# Dimension Leaps

Many mathematical areas have a notion of "dimension", either rigorously or naively, and different dimensions can exhibit wildly different behaviour. Often, the behaviour is similar for "nearby" dimensions, with occasional "dimension leaps" marking the boundary from one type of behaviour to another. Sometimes there is just one dimension that has is markedly different from others. Examples of this behaviour can be good provokers of the "That's so weird, why does that happen?" reaction that can get people hooked on mathematics. I want to know examples of this behaviour.

My instinct would be that as "dimension" increases, there's more room for strange behaviour so I'm more surprised when the opposite happens. But I don't want to limit answers so jumps where things get remarkably more different at a certain point are also perfectly valid.

• Who reads blogs anymore? They are soooo October-2009ish. – Loop Space Nov 13 '09 at 15:51
• Andrew, I don't see a question in there... – Ben Webster Nov 13 '09 at 16:05
• That's a very nice question. – Gil Kalai Nov 13 '09 at 17:34
• It was a mistake to hastily close the question and indeed I had a few more examples to mention when I will have time. Probably, (unless reopened,) I will simply revive the question in some way. – Gil Kalai Jun 24 '10 at 12:15
• @Gil, I've arbitrarily reopened this question. – Scott Morrison Jun 24 '10 at 15:40

The spaces of sequences of real or complex numbers, $(l^p,||·||_p)$, are not pre-Hilbert spaces unless $p=2$.
The max number of points interconnected (every-to-every) by lines of any curvature, such that no line crosses any other line. For $\mathbb{R}^2$ it is only 4 points (smth. like Mercedes symbol) - why 4 and not 3 or 5? How many points are possible to connect in such way in $\mathbb{R}^3$? (I suggest, infinite number, but it is interesting to look at a proof). What are some special properties of the Euclidean $\mathbb{R}^3$ such that the number of interconnected points jumps from 4 in $\mathbb{R}^2$ to infinity in $\mathbb{R}^3$?