Diminsions in Time and Space

Dimensions can be disputed, both how many they are, what each one is, and how they apply to our lives. A physics professor may disagree with a math professor on the details of the number and explanation of each dimension, which we will get to in a little while. Just to see what will happen, we have invited one of each to attend today's lecture:

I will begin with my rudimentary definitions of dimension:

1st Dimension: a point or line. The point or line has no width, or thickness, and can only exist in one direction.

2nd Dimension: Think about area. the second dimension is like a square or circle; flat, but with 2 directions: width and length. Circles, squares, basic shapes, and even complex shapes can all be 2 dimensional.

3rd Dimension: Now think about volume: cubes, spheres, swimming pools all have 3 directions: width, length, and height. Most of what we interact with on a daily basis is 3 dimensional. Actually, even a piece of paper has 3 dimensions, it is just very thin.

4rth Dimension: Although this can be disputed, Time is the generally accepted 4rth dimension. While time is not a spatial dimension, it can interact with the previous 3 dimensions. Water for example, when placed in a freezer will turn to ice. Was it always ice? As it progresses through time, it will change depending on its environment.

Our mathematician however, loves to complicate things: (and he loves graph paper)

     The dimension of a mathematical object is informally defined as the minimum number of coordinates needed to specify any point within it.
Huh? Lets ask him more slowly to define this:
     When drawing any object on a set of axis (graph paper), if you can refer to a point in your object using a single coordinate (such as 1,2) then the object is in the 1st dimension. Any object which requires 2 coordinates, will be in the 2nd dimension. A box for example, has area, and would need at least 2 points to define it. A sphere, would need 3 points to define it. And so on and so forth. 
Ahh, that makes a little more sense. This sounds very much like our informal definition above. Oh wait, I don't think he's done:
     However, there are strange cases of objects like a unit circle, which are 2 dimensional, but can be defined as being 1 dimensional! A point in a unit circle can be specified by two coordinates, but on a polar (circle graph) you only need one polor coordinate! Thus, it exists in 1 and 2 dimensional space.
 Hmm, I didn't know that. Oh, he's still talking:
     In vector analysis/matrices mathematics there can be many more than 4 dimensions. While these dimensions do not exist as we see, and they do not even define nth dimensional objects, they are used simply to compare points, figures, and sets to one another. 
Okay, I think he lost me in there somewhere...
     Fractal dimension is another example of a unique set of mathematics. Fractal dimension implies some objects may be between dimensions. For example, the perimeter of the Koch Snowflake we examined a month or two ago. The perimeter is a line, which by our definition, is 1 dimensional. However, that line weaves and swerves, and continues expanding indefinitely. This is too complex to be 1 dimensional, and yet has no area like a 2 dimensional object should have. Fractal dimensions have been put to use to define these unique case.
Ack! Okay, lets move on to the physist's, maybe he'll have some straight answers for us. Oh good, our physist is much more practical about all this: well for the time being anyway. 
     There are 3 dimensions which are clearly defined. The first dimension is a point or line, the second has area, the third has volume. Time is often referred to as the 4rth dimension, but is viewed as different in nature to the first 3.  We can *currently* only move through time in one direction.
He agrees with our basic definitions, and will consent that time is often referred to as the 4rth dimension. He can observe things changing over time, has no problem with this. 
He then launched into a 45 minute lecture about string theory and superstring theory, which involved 10 spacetime dimensions. Basically speaking, each new dimension is another force acting on our object. Gravity, for example can be a dimension, or the sun pulling on the planets, or the galaxy pulling on our solar system. This is a very rough introduction, but it led to a 90 minute argument between the mathematician, and our physicist. In the end, they both stormed out, but not before eating all the snacks. :P