# When Higher Dimensions Help

I'm interested in examples of situations that are easier in higher dimensions. To give a flavor of what I am looking for, here are two of my favorites:

(a) In dimensions 2 and higher, one can characterize the standard normal distribution (up to a constant multiple) as the spherically symmetric distribution with independent marginals. Obviously, such a characterization fails miserably in one dimension.

(b) In dimension 3 and higher, you can prove Desargues' Theorem using the incidence axioms. No such proof works in two dimensions (and there are non-Desarguesian planes).

What are some other nice results where having at least $n$ dimensions allows one to prove things or characterize things in ways that are not possible in fewer than $n$ dimensions?

• Maybe not a proof of something but still an interesting fact in higher dimensions: The Klein Bottle does self intersect itself in three dimensions but not in four. (en.wikipedia.org/wiki/Klein_bottle) – YukiJ May 15 '18 at 11:16
• Contour integration in $\Bbb C$ to evaluate some definite integrals over $\Bbb R$ that are otherwise fairly intractable. And another one: Let $J=\int_{-\infty}^{\infty}\exp(-x^2)dx.$ Then $J^2=\int_{\Bbb R^2}\exp (-x^2-y^2)dxdy.$ Now with a switch to polar co-ordinates, we easily find $J^2=\pi.$ – DanielWainfleet May 15 '18 at 11:59

Here is an example which is easier for higher genus. The Four Color Theorem for the complex plane is much harder than the analogue result for surfaces with higher genus $g\ge 1$. In fact, the maximum number $p$ of colors needed depends on the (closed) surface's Euler characteristic $χ$ according to the formula $$p=\left\lfloor {\frac {7+{\sqrt {49-24\chi }}}{2}}\right\rfloor .$$
The Poincaré conjecture is much easier to prove for its generalization to higher dimensions. Actually, the first proof was for dimension $n\ge 5$ by Smale in $1960$. Michael Freedman solved the case $n = 4$ in 1982 and received a Fields Medal in 1986. Grigori Perelman solved case $n = 3$ in 2003. This was still possible to prove, but barely so.