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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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