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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: http://mathoverflow.net/questions/5372/dimension-leaps2010-08-23
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    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".)2010-08-23
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    This question is what I'm going to link to, to get people interested in the Math Stack Exchange. Really great thing to ask.2010-12-12
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    Just out of curiosity, which of the 5+ -dimensional regular polytopes do you consider non-obvious?2011-08-18
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    @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!).2011-08-18
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    The vertices of the regular cross-polytopes ("hyper-octahedrons") can also be assigned simple coordinates, specifically all permutations of $(\pm1,0,0,\ldots,0,0)$. They can also be constructed iteratively in the same manner as the regular simplices, except that at each stage you add two new vertices instead of one. In fact, one of the simplest (IMO) parametrizations of the $d$-simplex is as one facet of the $d+1$-cross polytope, with all permutations of $(1,0,0,\ldots,0,0)$ as vertices. (Of course, you can then project it down to $\mathbb R^d$ if you want.)2011-08-18
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    @Ilmari. OK, that's simple enough. Anyway, accepting your point only sharpens the view of how singular the 3D case, with its far-from-obvious dodecahedron, really is.2011-08-18

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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 3.2010-08-23
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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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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»?2010-08-23
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    Yeah, "for example" probably fits better.2010-08-23
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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?2010-08-23
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    That's very interesting!2010-08-23
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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?2010-12-12
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    @Lao Tzu: Which part of my statement is unclear to you?2010-12-18
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    Are you saying that the 1-D case is "rich" in some sense?2010-12-18
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    @Lao Tzu: In the sense I described maybe yes. I think the word "special" would be more appropriate.2010-12-18
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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.2010-12-12