Is it possible to model $n$ dimensions with $n$ greater than 3?

Question

I've recently started a self-study of linear-algebra and the inability to visualize $n$ dimensions where $n$ is greater than three has been brought to my attention. I was aware of this limitation, but I had never really thought about it until I had a look at $\mathbb{R}^3$ vector spaces. So my question is: have there ever been any notable attempts at modeling $n$ dimensions? Obviously through computers.

Answer 1

For dimensions greater than $3$, I don't know how most people visualize this but for the space between points I think of a grid with the origin of a $3$ dimensional coordinate frame at every unit intersection.

This helps me in visualizing a line from say $(1,2,3,4)$ to $(3,1,-2,2)$ in $\mathbb{R}^4$, which is indicated by the line between red dots in the diagram. This diagram is good for up to $\mathbb{R}^6$. After that I start imagining these grid frameworks located at each unit intersection on another larger grid and so on.

I can visualize every vector as a $3$D line but at different coordinate frame locations. I don't know whether this will help or confuse you.