i don't understand how that's not simply two dimensions (because the horizontal and vertical measurements are all in the same 'plane' for want of a better term).
is this simply a difference between the layperson's idea of a dimension and the mathematician's?
alasdair
Yes, sort of. I was trying to underline the difference between the mathematical idea of "dimension" and the particular idea of physical
spatial dimension. Physical space has dimensions, and that's what people often talk about, but lots of other things have dimensions.
Here, the key thing is that we're talking about
the configurations of rectangular windows on the screen, not the
locations of points on the screen. The location of points on the screen is two-dimensional, because you only need two numbers to label any point. Just x and y, or alternatively you could use r and theta -- the distance from the corner of the screen r, and the angle from that corner theta. But
the configuration of a rectangular window is different. Two numbers won't do it there. Whether you pick your two numbers to give the location of a corner of the window, or the center of the window, or whatever, they're still not enough to describe it fully. You can always come up with lots of different windows that will have those same two numbers (the same corner location, or center location, or whatever). You need 4 numbers. So the set of rectangular window configurations is 4-dimensional. Yes, each point in a window only "lives" in a 2-dimensional space. But because windows are more complicated objects than points, the set of configurations of windows is 4-dimensional.
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The point is that 'dimension' is a bigger concept than just spatial dimensions, though it applies to space too. Probably I picked a bad example, since it involved space. You can have dimensions without involving physical space at all. For example:
musical tones. Physically, sounds are pressure waves in the air. They have frequencies & wavelengths, like any wave. For example, the frequency of a low A is 110 Hz. Higher tones are higher frequencies; middle C is 261.63 Hz. So if we talk about the space of different tones, then that's a 1-dimensional space. You only need 1 number, the frequency, to define a tone.
You could also talk about the space of sounds made with just one pure tone. Besides the frequency of the tone, the only other thing making up the sound is its loudness, or amplitude. So there's a 2-dimensional space -- frequency of the tone and its amplitude. Pure tones, however, aren't very common -- they're those annoying sounds you hear when you're having your hearing tested. What about notes? Well, a note is not just a pure tone -- it involves many different frequencies of sound all together at once. What defines a musical note is that every note is made up of a "tower" of frequencies
f, 2f, 3f, 4f, 5f, .... When you play a low A on any instrument, you get not just the 110 tone, but also a 220 tone, a 330 tone, a 440 tone, and so on.
So what is the dimension of the
space of sounds made by one note? Well, there's the frequency f of the note. That's 1. Then there's how loud the tone at f is. That's 2. Then there's how loud the first "overtone" at 2f is. That's 3 dimension. But then there's how loud the second overtone at 3f is, and at 4f, and so on for all the others. But there are an infinite number of those overtones! So that's an
infinite-dimensional space.
