Two sine waves at right angles, at frequencies that are almost but not quite the same.
Two sines, one pen
x = A · sin(a·t + δ)
y = B · sin(b·t)
Nothing more. Drive the horizontal position by one sine and the vertical by another,
and the pen traces a Lissajous figure — named for Jules Lissajous, who produced them
in 1857 by bouncing light off tuning forks, and independently found by Nathaniel
Bowditch in 1815.
For most of the twentieth century they were what an oscilloscope showed when you fed
one signal into X and another into Y, and they were the standard way to compare two
frequencies before anything digital existed.
A tenth of the period. One sweep, no returns.
The shape depends only on the ratio a : b and the phase δ.
If the ratio is rational, the curve closes: after lcm periods both sines are back
where they started simultaneously, and the pen retraces its own path forever. If it is
irrational, the curve never closes and eventually fills a rectangle.
The woven renderings use ratios of consecutive integers:
| Rendering | a : b | Lobes across | Lobes down |
|---|
| The lead | 5 : 4 | 5 | 4 |
| Ribbon | 3 : 2 | 3 | 2 |
| Dusk | 3 : 2 | 3 | 2 |
| Ghost | 4 : 3 | 4 | 3 |
| Ember | 7 : 6 | 7 | 6 |
| Tide | 5 : 4 | 5 | 4 |
The lobe counts are the frequencies. That is the reading a technician took off an
oscilloscope: count the lobes on each axis and you have the frequency ratio, without
measuring anything.
Consecutive integers are the interesting case, because the two frequencies are close
enough that the figure nearly repeats each cycle and the near-misses produce the woven
appearance.
Orbit is the limit of that idea: 50 : 51 is as close to equal as makes no visual
difference, so each period almost repeats the last and the misses open into a slow
precession. The pen needs fifty periods to return, and in between fills the frame with
every phase the figure passes through.
A quarter. The pen has begun folding back over its own path.
Half the period. A Lissajous figure with a quarter-turn phase offset is symmetric about
both axes, and by half the period both halves have been drawn — because the second half
retraces the reflection of the first.
Phase decides how open it is
The lead is drawn at δ = π/2 — a quarter cycle. That is the value that makes the
figure maximally open: at δ = 0 the same ratio collapses to a curve that touches its
own bounding box at the corners, and the lobes close up into a much thinner figure.
At a = b this is exactly the difference between a circle and a diagonal line: same
two sines, one number apart. Ghost renders the difference directly — the same 4 : 3
figure twice, the collapsed form at δ = 0 dim beneath the open one at δ = π/2.
Three quarters.
The full period. The pen ends exactly where it began, with the same velocity, and would
trace the identical path for as long as it was allowed to.
The period is taken as 2π, which closes correctly for coprime integer ratios; the
step count is what makes the polyline smooth rather than what makes it close — from
8,000 for the sparse figures to 45,000 for Orbit’s fifty-period sweep.
Colour follows the pen
Every segment is coloured by where it sits along its own trace. The flat renderings —
the lead’s layered ink, Ghost’s two figures — hold one colour per pass; the gradient
renderings sample a short palette of two or three stops, so the colour moves
deliberately from one end of the trace to the other instead of cycling through the
spectrum. Ribbon fades the stroke itself: the open figure ends the way pen pressure
ends, with the line pressed in and lifted off.
Determinism
Nothing is seeded and nothing is random. Every point is a closed-form function of its
index, computed in single precision and drawn in order.
