06 / Physics · chaos · sound
A Thousand Double Pendulums
A thousand pendulums start a millionth of a radian apart. Eight seconds later, each goes its own way.
- Pendulums
- 1,000
- Initial difference
- 10⁻⁶ rad
- Integration
- Runge-Kutta 4, 20 steps per frame
- Format
- 1080 × 1920 · 30 fps · 25.5 s
- Sound
- 96 voices, one per tip
It has sound: turn it on in the video controls.
The piece
Between the first and the last there is a millionth of a radian: at the tip of a one-meter arm, one micron, dozens of times thinner than a hair.
For eight seconds they move as one. Then that invisible difference doubles again and again until each pendulum follows its own path. That's chaos: a deterministic system, with no randomness at all, where a tiny error grows exponentially.
In the end, time rewinds and the thousand become one again. The equations lose no information; we are the ones who can't measure that precisely.
Play
Pick the difference.
Release 300 pendulums with any initial difference you like. The smaller it is, the longer they take to split, but any difference above zero splits them eventually.
- Time
- 0.0 s
- Separation
- 0 pm
ε is the angle difference between the first and the last pendulum. Each arm is 1 m long, so 10⁻⁶ rad is one micron at the tip.
Behind the scenes
Behind the thousand pendulums.
The reel runs at 0.6× and each layer is explained while you see it.
- 01
Physics
The exact double-pendulum equations, integrated with 4th-order Runge-Kutta and 20 steps per frame. Energy is conserved to better than one part in ten million.
- 02
A thousand pendulums
The first and the last start a millionth of a radian apart. The distance between their tips doubles roughly every half second, from 0.3 microns to over a meter.
- 03
Light
Each arm is a row of light points that add up; tone mapping, bloom and grain go on top.
- 04
Rewind
Integrating backwards never returns to the start: chaos amplifies even rounding error. So the stored states are replayed.
- 05
Sound
96 voices track the height of 96 tips: a unison while they match, a choir when they split.
Scope and limits
What the video claims, and what it doesn't.
- The pendulums are ideal: point masses on rigid, massless, frictionless arms.
- The simulation runs at 0.85× real time.
- The browser demo uses 300 pendulums and a coarser integration step than the video.
Sources
