Pardon me if this is inappropriate. I have no mathematics background, so this may be a naive question.
Why do we say that space is three dimensional? What does it mean?
For example, people use x, y and z axis and positions along each of these axes to describe a point in space (Just considering space). What is special about these axes being perpendicular to each other?
Is it possible to have a different system where the location of a point can be accurately and uniquely described by say two numerical variables instead of three?
I hope my question is clear.
Why do we say that space is three dimensional? What does it mean? Because the least number of vectors we may scale (multiply by a constant; for instance, I may scale $2$ by $3$ to obtain $2 \times3 = 6$) and add together to arrive at any point in 3D space is $3$. If you are unfamiliar with vectors, here is a simple analogy: pick any point in space. How do you get there? Surely, you may walk straight there, but let us say you want a simple way to describe how you would get anywhere. If that was the case, you would say that to get anywhere, you are doing a combination of the three: going up, going right, and going forward. What about down, left, and backward? You are simply doing the former 3 motions in reverse; negative, if you will. If I only have 2 directions (for instance, if I may only go forward and up) then there are points to my left and right which I cannot get to. If I may only go forward and to the right, there are points above me I cannot reach! With exactly three directions, I may get anywhere in 3d space.
The analogy carries over for two dimensions. By only going up and to the right, I may get anywhere on a 2D graph. But, if I could only go in one direction (for instance, say I may only go to the right, or perhaps I could only move parallel to a line) then I could only move along a single line, not the entire 2D space. However, by adding a second possible direction of motion, I may now find some combination of these two motions (keeping in mind these can be 'negative' in other words, backwards in that direction) to get anywhere.
What is special about these axes being perpendicular to each other? We like to describe points as $(a,b,c)$ in 3D space because it tells us exactly where they are; $a$ units in the $x$ direction, $b$ units in the $y$ direction, and $c$ units in the $z$ direction. This is analogous to the "up, right, and forward" analogy above; there are three basic ways to move that I can use to reach any point in space. Why are these axises (axes?) perpendicular? Well, consider if they were not. For instance, say I could only move up, forward, or left AND up (at the same time). How would I get directly to my left? I cannot simply say 'go left one unit' because that is no longer an option. I would have to say 'go left AND up, for a distance of $\sqrt{2}$ (because this is the diagonal length of the unit square) and go up $-1$ units." That works, but it is more complicated; why not just let the third allowed direction of motion to be independent of the other two, which allows me to simply say 'go LEFT one unit'?
Is it possible to have a different system...? Sure! Why not the Cartesian coordinate system, where we assign points a position based on their $x$ value and $y$ value? If you mean 3D space, there is not a way to do this. The best you can get is a line. Consider a new coordinate system in 3D space, call it $(\phi,\theta)$ where $\phi$ is the angle made with the $z$ axis, and $\theta$ is the angle made with the $x$ axis. This will orient us along some line, but we cannot determine where along that line our point would be! So, if the length is indeterminate, all we can produce is a line, where any point along that line makes a line with the origin that satisfies our angles $\phi$ and $\theta$. If we wanted to produce only a point, we would introduce a third variable, call it $\rho$. Then, we may shove this into our coordinate system and make it $(\rho, \phi, \theta)$ where $\theta$ and $\phi$ give us the same line, but now $\rho$ describes 'how far along that line we go, with negative values taking us backward,' thus giving us exactly one point.
In fact, $( \rho, \phi, \theta )$ is the coordinate system used in 'spherical coordinates' to describe the position/shape of things in 3D space that is sometimes used instead of traditional $(x,y,z)$ coordinates. Relevant Wikipedia page: https://en.wikipedia.org/wiki/Spherical_coordinate_system.