Lissajous Knot
(2.5 · sin(3u + t), 2.5 · sin(4u), 2.5 · sin(5u + t/2))
Open in the app The dials and keys named below are the app's.
What it draws
All three coordinates are sines of the same u, at frequencies 3, 4 and 5 and amplitude 2.5. The strand therefore stays inside a box five units on a side, and it closes after one turn of u. Each flat shadow of it is a plane Lissajous figure: Front shows 3 against 4, Top shows 3 against 5.
The second expression, 2.5sin(4u), is the height, and it is the one coordinate with no t in it. The strand keeps its profile of four rises and four falls while the clock advances the phase of x at one radian per second and the phase of z at half that. The two rates put the whole motion on a cycle of 4π ≈ 12.57 seconds at Speed 1.
When it is a knot
Three sinusoids of one parameter, with frequencies that are coprime in pairs, trace a Lissajous knot, provided the phases keep the strand clear of itself. The test is that no combination n₁φ₂ − n₂φ₁ of a pair of frequencies and their phases is a whole multiple of π. At t = 0 it fails badly. Both u = 0 and u = π send all three sines to zero, so the strand passes through the origin twice and is pinched, not knotted. The same happens at a scattering of later instants, among them t = π/4 ≈ 0.79. Between them the strand misses itself everywhere and the picture is a genuine knot, drifting through a family of them as the phases turn.
History
Lissajous knots were introduced in 1994 by M. G. V. Bogle, J. E. Hearst, V. F. R. Jones and L. Stoilov, who asked which knots can be drawn by three independent harmonic oscillations. Not every knot can be: the form forces symmetries that many knots do not have.
Try
- Rewind sends the clock back to zero, where the strand pinches through the centre twice and stops being a knot.
- Front flattens the strand onto its 3:4 plane Lissajous shadow, the picture two tuning forks would draw.
- Turn on trace: the pen walks the strand, which is the way to check it is a single closed loop.
- Give the height a clock of its own, 2.5sin(4u + t/3): now all three phases move and the knot tumbles rather than sliding.