One-dimensional, two-dimensional, three-dimensional, four-dimensional space, and, in general, $\Bbb R^n$

Question

Often with high school students I find myself confronted with the concept of one-dimensional ($\Bbb R$, where $(x)$ it is a coordinate of a straight line), two-dimensional $\Bbb R^2=\Bbb R\times \Bbb R$ (cartesian plane, $(x,y)\in\Bbb R^2$), three-dimensional $\Bbb R^3$ (like triple coordinates in the space $(x,y,z)$), four-dimensional (our dimension) $\Bbb R^4$ with $(x,y,z,t)$ where $t$ is the time, and $t>0$, and $\Bbb R^n$ spaces.

What idea could be valid to better motivate the meaning of $\Bbb R^n$?

Answer 1

Suppose you have $N$ particles, and to describe each of their location, you need a point in $\Bbb{R}^3$. So, $\xi_1= (x_1, y_1, z_1)\in\Bbb{R}^3$ is the point where the first particle is situated, $\xi_2 = (x_2,y_2,z_2)\in \Bbb{R}^3$ is where the second is situated and so on up to $\xi_N$. If you want to talk about the whole system simultaneously, you need to keep track of everything! You need to know $(\xi_1, \dots, \xi_N)= ((x_1, y_1, z_1), \dots, (x_N,y_N,z_N))$. How much information is there here? Just count: we have $x_1,y_1,z_1, \dots, x_N,y_N,z_N$, which is a total of $3N$ numbers (i.e we're not dealing with $\Bbb{R}$, or $\Bbb{R}^2$ or $\Bbb{R}^3$, but we're dealing with $\Bbb{R}^{3N}$).

So, depending on how much "information" you want to keep track of, you need a higher dimensional space. So, even though we are accustomed to a 3-dimensional world, it makes a lot of sense (to me anyway) to start dealing with $\Bbb{R}^n$ generally for any integer $n\geq 1$.

Answer 2

As requested from the comments:

For me, by far, the best "description" is the one peek-a-boo described. Think of a point in $\Bbb R^4$ as a point in $\Bbb R^3$ along with its temperature. A point in $\Bbb R^5$ is a point in $\Bbb R^3$ along with a temperature and color. Etc.

In theory, you can keep going using pressure, density, intensity, etc.

I'm not sure this will help anyone actually solve problems, but it's a nice mindset to have for an introductory multi-dimensional calculus class, for example. Often times those courses mostly deal with $\Bbb R^3$ and $\Bbb R^4$, and I remember being comforted by having some mental image for $\Bbb R^4$.

Maybe this is off-topic now, but I remember being frustrated for a while because I couldn't "visualize" $\Bbb R^4$. Now I'm of the mindset that nobody can, or rather that "visualizing $\Bbb R^4$" is meaningless because we live and evolved in three apparent dimensions. So I don't think one can do much better than describe $\Bbb R^n$ in terms of a more familiar space, and it might help new students to know that they aren't necessarily lacking some special intuition that everyone else has.