A visual essay
Atlas of Chaos
Pendulums, weather and dice: eight pictures about the strange place where simple rules stop being predictable, and where, surprisingly, order comes back.
Waldeck Schützer · UFSCar
Your system asks for reduced motion: the pictures change with your scrolling but do not animate by themselves.
Scene 1 of 8
A swing you can predict
One rod, one weight. Give it a push and its whole future is written in a single formula: it swings back and forth, and the time of each swing depends only on the length. Physicists love this pendulum because it behaves.
Scene 2 of 8
Add one joint
Hang a second rod from the first. Nothing else changes: same gravity, same Newton. Yet the lower weight now draws loops, hesitates and lurches. Its path never repeats, and no simple formula writes it down.
Scene 3 of 8
Two nearly identical starts
Release two pendulums that differ by one millionth of a radian, far less than anything you could measure. For a while the pink and the blue dance together. Then they part ways, and soon they have nothing in common.
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A map of every start
Each point of this square is a starting position of the two angles, coloured by how long the pendulum takes to flip over the top. Where colours form smooth regions, small changes do nothing. Where they turn to static, the future depends on digits you cannot measure. Keep scrolling and the map zooms in: the static never goes away.
Scene 5 of 8
A weather model that would not settle
In 1963 Edward Lorenz simplified the equations of air heated from below to just three. The solution never repeats and never escapes: it winds around two lobes forever, a butterfly drawn by a single moving point.
Scene 6 of 8
How far ahead can we see?
Start two Lorenz points a billionth apart and measure the gap. It grows by the same factor every few moments, a straight line on this logarithmic chart, until it saturates at the size of the butterfly. Each extra digit of precision buys only a little more time: that is the forecaster's horizon.
Scene 7 of 8
Order lives next door
Two plain oscillations, one left–right and one up–down, with frequencies in the ratio a : b. When the ratio is a fraction the curve closes into a tidy knot; when it is not, the curve never closes and slowly fills the whole square. Same ingredients, opposite behaviours.
Scene 8 of 8
Order from dice
Pick one of three corners at random and jump halfway towards it. Repeat. Every step is a coin toss, yet the dots assemble, exactly, into the Sierpiński triangle. Randomness with a simple rule can draw perfect geometry, and that is the other half of the story of chaos.