A short, friendly explanation of the shapes on this site: how a very simple rule can produce something that never repeats, yet never falls apart.
Take a single point on a sheet of paper. Now apply a rule that tells it where to jump next, and apply the same rule again, and again, millions of times. The rule is tiny. The famous one by Clifford Pickover reads:
x → sin(a·y) + c·cos(a·x)Four numbers a b c d decide everything. Mark every place the point lands, and make the mark a little brighter each time it lands there again. That is all the picture on the front page does.
Wherever you start the point, after a few jumps it ends up in the same region and stays there. The shape attracts it, the way a valley collects every marble rolled into it. Many systems settle into something dull: a resting point, or a loop that repeats forever, like a pendulum or a planet’s orbit.
A strange attractor never settles into a loop. The point never lands twice in exactly the same place, and yet it never leaves the shape. Zoom in and you find more structure, layer inside layer: a fractal. Order and disorder at the same time. This is called deterministic chaos: nothing is random, every step is fixed by the rule, but you still cannot predict where the point will be far ahead.
The reason is sensitivity. In the early 1960s the meteorologist Edward Lorenz restarted a weather simulation with numbers rounded to three decimals instead of six, and after a while the forecast was completely different. In 1963 he boiled the problem down to three equations. The difference you put in doubles again and again until it is as large as the whole system. Here are two points that start one millionth apart:
For a while the two lines are one. Then they part, and from then on they have nothing to do with each other, although both follow the same rule. This is the famous butterfly effect, and the Lorenz attractor itself even looks a little like a butterfly.
On the front page you can pick one of them and press another one for new numbers. The Attraktor-Expedition turns the search for good numbers into a game, and every find can be heard as well as seen.
Strange attractors show up in weather, in a dripping tap, in heart rhythms, in electronic circuits and in the tumbling of small moons. They are a reminder that simple rules are not the same as simple behaviour. And they are beautiful, which is the reason this site has been drawing them since 1999.