OK, so what is a circle? What are imaginary numbers?

What is this:

pi*r^2 = Area, and volume is derived.

Start with a finite set of things in crowded together within an environment.  They have a center and must be connected. The approximation to Pi is a result of that condition.  The model of the circle we humans crated is a grammar, pencil and paper, and it means: What happens if the number of crowded things goes to infinity in a small space. But be careful, nature takes the square because of causality and finite systems.  Hence, the Shannon Nyquist rate makes causality work and generate optimum divergence of neighboring combinations, and that makes pi. In a non uniform environment, the sample rate would vary, or the system would superimpose two segmented divergence processes.

a + ib, and imaginary number. It means a recombinations happened and b recombinations did not. The total number of actions is partitioned into those that have happened and those that have not happened. a^2 + b^2 is a number of thing optimally  packed to maintain connectivity. One is a packing of actions that did not happen, the other a packing of things that did happen. The unit circle which counts both action that have happened and actions that have not assumes the unactions and the actions were optimally packed together.

What is -i?

That tells us that if exp(-i) is to be the inverse of exp(i) then those actions that have not happened,i, will always have not happened.  Its about happenings (recombinations); they are [ future, past], and [never, always]. Kind of like a Feynman diagram. I have been looking into these lately because they resemble and exchange between two Lucas angles.

Imaginary numbers and Newton calculus.

As I often point out, Newtons calculus made the premise that all power series converge uniformly.  That implied a very simple grammar, if you used the transcendentals for symbolic multiply.  But that assumption thus needed to include actions that have not happened, so the grammar balances.

Lucas polynomials:

The Lucas polynomials, for example, at x=1, all the necessary combinations have been made to generate the Lucas numbers.  But at x < 1, the polynomials generate the actions taken and the actions not yet taken. Since the hyperbolic process is a continual approximation process we would expect some wandering about the solution set, government by the Lucas polynomials.  So take the polynomials for some order, n; and compute its derivative as a combination of Pn and Pn-1, perform the cosh^2 - sinh^2 = 1, and find the solution set; a path of combination of actions taken and actions yet to happen.

Folks, we are doing math differently and these are exciting moments.