# Thomas Attractor

Flow · (ẋ, ẏ, ż) = f(x, y, z, t)

`(sin(y) − 0.2x,  sin(z) − 0.2y,  sin(x) − 0.2z)`

[Open in the app](https://www.wavelace.com/app#p=80) · [This page](https://www.wavelace.com/presets/thomas-attractor)

### The system

The three lines are one rule written three times, passed round the variables: a sine of the next coordinate, less a damping `0.2` on this one. Other flow presets exchange the textbook's y and z, because the app is y-up and the classic systems are written z-up; the cycle here is symmetric, so this one needs no swap. Exchanging y and z only reverses the cycle, and the picture is the published attractor mirrored.

### A lattice of saddles

A rest point needs `sin y = 0.2·x` and its two cyclic partners. Since a sine never exceeds 1, every rest point lies inside `|x|, |y|, |z| ≤ 1/0.2 = 5`, which is exactly the box `Span of x, y, z` shows. There are twenty-seven of them in that box. Three sit on the diagonal `x = y = z`, at `0` and `±2.596`, where `sin x = 0.2·x`. At the origin the eigenvalues are `0.8` and `−0.7 ± 0.866i`: pulled in on a spiral, pushed out along one line.

Every one of the twenty-seven behaves like that. A seed spends its time being caught by one, wound around it and flung toward the next. Thomas called the result labyrinth chaos: a walk through a lattice of doorways rather than a fold. The damping decides how far the walk goes. At `0.2` it is strong enough to keep the seeds to one cell of the lattice, roughly `−1.3 < x < 4`. The stretching there is slow, a few hundredths per second against about 0.9 for the Lorenz preset, so the threads keep their bundle far longer.

### History

René Thomas proposed this system in 1999 as the simplest cyclically symmetric flow, reading it as a particle moving through a three-dimensional lattice of forces against a friction of `b`.

### Try

- Change all three `0.2`s to `0.1`: less friction, and the seeds break out of their cell to wander the lattice, out past `x = ±6` and beyond the box.
- Raise them instead to `0.4`: friction wins and every seed drifts to the diagonal rest point at `x = y = z = 2.13`.
- `Top` in the deck: the same tangle seen from above. A third of a turn about the diagonal `x = y = z` carries the figure onto itself, which is the cyclic symmetry of the three expressions.
- Set `Seeds` to 1 and follow one path through the maze from cell to cell.

### Read more

- [Thomas' cyclically symmetric attractor](https://en.wikipedia.org/wiki/Thomas%27_cyclically_symmetric_attractor)
- [Attractor](https://en.wikipedia.org/wiki/Attractor)
- [Lyapunov exponent](https://en.wikipedia.org/wiki/Lyapunov_exponent)
- [Chaos theory](https://en.wikipedia.org/wiki/Chaos_theory)