Is there any distinction between a line along the $X$ axis in a $1\mathrm d$ space as opposed to one in a $3\mathrm d$ space? Is there any distinction between a line along the $X$ axis in a $1\mathrm d$ space as opposed to one in a $3\mathrm d$ space? I was wondering if they were both the same since they have the same attribute, but are placed in a different space. Or is it? I am no mathematician, so I wanted to ask this. This is either a really dumb question or a really good one, but I don't know which it is.
 A: There isn't any particular difference in terms of the geometry of the line, however one could say, for example, that the line in $1$ dimension can be thought of as all multiples of the number $1$, whereas the line in 3 dimensional Euclidean space (which we call $\mathbb{R}^3$) is all the multiples of the vector $\begin{pmatrix}1\\0\\0\end{pmatrix}$.
This can give us a little more information about the line, especially with regard to perspective. If you imagine positioning yourself so that you are looking towards a line in 3 dimensional Euclidean space that is in the $xy$ plane, but your line of sight is going right towards the origin (and is normal to the origin), then you would not be able to determine whether or not the line had any "depth". In essence you are "projecting" this 3 dimensional line onto the plane, at which point it would end up looking like the $x$-axis, even if it really isn't.
So if a line really looked like this:

It could end up looking like this, if you are looking at it in a specific place:

A: Yes, obviously there is some distinction, otherwise we would be unable to distinguish between them.
One of these distinctions is that for the one-dimensional line, it really doesn't make sense -- indeed, it's impossible! -- for denizens of the line to go out of it, whereas for the three-dimensional line, denizens of the space can possibly step out of this line and observe it from the outside.
