, say) and numbers on another side (1 to 10 again), and then meeting to create different numbers in the table space. This is a 2D variable comparison space.

But remember that those variables, such as in that times table (though they're not scientific variables) are discrete - as in, they only have certain values, such as 1, 2, 3, 4, etc. We can link our tables to what is known as Cartesian space. Cartesian space is the xy plane in which we deal with most graphs. And the only difference between a two-way table and Cartesian space is that in the xy plane, the variables (x and y) are continuous, whereas in a table, the variables are discrete. In a table, the values are simply '1,2,3' but in xy space the variables could use values '1,2,3' or go deeper and use '0.00001, 0.00002', etc. There are an infinite number of values in Cartesian space.

You can see the comparison here, and it's also clear to see that variables are compared to each other in both systems as being a simple 2D shape - extending that 1D line outwards to form two sides of a square.

Now, if you were doing proper science, you'd perhaps want to compare three variables. And just as comparing two variables requires a 2D table space, comparing three requires a 3D space. You extend that grid of values out into a third dimension, creating a grid cube of sorts. Imagine you had ten values in Variable A, ten values in Variable B and ten values in Variable C. Plotting A against B will give you a table of 100 values, and plotting those two against C will multiply that number by the number of values in Variable C, ie. 100x10 = 1,000.

Knowing this simple maths, we can easily plot our first three variables - A, B and C - against a fourth, D. If D also has 10 values, then clearly we have 1,000x10 = 10,000 values in our grid space. We can't represent it graphically in a 2D space like we can a 3D table, but it's still possibly to work out - as we just did - how many values would be in it according to how many values our fourth variable has.

Similarly, we can plot the first four variables against a fifth, E. If E has ten values, then we know that the resulting 5D grid space has 10,000x10 = 100,000 resulting values. The five variables intersect at 100,000 points, and it's simple to find out what those are. Say A=1, B=2, C=3, D=4 and E=5. Therefore the resulting value is 1x2x3x4x5 = 120, and the highest possible value is when all five equal 10, so 10x10x10x10x10 = 100,000. See? Not so complicated stuff after all.

Plotting a 3D ABC table against a 2D DE table
So now let's return to Cartesian space. Descartes, who was the dude who created Cartesian space, originally accommodated it for three variables. In our previous example, we had Variables A, B and C creating a 3D table space. In Cartesian space, we have x, y and z. xy is the 2D space you see to your left, plotting x against y. Just like in the table, we can then plot that 2D space against a third variable, z. This gives us a 3D space to work in, where people can place and manipulate cubes, spheres and other things. However, the genius of it is that we can also place one- and two-dimensional shapes in 3D space, such as lines and squares. The only difference is that instead of a point being defined by (x,y), its coordinates include a third variable: (x,y,z).

Side note: It's ironic to think that we have an infinite number of values in the xy space, then when we plot them against a third, infinite variable (z), we are left with ∞x∞ values. Which turns out just to be infinity again.

But, just as 1D and 2D objects can be displayed in a 3D space, that same 3D space, full of complex 3D objects, can be displayed accurately in 2D spaces. The only problem is, 2D spaces only plot two variables against each other. We need three. So we plot x against y on one 2D graph, then y against z on another 2D graph, therefore showing the value and location of all the points in all three dimensions. This solution needs two graphs as opposed to one, and it's much harder to understand the location and size of shapes, but it could mean the difference between thinking two points as the same and realising that they are incredibly far away in the z plane.

You can see there I'm using the point (2,2,2) as an example. Luckily for us, the xy coordinate (2,2) has a z coordinate of 2. That's not too hard to comprehend. Take a point at 2 on a 1D plane, extend it into a 2D plane by 2, then extend it out again into another, 3D plane, creating a (2,2,2) coordinate.

But what if we want to show a line in this 3D space? I'm not going to try showing it in the xyz space in a 2D format as above, because it's hard to tell where the points are, so I'll use two 2D graphs to show one 3D graph. Here we can see a line that runs from (2,2) to (3,3). Quite a basic line, right? Well, we may misinterpret it! If we then show that line in another graph, plotting y against z, then we see that it runs from (2,2,2) to (3,3,17). It's longer than we thought. NB. we could also have plotted x against z, the line runs from one point to another in that plane too. It's just a different way of showing where the point lies. However, for future developments (it gets complicated later on), we will follow a rule to plot the last variable against the next one, ie. in xyz, plot xy and yz, not xz. You'll see why later.

So that's alright to understand. Coolio. But, when we were dealing with tables way back at the start of the post, we extended our 3D table space by presenting another variable, D, and plotting A, B and C against D. ie., the space was called ABCD. It turns out that you could do this pretty easily in Cartesian space too, though I don't think Cartesian space was designed to deal with a fourth variable. Our fourth variable is the value of a point, or line, or whatever, in a fourth dimension. 'Fourth dimension!?' you say, 'that's crazy talk! How can we ever interpret or understand that?'. Well my little non-believer, we can. We can't display a 3D graph on a 2D screen (such as the one you are reading from), because each point would correspond to an infinite number of points if that 3D graph were real. That's why I said it was bad form to keep using that '3D' graph to show xyz spaces. So I used two 2D graphs to = a 3D graph instead.

And we can do just the same with a 4D graph. Only, because we now have four variables, we have to show three variables on one graph and two variables on another. We use a common variable that appears on both graphs to link them together. So, let's take a look at this:

This is a 2D (just about) representation of a 4D space. The first graph is easy; it's a simple 3D graph (yes, in a 2D format, but just ignore that for now). We're plotting x against y (as per usual) against z. That gives us a three-dimensional space. But what if that point, (2,2,2), had another value? A value in a fourth dimension? Well, just as we took the y from xy and plotted it against the new variable, z, we're gonna take the z from xyz and plot it against our new fourth variable, w. (Wikipedia uses w, BTW, it's no personal thing). So we plot a second, 2D, graph of zw. And from that we can see that the point (2,2,2) has a value of w=2. Therefore its xyzw coordinate is (2,2,2,2). It's quite simple, really.

And, just as before, we can use multiple graphs of the same multidimensional space to show the true values of lines. Just as before, I'm going to switch back to only using 2D graphs when I'm representing lines in higher dimensions, because otherwise it's really hard to display it properly. So, I've used three 2D graphs here. We need to show an xyzw space, so we plot xy (you'll be sick of that by now), yz (remember that?) and zw. Once again, we see that the first few dimensions are very deceptive. In the xy plane, we see the points on the line are (2,2) and (3,3). In the xyz plane very little changes as we see their coordinates are (2,2,2) and (3,3,3). However, extend that line into a fourth plane, a fourth dimension, and we see that the points are (2,2,2,2) and (3,3,3,20). That second point is in fact further away than we thought.

But does that make the line longer ? We can still measure it in cm, it's still a 1D shape, but in a 4D plane. Perhaps. There's a bit too much mindfuckery there for me.

So now we reach the final throes of this horrendously long post. But before we go, I have one final thing to show you. We've seen how easy it is too understand and graphically represent 4D space, whether they exist physically or not. So, using our advanced understanding of simple 2D and 3D graphs, we can therefore represent even higher dimensions. Be prepared to be confused.

So now we're plotting five variables against each other. Remember we need to display all five, but repeat one to link the two graphs together. That means we need a series of graphs to display six variables. Therefore, we need two 3D graphs. One one graph, we plot our usual 3D space of xyz. On the second graph, we continue z from the first to link the two. We plot z against w (the fourth dimension) against our newbie, ŵ. The first graph tells us the point is at (2,2,2), and the second graph tells us it's at (x,y,2,2,2). Combine the xyz and zwŵ spaces together and we can produce a (x,y,z,w,ŵ) coordinate of (2,2,2,2,2). It's surprisingly simple to do!

I won't go on, because from now on it's all boring and samey. Need to display an eighth-dimensional point? Sure thing! Eight dimensions means (theoretically, just choosing symbols now) (x,y,z,w,ŵ,ŵ',ŵ'',ŵ'''). That means plotting graphs xyz, zwŵŵŵ'ŵ'', and ŵ''ŵ'''. And that's all there is to it, for an infinite number of dimensions, forever and ever. Of course, this is all purely mathematical, and hypothetical, so that means that there may not be a 58th dimension, however many shapes you can represent in it. Different theories give us different values as to how many dimensions actually exist (ignoring time, which isn't representable in Cartesian form, haha); some say 16, some say 20. If you ask me, the nature of the universe seems infer that there would be an infinite number of dimensions, partially because we can work with an infinite number of dimensions graphically (as shown above). I also think that something so key to the being of the universe shouldn't be so finite, so human-defines as simply 16. To me, it seems crazy to put a cap on the number of planes in which matter - and antimatter and presumably dark matter - exists and is perceived.

But that's a rant for another day.



Alistair said...

Very interesting John,

Just as an aside, calculating the length of a four dimensional line would not be difficult. We just apply Pythagoras' Theorem in the fourth dimension.

So the length of the line is:


Alistair said...

One of the main problems with infinite dimensions is that mathematically it can work as we are able to theorise in n dimensions and let n tend to infinity. In physics it doesn't really work as then there are problems involving infinite energy :/

That is why they theorise caps

John said...

Ah, I understand that. So it would be just as easy to work out the area of a square in 4 dimensions or a volume of a cube. Or maybe the hypervolume of a tesseract, if it only had sides of length, say, 2cm, so 2x2x2x2=16cm^4. That kinda thing.

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