PI HIDING IN ITERATION
A famous limit
Approach c=1/4 along the real axis using carefully scaled distances and count iterations before escape. After appropriate normalization, the counts approach π. This comes from local asymptotic dynamics near a parabolic point, not a hidden geometric circle.
GO DEEPER
2025 generalization
A 2025 paper titled Pi in the Mandelbrot set everywhere gives a broad generalization of this phenomenon across bifurcation points.
Sources & further reading:arXiv — Pi in the Mandelbrot set everywhere