Chladni Figures
Grains settling on the nodal lines of a plate as its driving frequency sweeps.
About this piece
In 1787 Ernst Chladni sprinkled sand on a metal plate and drew a violin bow along its edge. The sand jumped off everywhere the plate was moving and piled up along the lines that stayed still — the nodal lines. Every resonant frequency has its own set of them, and they are the reason a standing wave is something you can look at rather than only calculate.
The standard two-mode model of that nodal set on a square plate is
f(u, v) = sin(nπu)·sin(mπv) − sin(mπu)·sin(nπv)
and the figure is the curve f = 0. A grid of points is sampled across the plate and a grain is kept only where the sample sits close to that curve. What you are looking at is the nodal set itself, grain by grain — not an outline traced around it.
“Close” deserves a note, because the obvious test is wrong. Thresholding on |f| gives trails that are fat where the plate is loud and hairline where it is quiet, since |f| climbs at different rates in different regions. The distance from a point to the curve is |f| divided by the length of the gradient, and both partial derivatives of f are available in closed form, so that is what is tested here. The trail comes out a constant 1.35% of the plate wide everywhere.
What to look for
The sweep holds on each whole-number pair for about five and a half seconds — 5:3, 7:4, 6:4, 4:7, 8:5, 5:6, 6:3, 3:8 — then takes about two seconds to dissolve into the next, as if the driving frequency were being turned. The crisp symmetric states are the classical figures; the states in between are not.
Two lines are always there whatever the mode: the plate edge, and the diagonal u = v. Both fall straight out of the formula — swap u and v and f changes sign, so it must be zero where they are equal.
Honest limit: this is the square-plate model drawn across a rectangular frame, so the figure is stretched to whatever aspect your window has. A real rectangular plate resonates at a different set of modes. It is a faithful picture of the equation, not a simulation of a specific piece of metal.
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More from the gallery
- Plate 12 Strange Attractor One de Jong orbit, drawn until its dust becomes a shape.
- Plate 13 Apollonian Gasket Tangent circles packed into every gap, turning slowly.
- Plate 14 Frost Dendrites Wandering particles freezing onto a growing crystal.
- Plate 15 Voronoi Drift Cell walls redrawn as their seeds drift and relax.
- Plate 16 Hilbert Weave One unbroken line that visits every cell of the plate.
- Plate 17 Turing Bloom Two spreading chemicals growing a coral reef out of nothing.