The Mandelbrot Set Contains the Logistic Map

Slice the most famous fractal along its real axis and population biology’s route to chaos appears inside it, bulb for fork.

The population model’s route to chaos, flipped and laid along the Mandelbrot set’s real axis: each fork drops onto a joint between bulbs. Figure by the author.

Two of the most reproduced images in mathematics come from fields that have little to say to each other. The first is the Mandelbrot set, the black, warty silhouette with a flaming halo that has hung on dorm-room walls since the 1980s and still turns up as a worked example in NumPy’s own tutorial. The second comes from population biology: the bifurcation diagram of the logistic map, a branch of lines that splits in two, splits again, and then dissolves into a grey smear, drawn to show how a simple model of breeding animals slides from a steady state into chaos. One is a picture of complex numbers, the other a picture of rabbits or moths.

Flip the second picture so that it runs right to left, stretch it to the right scale and lay it along the horizontal axis of the first, and the two line up. The tree’s first split lands on the joint where the big heart-shaped body of the Mandelbrot set meets the disc to its left, the second split on the joint between that disc and a smaller bud, and the splits crowd together where the buds shrink to a vanishing chain. A picture built from the arithmetic of complex numbers has no obvious way to know how a…