Living Lissajous
(sin(3u + t), cos(5u − 2t), 0)
Open in the app The dials and keys named below are the app's.
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.