Three-dimensionality

It seems surprising that space is three-dimensional. Why so few dimensions?

An anthropic answer seems implausible. Anthropic considerations might explain why we don’t have one or two dimensions—perhaps it’s hard to have life in one or two dimensions, Planiverse notwithstanding—but thye don’t explain why don’t have thirty or a billion dimensions.

A simplicity answer has some hope. Maybe it’s hard to have life in one and two dimensions, and three dimensions is the lowest dimensionality in which life is easy. But normally when we do engage in simplicity arguments, mere counting of things of the same sort doesn’t matter much. If you have a theory on which in 2050 there will be 9.0 billion people, your theory doesn’t count as simpler in the relevant sense than a theory on which there will be 9.6 billion then. So why should counting of dimensions matter?

There is something especially mathematically lovely about three dimensions. Three-dimensional rotations are neatly representable by quaternions (just as two-dimensional ones are by complex numbers). There is a cross-product in three dimension (admittedly as well as in seven!). Maybe the three-dimensionality of the world suggests that it was made by a mathematician or for mathematicians? (But a certain kind of mathematician might prefer an infinite-dimensional space?)

Magnifying sets of real numbers

If S is a set of real numbers and a is a real number, let aS={ax:x∈S} be the set you get by magnifying S by a factor a.

Here's a funny thing. Some sets get bigger when magnified and some get smaller. For instance, if we take the interval [0,1] and magnify it by a factor of two, we get the interval [0,2], which is intuitively "twice as big". But if we take the natural numbers N and magnify them by a factor of two, 2N will be the even natural numbers, and so 2N will intuitively be "twice as small" as N.

Next observe that if R−[0,1] is the real numbers outside the interval [0,1], then 2(R−[0,1])=R−[0,2] is smaller. Magnifying a set by a factor greater than 1 magnifies both the filled in parts of the set and the holes in the set. The effect of this on the intuitive "size" of the set will depend on the interaction between the holes and the filled in parts.

And if we take the Cantor set C, then magnifying it by a factor of three makes the set be intuitively twice as large. I.e., 3C=C∪(2+C). This makes it very intuitive that the dimension of the set is log 2 / log 3 (which is indeed its Hausdorff dimension). For intuitively if we have an n-dimensional set, and we magnify it by a factor of a, its size is aⁿ. So if n is the dimension of the Cantor set, then 2=3ⁿ, and so n is log 2 / log 3.