Extension of Up/Down/Right/Left in n dimensions

Question

I'm wondering if there are extensions of the right/left/up/down concepts in n dimensions, i.e. "if $x_0$ points ahead of me, I'm looking LEFT/RIGHT along $x_1$, UP/DOWN along $x_2$, XXX/YYY along $x_3$". I can't find anything, even just ideas from random people.

I'm sorry if this is off-topic but I think the question is best in math than in linguistics as it's the math concept I'm looking for.

Answer 1

I found the answer: it's Ana / Kata for the 4th dimension

They mean up and down in Greek

Answer 2

In $n$ dimensions we can represent the directions or dimensions as vectors.

Starting with $n=0$ in $n$ dimensions the $n$th vector $\vec x_n$ can be expressed as $$\vec x_n=\begin{bmatrix}x_0 \\x_1 \\x_2 \\x_3 \\x_4 \\\cdot\\\cdot\\\cdot \\x_n \\\end{bmatrix}$$

Answer 3

The extension of the concepts UP/DOWN and LEFT/RIGHT is, if you describe a vector (or point, once you have an origin) thanks to its coordinates $ (x_1, ..., x_n) $ :

$x_i \geq 0 / x_i \leq 0$.

I have never seen a specific term for this, and I don't think it exists. It is up to you to define it and make people use it (something like Up($i$)/Down($i$)).

In what context do you need this?