arxiv:2505.07138

Pi in the Mandelbrot set everywhere

Published on May 11, 2025

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Abstract

The numerical phenomenon of π appearing at parameters c = 1/4, c=-3/4 and c=-5/4 in the Mandelbrot set 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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