Difference between revisions of "Dimension"

Revision as of 17:18, July 5, 2008

The dimension of a geometric object is the number of independent parameters needed to specify a point on that object.

For example, a line is 1-dimensional, since if you fix an initial point on the line, and choose an orientation for the line (a notion of right and left), then every point on the line may be identified in terms of a single parameter: that point's distance to the right or left from the initial point.

Similarly, the surface of a sphere is 2-dimensional, since one requires 2 independent parameters to specify the location of a point on the surface: the latitude and longitude.

In classical physics, the set of all events is 4-dimensional, since in order to specify an event, one needs to know both the position of the event in space (3 parameters), and the time the event took place (1-parameter).

Mathematical Definition

A real vectorspace V said to be n-dimensional if V has a basis of size n. Note that this implies that V is isomorphic to n-dimensional Euclidean space. Thus, in agreement with the heuristic definition of dimension given above, V is n-dimensional if every point of V is uniquely determined by n real numbers.

A manifold M is said to be n-dimensional if the tangent space to M at every point is an n-dimensional vectorspace. Note that this implies that M can be covered by local coordinate systems where each coordinate system looks like a region of n-dimensional Euclidean space. In other words, locally, every point on M can be specified using n independent parameters.

Extra Dimensions in Physics

Some models in high-energy physics postulate that space-time has more than 4-dimensions. For example, in the Kaluza-Klein model, a point is specified by a point in 4D-spacetime, plus a point on a circle attached to that point in 4D-spacetime. Though this model is currently out of favor, it illustrates the basic flavor of additional spatial dimensions in the other models; for example, in string theory.