# Living Lissajous

Curve · (x, y, z) = f(u, t)

`(sin(3u + t),  cos(5u − 2t),  0)`

[Open in the app](https://www.wavelace.com/app#p=39) · [This page](https://www.wavelace.com/presets/living-lissajous)

### What it draws

The two moving coordinates are `sin(3u + t)` across and `cos(5u − 2t)` up, the second expression being the height. Both swing between `−1` and `1`, so the figure fits a box two units wide and two tall before `Scale`. The third expression is `0`, which keeps the whole figure flat in the vertical plane that `Front` looks straight at.

The frequencies are `3` across and `5` up: three cycles of `x` and five of the height for one turn of `u`. Because `3` and `5` share no factor the curve closes after that single turn. It touches each side of the box three times, and the top and bottom five times each. This is a Lissajous figure, the picture drawn by two perpendicular oscillations of different frequency.

### Why it keeps changing

The clock adds `t` to the phase of `x` and subtracts `2t` from the phase of the height. Not all of that is visible. Sliding `u` along by some amount `a` adds `3a` to the first phase and `5a` to the second, and that is only a relabelling of the same drawing. The one combination such a slide leaves alone is `5` times the first phase minus `3` times the second, which here is `5t + 6t = 11t`. So the shape depends on `11t` and nothing else. It returns to itself when `11t` has gone once around, every `2π/11 ≈ 0.571` seconds at Speed 1. What looks like endless invention is one short cycle through the family of 3-against-5 figures, running eleven times faster than either phase alone.

### History

Nathaniel Bowditch drew these curves in 1815 with a compound pendulum. Jules Antoine Lissajous studied them in detail in 1857, bouncing a beam of light off mirrors mounted on two tuning forks so that a fork out of tune showed as a drifting figure. The curves carry his name.

### Try

- `Front`: the curve lies in the plane facing the camera, and straight on it is the textbook figure.
- Raise `Scale`: an amplitude of `1` leaves the figure small in a plot six units across.
- Hold one phase still, `cos(5u)` for the height. Only `x` drifts now, the surviving combination is `5t`, and the cycle slows to `2π/5 ≈ 1.26` seconds.
- Lift it out of the plane: put `sin(4u)` where the `0` is, and the flat figure becomes a Lissajous curve in space.

### Read more

- [Lissajous curve](https://en.wikipedia.org/wiki/Lissajous_curve)
- [Jules Antoine Lissajous](https://en.wikipedia.org/wiki/Jules_Antoine_Lissajous)
- [Nathaniel Bowditch](https://en.wikipedia.org/wiki/Nathaniel_Bowditch)
- [Harmonograph](https://en.wikipedia.org/wiki/Harmonograph)