History: worlds nobody saw for a hundred years

Julia, 1918: a formula without pictures

The French mathematician Gaston Julia was badly wounded in the First World War. Back at work, he took up a question: what happens if you repeat z = z² + c again and again. In 1918 his work on iterating functions won a prize of the French Academy of Sciences. Pierre Fatou was working on the same question at almost the same time.

They proved remarkable things about the sets that are now called Julia sets. But neither Julia nor Fatou ever saw them: to draw such a set, the formula has to be repeated millions of times. For sixty years these worlds existed only in formulas.

Mandelbrot, 1980: the first printouts

Benoit Mandelbrot, a student of Julia at the École Polytechnique, worked at an IBM research centre — he had computers. In 1980 he drew a map of all the Julia worlds at once: for every c, does z run away or not. The first pictures came out on a printer in characters, an asterisk being a dot in the set.

This is what the first pictures looked like (1980): the computer printed the set in characters on a printer, an asterisk is a dot in the set. Our 128 × 128 screen shows more detail.

The word fractal was also coined by Mandelbrot — from the Latin fractus, "broken". In 1982 Adrien Douady and John Hubbard proved that the Mandelbrot set is connected — one piece — and named it after Mandelbrot. The "Douady rabbit" from lesson 6 is named after Adrien Douady.

Home computers: wait a minute

In August 1985 Scientific American put the Mandelbrot set on its cover and explained how to write the program. People ported it to everything that could compute: the ZX Spectrum, the Commodore 64, the Apple II. Without a multiplier circuit the program multiplied with shifts and additions, and the picture appeared line by line — in minutes, in colour in hours. Our machine follows the same path, only its multiplier is all at once, made of transistors.

Barnsley, 1988: nature in four rules

In his book "Fractals Everywhere" (1988) Michael Barnsley showed that a fern leaf is four rules of moving and turning plus chance picking them. He believed photographs could be compressed this way: store rules instead of dots. The compression stayed rare, but the Barnsley fern became the best-known example of a living shape growing out of simple rules.

Barnsley's fern: the same chaos game, only instead of "a step half-way to a vertex" there are four rules of shifting and turning. Random dots form a fern leaf.

IBM is a trademark of IBM Corp.; ZX Spectrum, Commodore 64, Apple II and Scientific American are trademarks of their respective owners. This kit is not affiliated with or endorsed by them; the names are mentioned to tell the story.