Archimedes & Euler

A complex function that is its own derivative normalized to one at zero implicitly defines the famous Archimedean and Euler constants of circular motion and exponential growth. Even in a world of strong gravity, where the ratio of a circle’s circumference to its diameter noticeably varied from place to place, this exponential function and the axioms of mathematics would generate these same transcendental numbers.

Assuming the Taylor series expansion

where $z = x + i y \in \mathbb{C}$ with $x,y \in \mathbb{R}$ and $i = \sqrt{-1}$, the derivative condition

coupled with the normalization condition

implies

Numerically plot this expression to discover two jewels. As in the figure, the exponential function maps the imaginary axis to the unit circle $f(i\mathbb{R}) \rightarrow \mathbb{S}$, with negative real parts forcing complex numbers inside (red) and positive real parts forcing complex numbers outside (blue). The function is exponential on the real axis with e-folding time 1, so $f(1)=e$, but periodic on the imaginary axis with period $\tau$, so $f(z+i\tau)=f(z)$. Specifically, as it maps $1$ to $e$, it maps horizontal strips of height $\tau = 2\pi$ onto the entire complex plane, where

and

are associated with Archimedes and Euler.