# Figure-8

Bodies · mechanical systems

```
gravity
m=1  x=-0.97000436  y=0.24308753   vx=0.46620369   vy=0.43236573
m=1  x=0.97000436   y=-0.24308753  vx=0.46620369   vy=0.43236573
m=1  x=0  y=0  vx=-0.93240737  vy=-0.86473146
```

[Open in the app](https://www.wavelace.com/app#p=82) · [This page](https://www.wavelace.com/presets/figure-8)

### The setup

Three equal masses pull on each other in the floor plane. Two of them sit at `(∓0.97000436, ±0.24308753)`, and both are given the same velocity, `(0.46620369, 0.43236573)`. The third starts at the origin with exactly minus twice that velocity. The three momenta cancel, so the centre of mass stands still at the origin for good.

### The choreography

All three bodies run the same closed curve, one third of a period apart. At `t ≈ 2.11` each body stands where its neighbour started. By `t ≈ 6.33` the whole state has come back to its beginning.

The curve is a flattened eight, `2.17` across and `0.72` tall, with the crossing at the centre of mass. The nearest two bodies ever come is `0.687`, so there is never a close encounter to resolve. The eight is still the eight after three hundred seconds, some forty-seven turns of it.

### History

Cris Moore found the orbit numerically in 1993, by minimising the action over braided paths. Alain Chenciner and Richard Montgomery proved in 2000 that it exists. Carles Simó then computed the initial conditions to the precision this spec carries, and went looking for other choreographies. It remains the famous exception in a problem with almost no closed solutions.

### Try

- `Top` in the deck: the eight seen flat, the way it is always drawn.
- Nudge one body sideways, `x=0.98` on the second line. The three stay bound, but the eight stops closing and drifts further off its start every turn.
- Make one body heavier, `m=1.05` on the first line. The choreography fails outright and a body is thrown clear out of the plot within a minute.
- `Rewind` puts the clock back to zero and it all repeats exactly.

### Read more

- [N-body choreography](https://en.wikipedia.org/wiki/N-body_choreography)
- [Three-body problem](https://en.wikipedia.org/wiki/Three-body_problem)
- [Newton's law of universal gravitation](https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation)
- [Runge–Kutta methods](https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods)