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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: godplaysdice.blogspot.com/2009/07/… –  Qiaochu Yuan Nov 13 '09 at 15:47
Who reads blogs anymore? They are soooo October-2009ish. –  Andrew Stacey Nov 13 '09 at 15:51
Andrew, I don't see a question in there... –  Ben Webster Nov 13 '09 at 16:05
Time to put this one to bed. (ie, time to close it, I deem.) –  Andrew Stacey Jun 23 '10 at 18:07
@Gil, I've arbitrarily reopened this question. –  Scott Morrison Jun 24 '10 at 15:40
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33 Answers

This is a very nice question. Let me describe the very different behavior of convex polytopes in 3-dimensions compared to convex polytopes in higher dimensions. In three dimensions we have the following facts:

1) Every triangulation of $S^2$ and, more generally every realization of $S^2$ by a polyhedral complex are combinatorially equivalent to the boundary complex of a convex polytope.

2) Every polytope is combinatorially equivalent to a rational polytope - namely to a polytope all whose vertices have rational coordinates.

3) Every automorphism of the face lattice of a convex polytope can be realized by a rigid motion of a combinatorially equivalent polytope.

These statements follow or extend a well known theorem of Steinitz. They are related to the Koebe-Andreev-Thurston circle packing theorem. They all fail very strongly in dimension 4 and higher.

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Brownian motion is recurrent in dimensions 1 and 2, and transient in three or more dimensions.

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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
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