Mad science: changing pi

[Radar]

Now I want to see the algebra constructed around a field whose numbers allow this property to vary. I suppose it's natural to start with the complex numbers since that has a relationship between derivatives and values if you're working with harmonic functions... And it also connects products to rotations.

So maybe something like extended complex numbers, where instead of representing a 90 degree rotation, i represents either a greater or lesser angle of rotation? So instead of i^4 = 1, you have i^(4+alpha) = 1, with alpha indicating how detuned things are from Euclidean. Then keep all the complex derivative/harmonic function stuff the same, and see what you get when you write down f(z)'' = -f(z) in that space?

Maybe this is overcomplicating it and if you just have a curved metric over x, its enough. Like solving the wave equation in flat space gives you sines and cosines, but the same equation on a sphere gives you spherical harmonics.

This seems to create a complex space spanned on a cone (with 0 at the pointy top) or whatever the hyperbolic counterpart would be. So you would have a non-differentiable point and a fairly well understood curvilinear space everywhere else. Would require a different definition of the scalar product though as I*(-I) would not be real, but in non-Euclidean spaces that is pretty standard.

Interestingly, there are ways to make weirdly long circles in complex spaces that do not need any redefinition of I. Any multivalued complex function contains specific points (either with value 0 or a singularity) around which the full angle (as in the ratio between a circle's circumference and diameter) is a multiple of 2 pi. For example, simple square root is one of those functions. Any monomial with non-integer power will induce such a space with the number of branches you need to loop around depending on the denominator in the power. Typically z^something irrational is considered to have infinitely many branches that do not loop around, but if you force things to fit onto a single branch, there would some serious shenanigans needed with the arguments as we would have to skip some part of the complex plane or repeat just a part of it in order to connect the values smoothly. How to define such an operation, I do not know to be honest.

Hmm... maybe something weird can be done with modulo rings over real or complex numbers?