Infinite Coastline
The camera is already falling into Seahorse Valley when the page opens, and it keeps going: past a thousand times, where ordinary graphics arithmetic runs out, and on to several billion, where a complete copy of the whole set waits in the dark. Move the pointer and a second picture redraws itself live — the Julia set for that exact point, because every point of the Mandelbrot set is a map of one.
Take a number c, start at zero, and repeat one step: square it and add c. For some values of c the result stays small for ever; for others it runs off to infinity. Colour the first kind black, and the shape you get is the Mandelbrot set. Benoît Mandelbrot printed the first recognisable pictures of it at IBM in 1980, on a line printer, and the speckles at the edge looked like dirt on the paper until he zoomed in and found they were more copies of the set.
The surprise is how much one line of algebra contains. The edge never smooths out: magnify any piece of it and there is more coastline, seahorses and spirals and lightning forks, and somewhere in every neighbourhood a tiny complete copy of the whole set, attached by filaments thinner than anything you could draw. In 1998 Mitsuhiro Shishikura published the proof that the boundary is as rough as a curve in the plane can possibly be: its fractal dimension is exactly 2. The coastline is infinitely long, yet it encloses a finite area, about 1.5066 — a number nobody has a formula for, only estimates, which is why the page measures it by throwing random darts and counting the ones that land inside.
The second picture is the oldest half of the story. Before computers, Gaston Julia and Pierre Fatou studied what the same step does when you hold c fixed and vary the starting point instead. Each c has its own Julia set, and the Mandelbrot set is the catalogue: if c is inside it, its Julia set is one connected piece; if c is outside, it shatters into dust. Near the edge, the Julia set looks like the Mandelbrot set does at that very spot, which is why moving the pointer through Seahorse Valley grows seahorses on the right.
The same arithmetic turns up wherever a process feeds its output back as input: population models that tip from steady to chaotic, the period-doubling route to turbulence, the stability of numerical methods. The pictures themselves became a staple of computing, partly because they are a stern test of it — a deep zoom is a measurement of how many digits your machine really carries, and you can watch this one reach its limit.
Four rules about neighbours, four million cells, and no designer — yet it builds guns, spaceships and, in principle, a working computer.