Strange Attractor
One point, iterated forever through two lines of arithmetic, until where it has been becomes a shape.
About this piece
Everything on this canvas comes out of two lines that the Dutch physicist Peter de Jong contributed to A. K. Dewdney’s Scientific American column in the late 1980s:
x′ = sin(a·y) − cos(b·x)
y′ = sin(c·x) − cos(d·y)
Take one point, put it through that pair, put the answer back in, and keep going. There is no random number anywhere in this piece. The orbit never repeats, never leaves the square from −2 to 2, and never settles down — that combination is what the word strange is doing in the title.
The picture is not the path. At a couple of thousand iterates a frame the path is illegible dust. The picture is where the orbit spends its time: run it long enough and the visits pile up into folded sheets, bright along the creases where the map squeezes and faint across the sweeps where it stretches. So the canvas here is never cleared. Each frame adds its iterates to what is already there and decays the whole image by 3.5% first, which means roughly the last thirty frames — half a second of orbit, about 400,000 points at a desktop size — are visible at any moment.
What to look for
The bright hairlines are caustics: places where the map folds the plane back on itself, so points from two different regions land on top of each other. They are the same thing as the bright curves at the bottom of a swimming pool, and for the same reason.
Coral marks the long jumps. Every iterate is measured against a slowly-updated running mean of how far the orbit has been moving, and the top fifth or so — the sweeps between folds, never the folds themselves — is drawn in the warm ink. That ratio holds at about 21% whatever the parameters are doing, which is why the piece stays cobalt-with-accents instead of swinging between all-blue and all-orange.
a, b, c and d drift, so the figure breathes rather than sitting still. They ride four sine terms whose frequencies are in the ratio 1 : 2 : 3 : 5 off one base, which makes the whole parameter path exactly periodic — it closes after 151 seconds and repeats. That is a deliberate choice, not a stylistic one. Most of de Jong’s parameter space is not chaotic: wander into the wrong corner and the orbit falls into an attracting cycle, and the plate collapses to a handful of dots. A closed path is a path you can check, and this one was — sampled at 240 points around its full period, the sparsest frame still occupies 1,730 cells of a 160×160 grid.
Honest limits: the plot is stretched to whatever aspect your window is, so the figure is wider on a monitor than on a phone — an attractor has no canonical shape, but it does have a canonical aspect, and this is not it. And the accumulation makes this the most expensive piece in the gallery on an old machine; if you want it cheap, give the iframe a smaller box, because the iterate count scales with area. Reduced-motion visitors get a settled still instead of the loop.
Curious how the loop and canvas fit together? Read how it works →
More from the gallery
- Plate 13 Apollonian Gasket Tangent circles packed into every gap, turning slowly.
- Plate 14 Frost Dendrites Wandering particles freezing onto a growing crystal.
- Plate 15 Voronoi Drift Cell walls redrawn as their seeds drift and relax.
- Plate 16 Hilbert Weave One unbroken line that visits every cell of the plate.
- Plate 17 Turing Bloom Two spreading chemicals growing a coral reef out of nothing.
- Plate 18 Penrose Tiling Two rhombs tiling the plane in a pattern that never repeats.