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The Banach--Tarski theorem applies only in the case of three or more dimensions. In 3D, there are five regular solids, two of them being not at all obvious, and the 4D case is also interesting; but the higher-dimensional cases each yield just three "solids", only one of which isn't obvious. In dynamics, particles tend to coalesce in one or two dimensions, while in four or more dimensions they tend to disperse drearily; only in 3D do they move freely but with significant local interaction. And the Poincare conjecture proved to be much harder in the 3D case than in the others.

So the question has three parts: What other examples are there of 3D richness? Are there any underlying reasons for it? And are there fields where richness begins at a higher, but still small, number of dimensions? I would be particularly interested to learn of any theorems that hold in just a particular number (not 0, 1, or 2) of dimensions.

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Most of the examples already given in this question are already contained in this MO question: – damiano Aug 23 '10 at 8:39
Careful - the smooth Poincare conjecture is settled in all dimensions except for dimension four. (In particular it is false in high dimensions. Google "exotic sphere".) – Sam Nead Aug 23 '10 at 13:05
This question is what I'm going to link to, to get people interested in the Math Stack Exchange. Really great thing to ask. – isomorphismes Dec 12 '10 at 7:13
Just out of curiosity, which of the 5+ -dimensional regular polytopes do you consider non-obvious? – Ilmari Karonen Aug 18 '11 at 12:47
@Ilmari: In $n$ dimensions,the regular ($n+1$)-simplex obviously generalizes the $n=2$ (equilateral-triangular) and $n=3$ (regular-tetrahedral) cases. The $2^n$ vertices of a hypercube can easily be modelled by assigning to them all possible coordinates $(x_1,\dots,x_n)$ with $x_i\in$ {$-1, 1$} ($i=1,\dots,n$), obviously generalizing the square and cube. But its dual, like the octahedron, is less obvious (at least for those, like me, who don't find duality obvious!). – John Bentin Aug 18 '11 at 18:35

This is a well-known phenomenon among topologists, and although I'm not an expert, I'll give one standard answer: 3- and 4-dimensional topology are very different from topology in 5 or more dimensions because surgery theory works in 5 or more dimensions. In 3 and 4 dimensions one does not have enough "wiggle room" for surgery theory to be effective and this is responsible for some anomalous behavior. This leads to some remarkable phenomena connecting 3-manifold topology to other branches of mathematics, some of which are listed in the Wikipedia article, and it makes the 4-dimensional case rather special as well.

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Exotic $\mathbb R^4$

There are infinitely many non-diffeomorphic smooth structures on the topological space $\mathbb R^n$ if and only if $n=4$. (Otherwise there is only one diffeomorphism class.)

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Thank you. It seems that 4 dimensions are as interesting as 3. – John Bentin Aug 23 '10 at 13:38

There is a very specific reason why one needs 3 dimensions or more for the Banach Tarski paradox. In dimension 3 or higher one can make rotations in independent directions, and so the group $SO(3)$ of rotations of space contains a copy of $F_2$, the free group on two generators. This fact is what underlies the Banach Tarski paradox. (The group $F_2$ is not amenable.)

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Hurwitz's theorem

The only normed division algebras over the real numbers appear in dimensions 1, 2, 4 and 8.

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One example of $n \geq 2$-D richness is simply the fact that $n$-by-$n$ matrices don't commute.

This is trivial, but it means for example that every nonabelian finite group has an irreducible representation of degree $>1$.

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Why «in particular»? – Mariano Suárez-Alvarez Aug 23 '10 at 22:14
Yeah, "for example" probably fits better. – Akhil Mathew Aug 23 '10 at 23:23

I think it's to do something with that in dimension 4 and above, you can have 2 commuting rotations about different axes (e.g. rotating around the xy-plane or zw-plane), so that things can be 'broken up' into separate pieces.

Related is the fact that you need 3 parameters to describe a rotation in 3-space, but 6 in 4-space.

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More "degrees of freedom" in hyperspace, then? – J. M. Aug 23 '10 at 11:00
That's very interesting! – John Bentin Aug 23 '10 at 13:44

The following is just a toy example and of course not as deep as the examples already mentioned:

One-dimensional space is special in the sense that $\mathbb R^1\setminus\{0\}$ is not connected---it has two connected components. In all other dimension $n\neq 1$, the space $\mathbb R^n\setminus\{0\}$ is connected.

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What are you trying to point out? – isomorphismes Dec 12 '10 at 6:46
@Lao Tzu: Which part of my statement is unclear to you? – Rasmus Dec 18 '10 at 1:03
Are you saying that the 1-D case is "rich" in some sense? – isomorphismes Dec 18 '10 at 7:32
@Lao Tzu: In the sense I described maybe yes. I think the word "special" would be more appropriate. – Rasmus Dec 18 '10 at 8:27

I'm going to be a bit contrary: is this really true? Or is it just that we started studying the 3D case before we could study anything higher-dimensional, and that our brains are obviously optimized for thinking in three dimensions?

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Kudos for being contrarian but the 20th century saw a rapturous study of $\geq 4$-D. – isomorphismes Dec 12 '10 at 6:45

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