History: patterns that repeat themselves

A triangle from 1915

In 1915 the Polish mathematician Wacław Sierpiński described a figure: take a triangle, cut an upside-down triangle out of its middle — three smaller ones remain; cut the middle out of each of them — and so on forever. A part of the figure looks like the whole — such figures would later be called fractals.

step 0step 1step 2step 3
Sierpinski's construction (1915): cut the middle out of a triangle — three half-size triangles remain; cut the middle out of each of them… After infinitely many steps you get a fractal.

The same pattern had long been hiding in Pascal's triangle: colour its odd numbers — and you get the Sierpinski triangle.

11112113311464115101051161520156117213535217118285670562881
Pascal's triangle: each number is the sum of the two above it. Colour the odd ones — and the Sierpinski triangle appears. That is exactly what the rule (x AND y) = 0 computes.

Arcade machines without a processor

The first arcade machines — Space Race and Pong (1972), Breakout (1976) — had no processor. The picture on the screen was drawn by raster counters: they ran along the rows following the beam, and logic at every dot decided whether there was a ball, a paddle or a brick. Our machine works exactly that way: counters X, Y and a rule "is this dot lit".

Munching Squares

In 1962 at MIT a few lines of a program on the PDP-1 drew a pattern of "munching" squares — Munching Squares: a dot is lit if (x XOR y) < t, and t grows with every frame. It is one of the first computer animations. We build it not as a program but as logic.

The chaos game

In 1988 the mathematician Michael Barnsley showed that random steps towards the vertices of a triangle draw the Sierpinski triangle, and other rules draw a fern and trees. This is called the chaos game: order from chance.

1988, Barnsley:the same chaos game,other 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.

Next — Mandelbrot

In 1980 Benoit Mandelbrot obtained, on IBM computers, the first pictures of the set that now bears his name: z = z² + c, repeat until |z| runs away. The machine drew dot by dot — like ours. That is the next kit.

z = z² + c1980, IBM
The Mandelbrot set on a 64 × 64 screen — this is how the next kit draws it: for every dot repeat z = z² + c and see whether z runs away.
PDP-1 is a trademark of Digital Equipment Corporation; IBM is a trademark of IBM Corp.; Pong and Breakout are trademarks of their respective owners. This kit is not affiliated with or endorsed by them; the names are mentioned to tell the story.