2
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ 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) $\endgroup$ – YukiJ May 15 '18 at 11:16
  • 1
    $\begingroup$ 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 $J^2=\pi.$ $\endgroup$ – DanielWainfleet May 15 '18 at 11:59
3
$\begingroup$

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 $$ p=\left\lfloor {\frac {7+{\sqrt {49-24\chi }}}{2}}\right\rfloor . $$

$\endgroup$
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.