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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.

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There was a relevant discussion at God Plays Dice awhile ago:… – Qiaochu Yuan Nov 13 '09 at 15:47
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
@Gil, I've arbitrarily reopened this question. – Scott Morrison Jun 24 '10 at 15:40

33 Answers 33

The spaces of sequences of real or complex numbers, $(l^p,||·||_p)$, are not pre-Hilbert spaces unless $p=2$.

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Is this really 'dimension'? – Spencer Apr 18 '11 at 20:22
Well, the poster said: "Many mathematical areas have a notion of "dimension", either rigorously or naively, [...]", and thus I felt the p-notion to fit the question. – Jose Brox Apr 22 '11 at 11:06

Isn't the Frobenius theorem on real division algebras an example?

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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 $\mathbb{R}^3$?

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R^3 doesn't have any special properties in that sense. The issue is that curves only have codimension 1 in R^2, so they can separate the plane into multiple components, whereas they can't separate R^3. You could probably formulate a similar problem about using surfaces with boundary to connect 1-manifolds (circles or line segments) in R^3 and get a finiteness result for that if you really wanted. – Steven Sivek Dec 18 '09 at 17:42
There are a few finiteness results of that type -- for example Seifert surfaces for knots (orientable surfaces that bound a knot in 3-space). If the knot is a special type "fibers over S1" and if the Surface is required to be "incompressible" (minimal genus) then it's known to be unique. You can of course always complicate surfaces by adding handles. – Ryan Budney Dec 18 '09 at 21:38

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