Pi in the Mandelbrot set everywhere

TL;DR

This paper proves the occurrence of the mathematical constant pi at specific points in the Mandelbrot set, extending the understanding of this phenomenon to all bifurcation points with a conceptual proof.

Contribution

It provides the first proof of pi's appearance at certain parameters in the Mandelbrot set beyond known points, generalizing to all bifurcation points with a conceptual approach.

Findings

Pi appears at parameters c=-3/4 and c=-5/4 in the Mandelbrot set.

A conceptual proof is established for these phenomena.

The proof is generalized to all bifurcation points.

Abstract

The numerical phenomenon of $\pi$ appearing at parameters $c = 1/4$, $c=-3/4$ and $c=-5/4$ in the Mandelbrot set $\mathcal{M}$ has been known for over 30 years. In 2001, the first proof was provided by Aaron Klebanoff for the parameter $c=1/4$. Very recently in 2023, an even sharper result for $c=1/4$ was proved using holomorphic dynamics by Paul Siewert. This new proof also provided a conceptual understanding of the phenomenon. In this paper, we give, for the first time, a proof of the known phenomenon for the parameters $c=-3/4$ and $c=-5/4$, which is also conceptual, and we provide a generalization of the phenomenon and the proof for all bifurcation points of the Mandelbrot set.