Co-ordinate Systems

If we want to think about things geometrically, we need a way to describe where things are in space, where things are in relation to each other and how big they are.

To do that, we can use a co-ordinate system. Linear co-ordinate systems have two properties - units and dimensions. The dimensions correspond to how many different combinations you can make with positions in your space. We will often see dimensions referred to by the canonical variables , , and so on.

There exist infinitely many points in the other corresponding dimensions for a single point on one dimension. For instance, if you had a two dimensional space and you held constant at , there are still infinitely many points you could pick on the dimension. If you had a three dimensional space and held at , there are infinitely many points that you could choose in the dimension, and then after that, there are infinitely many points you could choose in the dimension.

In the first dimension you would just have a number line made up of every possible point:

In two dimensional space, you have a co-ordinate plane made up of every possible line:

In three dimensional space, you have a volume made up of every possible plane:

Dimensions above the fourth are a little tricky to visualize, but the pattern continues. If two-dimensional space is a plane consisting of consists of every possible line and three-dimensional space is a volume consisting of every possible plane, then think about what that means for four-dimensional space. Or even five-dimensional space.

The logic will generalize to an n-dimensional space

For the sake of simplicitly, we will assume that all co-ordinate systems use the same units, meaning that movement of one step along one dimension and that if you rotated a system such that the one dimension became another, the steps would correspond.