``...how .. often at those points where mathematics meets the real world that things get the weirdest.... the primes book I am reading reminded me that it is primes, this weird untamed sequence of numbers, and complex numbers...,and random irrational ratios...that find expression in the physical world and address some of our most interesting questions...'' ravi
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What primes book are you reading?
Here is something related to all that, the Riemann Zeta function. I don't really understand it, but it is fascinating. Go here:
http://en.wikipedia.org/wiki/Reimann_hypothesis
Go down a way and first look at the polar graph. Then go down some more to the data plot, or `Absolute value of the Z-function (blue). I spent a few hours contemplating this plot of direct computation solutions. Now try to see if you can imagine the connection between the two images, the polar form and the data plot form.
What I find fascinating is you can see a complex symetry in the absolute value plot. Not the easy refective one about the abscissa but the other one about the ordinant.
What these complicated symetries remind me of are those in a polar, point and circle construction. Go here for a little java appl that generates polars:
http://www.cut-the-knot.org/Curriculum/Geometry/PolePolar.shtml
You can't see it from these, but if you play with this construction for awhile, you will see it is a representation of a conformal transformation. And it is the symmetries of this transform that remind me of the data plot and the polar coordinate graph of zeta-function values.
Here is more advanced view. What's missing in the explanations, is that the inversion circle is obtained following the pole, point, circle construction. The conic sections and algebraic curves all have poles, points associated with them. This follows the hyperbolic view:
http://wwww1.kcn.ne.jp/~iittoo/us23_infi.htm
If you're at all interested in this...? I am writing from the repair shop, so this is a little sketchy.
CG