Editing Mandelbrot set (section)

=== Pi in the Mandelbrot set ===
There are intriguing experiments in the Mandelbrot set that lead to the occurrence of the number pi. For a parameter <math>c = -\tfrac{3}{4}+ i\varepsilon</math> with <math>\varepsilon>0</math>, verifying that <math>c</math> means iterating the sequence <math>z \mapsto z^2 + c</math> starting with <math>z=0</math>, until the sequence leaves the disk around <math>0</math> of any radius <math>R>2</math>. This is motivated by the (still open) question whether the vertical line at real part <math>3/4</math> intersects the Mandelbrot set at points away from the real line. It turns out that the necessary number of iterations, multiplied by <math>\varepsilon</math>, converges to pi. For example, for ''<math>\varepsilon</math>'' = 0.0000001, and <math>R=2</math>, the number of iterations is 31415928 and the product is 3.1415928.<ref>{{cite book |first=Gary William |last=Flake |title=The Computational Beauty of Nature |year=1998 |page=125 |publisher=MIT Press |isbn=978-0-262-56127-3 }}</ref> This experiment was performed independently by many people in the early 1990's, if not before; for instance by David Boll. 

Analogous observations have also been made at the parameters <math>c=-5/4</math> and <math>c=1/4</math> (with a necessary modification in the latter case). In 2001, Aaron Klebanoff published a (non-conceptual) proof for this phenomenon at <math>c=1/4</math><ref>{{cite journal |last=Klebanoff |first=Aaron D. |title=π in the Mandelbrot Set |journal=Fractals |volume=9 |issue=4 |pages=393–402 |year=2001 |doi=10.1142/S0218348X01000828 }}</ref>

In 2023, Paul Siewert developed, in his Bachelor thesis, a conceptual proof also for the value <math>c=1/4</math>, explaining why the number pi occurs (geometrically as half the circumference of a unit circle). <ref>Paul Siewert, Pi in the Mandelbrot set. Bachelor Thesis, Universität Göttingen, 2023</ref>

In 2025, the three high school students Thies Brockmöller, Oscar Scherz, and Nedim Srkalovic extended the theory and the conceptual proof to all the infinitely bifurcation points in the Mandelbrot set. <ref>https://arxiv.org/abs/2505.07138</ref>