Benoit Mandelbrot Looked at Garbage on an IBM Printout in 1980 and Found a Famous Shape

Benoit Mandelbrot was a Polish-French-American mathematician on staff at IBM's Thomas J. Watson Research Center in Yorktown Heights, New York, in 1980. He was working on what he was calling "fractal geometry" — a then-new framework for the kind of self-similar shapes that come up in Brownian motion, coastlines, and turbulence. The problem he was studying that February asked, in modern notation: for which complex numbers c does the sequence defined by zₙ₊₁ = zₙ² + c with z₀ = 0 stay bounded?

The set of complex numbers for which the answer is yes — the Mandelbrot set — had been formally defined and crudely drawn by Robert Brooks and Peter Matelski in a 1978 Kleinian-groups paper that essentially nobody read. On March 1, 1980, Mandelbrot ran a higher-resolution iteration on IBM's printer and got back the famous lobed black silhouette — a cardioid main body, a smaller circular bulb to its left, an infinite cascade of further bulbs surrounding both — surrounded by a fractal boundary of impossible intricacy. The first printouts had black dust around the boundary that the operator initially assumed was a printer fault. The dust was real.

The set's properties have absorbed more research than its physical-printer origins might suggest. The set is contained in a disk of radius 2 centered at the origin. It is connected — a fact Mandelbrot himself initially doubted, looking at the early prints. Its boundary has Hausdorff dimension 2, meaning the boundary locally fills a two-dimensional region as efficiently as solid plane, despite being topologically one-dimensional. Tangencies near the cusp at c = -3/4 approximate π: at iteration step ε = 10⁻⁷, the count of iterations before divergence multiplied by ε produces 3.1415928. The Fibonacci numbers appear in the periods of the bulbs surrounding the main cardioid. The set is, in working mathematicians' parlance, weird.