*Some* of the quantum theories have many dimensions, but so far only four (three space, one time) have ever been observed. This causes most physicists to at least question, if not doubt, the existence of the other seven.
A degree of freedom is a dimension in which a particle or other object is able to move. This impacts, among other things, the heat capacity of various substances and particles, since heat is vibration velocity and you can put more energy into slower motion if you have more dimensions you can move in. A good way to imagine degrees of freedom is to imagine yourself moving around on the Earth's surface; you have two degrees of freedom, because you have two dimensions in which you can move: north-south, and east-west. Now, imagine yourself with wings and the ability to fly. You now have three degrees of freedom, because you've added another dimension to your ability to move, that being elevation. If you give yourself a time machine that allowed you to move around in time the way you can move around in space, then you could add yet another degree of freedom to your movement.
I use the Earth's surface in this example for a reason: it's a two-dimensional surface, but it's limited in its extent. Thus, you don't quite have two degrees of freedom moving around on it, because if you go far enough in any one direction - namely, the circumference of the Earth - you will wind up in the same place you started. Now, imagine if the Earth's circumference were the length of your stride, so you could travel all of the way around it in one stride. The speculation in string theory is that the other dimensions are like this, not only limited in extent, but small enough that the quantum uncertainty in the position of a particle is as large, or larger than the extent of the entire dimension. Such a dimension would not contribute a degree of freedom to a particle's movement, since it can't really move. However, using our Earth analogy again, there are two points on the Earth where the east-west dimension does exactly this: the north and south poles. The idea, thus, is that there may be parts of the universe where these other dimensions are expanded, and thus do contribute more degrees of freedom, relative to what we know of and are familiar with.
If you get a chance, watch "The Elegant Universe" on Nova on PBS, which is a good presentation of string theory in all its eleven-dimensional glory. :-)If you have access to the internet (or can convince a librarian to let you stay for three hours because it's an educational program), here's the link:elegant_universe
Or search Google for "watch the Elegant Universe". Another favorite of mine when I was in high school was "The Mechanical Universe". It's a bit old, but you might be able to find it online.
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