Why does it always take n numbers to characterize a point in n-dimensional space (or does it)?

Question

I don't know if this is obvious and a dumb question or not, but, here we go. To characterize a point in 2-d space we can use standard $x,y$ coordinates or we can use polar coordinates. There are probably other ways to do it other than those two as well. It's very interesting to me that those somehow both require exactly two numbers—either an $x$ and a $y$ or an $r$ and a $\theta$. It seems like a magical coincidence to me that these two completely different ways to describe a point require the same number of numbers.

Then moving into 3-d space, there's the same thing. We can use $(x,y,z)$ or $(\rho,\phi ,z)$ (cylindrical coordinates) or $(r,\theta ,\phi)$ (spherical coordinates). These coordinate systems seem to be to function in vastly different ways, and yet they all take three numbers. It's a conspiracy.

So I mean on the one hand, it's intuitive that it should take three numbers to describe three dimensional space. On the other hand, I can't figure out why this should be true. So question a) why is this the case and question b) can we imagine a world where there were points in n dimensions and two coordinate systems that took different numbers of numbers to characterize points?

P.S. I don't really know what to tag this as.

Answer 1

Well, in a silly way, it only takes one number to characterize a point in $n$-dimensional space. It will be easiest for me to explain how to do this with an $n$-dimensional box $[0, 1]^n$. If we write out $n$ numbers $(x_1, ... x_n)$ describing a point in this box using their decimal expansions, e.g. take $n = 5$ and $$x_1 = 0.12345...$$ $$x_2 = 0.33333...$$ $$x_3 = 0.52525...$$ $$x_4 = 0.31415...$$ $$x_5 = 0.27182...$$

then I can write down a single number that describes all of them by interweaving their digits: $$y = 0.1353223217335414321853552...$$

(Strictly speaking this is a small lie because I have not described what to do with numbers like $0.1 = 0.0999...$ which have two decimal expansions, but this turns out to be easily fixable so I'll gloss over it.)

However, this is not a useful way to describe points in $n$-dimensional space; small changes in $y$ may result in large changes in the $x_i$ and it is a huge hassle to have to deal with this. More precisely, the map above fails pretty badly to be continuous, and in particular it is not differentiable, so we can't use calculus in a way compatible with this map (e.g. we can't compare integrals in the two coordinate systems using the multivariate change of variables formula).

If we want a differentiable map that allows us to go back and forth between two coordinate systems, those coordinate systems need to be describable using the same number of numbers. This follows from the fact that their derivatives (Jacobian matrices) need to do the same thing to the tangent spaces. To really understand this, first take a course in linear algebra, then a course in multivariable calculus. (I do not understand why these are usually taught in the other order.)

If we only want a continuous map in both directions, then it is not at all obvious that it still takes $n$ numbers to describe $n$-dimensional space, but this turns out to be true by a difficult theorem called invariance of domain.

Answer 2

The intuitive explanation: degrees of freedom.

Suppose you have 3 dimensional space, you can go 3 meters left, 2 meters forward and 5 meters up (if you have a jetpack ;-) ). You can see that these numbers are independent, you can change each of them freely without affecting the others. In physics we say that in 3-dimensional space you have 3 degrees of freedoms. The number of degrees of freedom is an invariant of the space described.

In n-dimensional space I guess you probably mean to $\mathbb{R}^n$. It has the standard basis $\{ e_1 , ... , e_n \}$ which is the linear algebraic equivalent of cartesian coordinates. One can prove that the number of basis elements is an invariant of the vector space, and this invariant number is called the "dimension" of the space.

Vector calculus expands that notion to non-rectlinear systems of coordinates.

In differential geometry, when one talks about manifolds of dimension $n$ they mean that they locally look likes $\mathbb{R}^n$ and since we can patch systems of coordinates, each has $n$ numbers, it follow that the manifold is described by $n$ parameters.