Low dimension geometry teaches us that unique cases happen when there isn't enough space. Phase transitions cannot occur in one dimension. One dimensional mathematics like Ising's solution for a magnet system; does not show a phase transition. In two dimensions or more it is possible to have a transition, as Onsager showed in 1942.
Apart from Statistical Mechanics there are other geometric arguments to consider. Hamilton's version of Mechanics can be studied with Symplectic Geometry. Even though the dimensions in this so called phase space, are not the dimensions of space, Symplectic Geometry needs an even number of dimensions. From the dimension point of view, we cannot have a one or three dimensional symplectic structure. I am not saying that the three dimensions of space are explained by the need of the symplectic mathematics to have an even number of dimensions, what I'm saying is that I haven't understood the implications of the number of space dimensions in the structure of reality.
The complex plane is two dimensional. Cartesian products of this structure also give us even dimensional spaces. Quaternions are four dimensional. Now I say Symplectic Geometry also needs even dimensions. I am trying to understand the significance of this.
Poincaré's conjecture in three dimensions was hard to prove. Odd dimensions again.
Imagine that after the big explosion we believe gave birth to all we know, there were all kinds of possibilities for the number of macroscopic dimensions. Why did we end with three space and one time?
This question may be deeper than why we don't have yet a Quantum Theory of Gravity?
And my thoughts go on.
New Proposal March 2011 arXiv
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