Dimensions

This is another of my sort-of ranty posts. I was listening to the radio the other day, and heard an interview of Lisa Randall. Professor Randal is a mathematical physicist who's work focuses on the implications of extra dimensions. Things like string theory have suggested the possibility that our universe has more than the three spatial dimensions + time, but that we are unable to perceive them. The physics of this are fascinating, but I'll leave physics discussions to people better qualified to talk about them. What bugged me was the host's inability to grasp the concept of what a "dimension" is; and that's a problem that I've seen in a lot of discussions.

Bad Sci-Fi has, for a long time, babbled about "alternate dimensions", or "parallel dimensions", where by "dimension" they mean, roughly, "Universe". That is a wretched misuse of the term, which has almost no connection at all with the real meaning of the term. That's what the host of the radio show thought dimension meant, and no matter how many times she pointed out that that was wrong, he just couldn't get past the scifi misdefinition of dimension. (He kept asking questions like "Would there be other dimensions be just like ours, except that the Nazis won WW2?")

Mathematically, dimensions have a lot to do with a field called topology. Topology is, in my experience, the most mind-bendingly difficult field of mathematics around. Fortunately, you don't need to know much about topology to understand what a dimension is.

I'm running in circles a bit here, but there's a reason for it. Dimensions are hard to explain; they're such a fundamental part of our normal experience that we don't really consider what they mean in our perception of the world - and yet the ways to define them without going into the formal math of things like topology need to connect to that subconscious comprehension.

So, to stop goofing around, what is a dimension?

One way of saying it is that it's a direction. Think of a two-dimensional plane: You can move anywhere on the plane - between any two points - by moving either forwards or backwards in two, perpendicular directions. Think of our three dimensional experience of the world: you can move between any two points in three dimensions by moving left/right; up/down; or forward/back. There are three fundamental directions, and thus three dimensions.

The problem with saying it's a direction is that it implies a few things that aren't necessarily true:

1. Our idea of direction is connected to our three dimensional world - and even within that, it's mostly connected to two dimensions. We mostly think of directions as north/south-east/west; sometimes we add up-down. Directions, intuitively, are things that exist within our three-dimensional range of perception. But adding another direction beyond those three spatial directions - what does that mean? It's adding a direction that we can't see, that we can't point in, that we can't move in. It just doesn't fit with our basic intuitive sense of what direction means.
2. Direction implies that a dimension is straight - that is, that moving in a direction, even if it's a direction beyond our three dimensions, has the properties of motion in a euclidean space: things like the fact that you can't get closer to something by moving away from it. Or you can draw parallel lines separated by some distance in that direction. Dimensions aren't necessarily straight or continuous.

A better way to explain, but a slightly less intuitive one is to not separate dimensions quite so much. The set of dimensions in a space is the number of pieces of information that you need to identify a unique location in that space.

On a plane, you can put down a coordinate grid, and identify any point on the grid with only two numbers. In a region of three dimensional space, you can specify any location in that space using only three numbers. But getting beyond three, we start to have some trouble.

Try thinking about the colors on your computer screen. Basically, for each pixel on your screen, there are three lights that can be turned on in different brightnesses: one red, one green, one blue. The color of each pixel is determined by the brightnesses of three lights for a pixel: if all of the lights are completely off, the pixel is black; if the blue light is on very bright, the red light is on halfway, and the green light is off, you'll get a bright bluish-violet; if the red light and the green light are on full brightness, and the blue light is on halfway, you'll get a sort of pastel green. So the colors of your computer screen have three dimensions.

Try thinking about the flavors of food. It's commonly asserted that we humans have the ability to perceive sweet, salty, bitter, and sour flavors. Every flavor that we can experience is some combination of sweet, salty, bitter, and sour. Therefore, flavor has four dimensions.

Try thinking about how to describe a personality. There's a common personality test (the Meyers-Briggs) that measures a personality in terms of four traits: extraverted vs introverted; sensing vs intuitive; thinking vs feeling; and judging vs perceiving. That's a four dimensional description of personality.

Now, try looking at this post. I start off in the beginning by spiraling around the subject of dimensions before changing direction and actually talking about what it means. You can say that there are two different dimensions in this post; and one of them is not straight. It doesn't move in a straight line; it squiggles around, never getting very far from where it started.

Most versions of string theory say that the extra dimensions are all curled up in tiny little loops with a particular kind of topology - just like my discussion of dimensions starts off with little loops. Professor Randal thinks otherwise; that the other dimensions might be just as big as the spatial dimensions we perceive. But either way, the dimensions are the mathematical/topological sense of how many bits of information we need to locate a single point in the universe.