Hello,

In my country, we learn 3D spaces in the final year of high school

The four dimension space is mostly use by physicists, I guess it is in relativity

As for mathematics, there is an infinity of dimensions. For example, matrices illustrate this, since their dimension can be any nonnegative integer n.

There are a lot of applications (in analysis, algebra or calculus...) where you use $\displaystyle \mathbb{R}^n$. An element of it is in the form $\displaystyle x=(x_1,x_2,\dots,x_n)$ where n is arbitrary.

And there are quite a lot of applications, but we don't quite use it for graphic representations.

Example :

The distance from a point $\displaystyle x=(x_1, x_2, \dots, x_n)$ to another point $\displaystyle y=(y_1, y_2, \dots, y_n)$ in $\displaystyle \mathbb{R}^n$ is defined as being :

$\displaystyle \left(|x_1-y_1|^n+|x_2-y_2|^n+\dots+|x_n-y_n|^n \right)^{\frac 1n}=\left(\sum_{i=1}^n |x_i-y_i|^n\right)^{\frac 1n}$

(you can check that this works for p=2 and gives the formula you know).