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arXiv · 2505.07138

Pi in the Mandelbrot set everywhere

Thies Brockmoeller·Oscar Scherz·Nedim Srkalovic
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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.

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Thies Brockmoeller, Oscar Scherz, Nedim Srkalovic. 2025-05-11. Pi in the Mandelbrot set everywhere. https://arxiv.org/abs/2505.07138

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