Pi in the Mandelbrot set everywhere
Dynamical Systems
2025-05-13 v1
Abstract
The numerical phenomenon of appearing at parameters , and in the Mandelbrot set has been known for over 30 years. In 2001, the first proof was provided by Aaron Klebanoff for the parameter . Very recently in 2023, an even sharper result for 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 and , which is also conceptual, and we provide a generalization of the phenomenon and the proof for all bifurcation points of the Mandelbrot set.
Cite
@article{arxiv.2505.07138,
title = {Pi in the Mandelbrot set everywhere},
author = {Thies Brockmoeller and Oscar Scherz and Nedim Srkalovic},
journal= {arXiv preprint arXiv:2505.07138},
year = {2025}
}