Technique · Geometry
Strange attractor
The never-repeating paths of the Lorenz, Rossler, Thomas, Aizawa and Halvorsen equations, drawn as glowing ribbons turning in space.
Strange attractor is drawn geometry, rebuilt as it changes. Every piece on Synapse Drip is made from a recipe that layers techniques like this one, and you can change any of it in the Studio.
Good to know
- Lorenz's equations were a drastic simplification of the way a warm fluid circulates; the whole model has only three variables, yet no solution ever repeats.
- Weather forecasts cannot be trusted much beyond about two weeks, however good the model, because small errors grow exponentially, as the paths here do.
- Otto Rossler found his simpler attractor in 1976, deliberately looking for the smallest set of equations that would be chaotic, and Rene Thomas described his symmetrical one in 1999.
- Edward Lorenz noticed sensitive dependence around 1961, when a weather simulation restarted from a printout rounded from 0.506127 to 0.506 gave a completely different forecast; his three-equation system followed in 1963.
- In a 1972 talk Lorenz asked whether the flap of a butterfly's wings in Brazil could set off a tornado in Texas; the picture of the attractor does look like a pair of wings.
The mathematics
- The paths start 0.010 apart; in the Lorenz system the gap between two paths grows by a factor of e (about 2.7) roughly every 1.1 time units, which is how a tiny difference ends up as a completely different path.
- The Lorenz attractor has a fractal dimension of about 2.06: it is a surface that is very slightly more than a surface, folded into infinitely many sheets that never touch.
- The path is found by a fourth-order Runge-Kutta integrator, which takes four estimates of the slope for each small step and combines them, so the drawn line follows the true curve far more closely than a simple step would.
In art
- Pictures of attractors were among the first beautiful images that came out of chaos theory; Clifford Pickover's 1990 book Computers, Pattern, Chaos and Beauty collected many of them.
- Julien Sprott's 1993 book Strange Attractors showed how to search for new chaotic equations by computer, and its pictures became a genre of generative art.
More ways to use it
Related techniques
L-system · Space colonisation · Diffusion-limited aggregation · Flow ribbons · Ridgelines · Curve families · Streamlines · Paint strokes