Preface: To be specific, the following is written about Euclidian space. However, I'm interested in what's going on in other spaces as well.
I've been told several times that $1d$ and $2d$ spaces are somewhat special, i.e. some mathematical properties tend to have abnormalities in these dimensions. These are some I know:
- (For $2d$) Complex analysis, which is a distinct topic in mathematics by itself, exists purely in $2d$;
- (From partial differential equation theory) The solution of the homogeneous wave equation differs depending on the dimension. For $3d$ the solution is the Kirchhoff formula where there is an integral over a sphere which causes a sharp front. For $2d$ the solution is the Poisson formula where the integral is over a disk which causes a non-sharp front. For $1d$ the solution is the D'Alembert formula where the integral is over a line which again causes a non-sharp front.
- (Again from partial differential equation theory) The fundamental solution of the Laplace operator is $\frac{1}{2\pi}ln|x|$ for $n=2$ and $-\frac{1}{(n-2)s_n|x|^{n-2}}$ for $n\geq3$. Unfortunately, I don't know the solution for $n=1$.
The questions:
- What causes these dimensions to be so special? Is there a hidden meaning?
- (For the sake of interest) If you know, please suggest some other examples of such abnormalities, if any exist.