What divides one dimension from another ?

[36grit]

What divides the dimensions?

What defines them ?

 

I can understand length, width, depth, and time, and these are all just a matter of perspective and relativeity.

 

How are these extra dimensions linked together?

[Xittenn]

I think the brief answer to this question is Linear Independence as defined by Linear Algebra, which is the same as and is taken for granted when we regard [math] R^3 [/math] space.

[ajb]

What divides the dimensions?

What defines them ?

 

Let us rearrange the questions....

 

What defines them ?

What divides the dimensions?

 

So, by "the dimension of a space" we mean the number of numbers needed to specify a point. A point on the line [math]R[/math] needs just one number to specify it: [math](t)[/math] (say).

 

The two dimensional plane, [math]R^{2}[/math] needs two numbers to specify any point: [math](x,y)[/math] (say).

 

Then, the space [math]R^{n}[/math] needs n-numbers to specify any point: [math](x_{1}, x_{2}, \cdots, x_{n})[/math].

 

In physics we encounter more complicated spaces than these, however these have the nice property that they locally (think of any small piece) look like [math]R^{n}[/math] for some n. These spaces are known as manifolds.

 

What divides the dimensions? In truth nothing. In general one will have to consider transformations ("changes of coordinates") which mix the initial separation as defined by our choice of coordinates. That is, there is no unique choice of coordinates, (there may be classes of coordinates singled out).

Edited March 8, 2011 by ajb