D
DrRocket
Guest
<p>I have noticed occasional confusion in some of the threads with the notion of dimensions and with manifolds. I won't talk about manifolds here, since one first needs to understand Euclidean spaces and dimensions to discuss them, and since they are a bit more complicated. I may tackle manifolds in another thread -- later.</p><p>But let's talk about dimension. Dimension is just a another way of saying "degree of freedom". You walk down the street and are free to go straight ahead, backwards, left or right. Basically you can describe your position on a map with two coordinates. So the surface of the earth on which you walk is basically a two-dimensional plane (we are presently not advanced enough to graduate from the flat earth society). You come to a building and climb a set of stairs, so you have found a third dimension. And surely everyone has been exposed to the idea that in order to call a meeting you need to specify the place (3 dimensions now) and time ( a fourth dimension).</p><p>So to go to a meeting you need the coordinates in a 4-dimensional space (note that the word "space" is used here as it is used by a mathematician or physicist and may have nothing to do with the usual 3 spatial dimensions). But suppose that you are not attending a business meeting but rather are attending the theater. To see the show you need to know not only the place (3 dimensions) and time (a 4th dimension) but also the ticket price (a fifth dimension).</p><p>If you have ever taken a thermodynamics course you have been exposed to the notion of state variables. Or you may have seen them in a course on mechanics and dynamical systems. Let's look at the the latter. To specify the state of a particle you need to know the position (3 variables) and the momentum (3 more variables). So the state space, or phase space, for a particle is 6-dimensional. Now suppose that you have N particles. The description of each requires 6 variables and there are N of them so the dimension of the phase space for the system is of dimension 6N. If you happen to be a control engineer, then the state space for control systems is another very similar example (this is basically where Rudolph Kalman got the idea for state space analysis).</p><p>Here's another way to look at it. If you consider the set of all real valued functions defined on a single point, then any such function is described by a single number and the set of all such functions is 1-dimensional. If you consider the set of all real-valued functions defined on two points, say 0 and 1 then a function is described by two numbers and the set of such functions is 2-dimensional. It you consider the set of all real-valued functions defined on the integers 1,2,3,...,N then that set is N-dimensional. And if you consider the set of all real-valued functions defined on all positive integers (or all integers for that matter) then that set is infinite dimensional.</p><p>Now let's go one step further. Consider the set of functions defined on the integers such that the infinite sum of the values when squared is summable as an infinite series. That set of functions is also infinite dimensional. But you can also use the value of such functions as a coefficient for the function that takes x to exp(-ix) where i is the square root of negative 1. If you do that you capture the theory of Fourier series, for functions that are periodic and that are are square integrable over the unit interval from -pi to pi. So you see that the notion of an infinite dimensional space can have some practical applications.</p><p>So you see, in reality you deal with higher dimensional constructions all the time. There is nothing strange about it, once you understand what a dimension really is. </p><p>If you play a similar game on a finite set of integers, you get the theory of the Fast Fourier Transform. Another application of higher-dimensional spaces.</p> <div class="Discussion_UserSignature"> </div>