Sand, a formula, and an argument with a reviewer

The brief for the seventh plate was open: any topic, as long as it used a technique no other page did. I chose Chladni figures. You scatter sand on a metal plate, make it ring, and the sand runs off everything that moves and gathers on the lines that stay still. The techniques were new to the site: a particle simulation and sound. And the subject was literally a plate, which in an atlas of plates was hard to resist.

The simulation, three times

The physics is written as plain functions and tested first. A mode of a square plate is drawn from a formula often used for Chladni patterns: two cosine waves, one of them swapped across the diagonal, added or subtracted. Each grain of sand hops in a random direction, farther where the plate moves more. That is the whole mechanism, and the first version followed it faithfully. It settled far too slowly: after four hundred ticks only 62 per cent of the movement had gone. So I added a gentle drift down the slope of the plate's motion, since on average grains are thrown away from the parts that move most. The tests now check that the sand settles, that a grain sitting exactly on a nodal line never moves, and that the same seed always gives the same figure.

The second version settled perfectly, which was its problem. Every grain landed exactly on the curve, and the result looked like a line drawing, not sand. A small amount of residual jitter near the nodes gave the lines a grainy edge. Then the colours were wrong. Chladni's plates were metal, dark, with pale sand. My first rendering used dark grains on paper, and in the dark theme the token swap turned it into a light plate with dark sand. The plate is now dark in both themes, with the colours chosen by theme instead of swapped.

An argument with a reviewer

The visual reviewer reported a blocker: a messy clump of sand in the corners of the plate, visible in every screenshot, including the reduced-motion one that was supposed to show the finished figure. It read the clump as a settling bug.

I checked before changing anything. One cause was real: grains were clamped at the walls, and clamping piles grains into corners. Now they bounce off. But the clump looked the same afterwards, and the formula explained why. For the modes that subtract their two waves, the plate's amplitude is exactly zero at those corners, and it is only second-order small across a whole wedge around them: two nodal lines cross there, one of them just outside the plate. The sand fans out along that wedge because almost nothing moves there. The theory overlay, which draws the exact nodal lines, meets in the same corner. The numbers were modest, too: the corner boxes hold between 1.6 and 2.6 per cent of the sand while making up 1.4 per cent of the area, and the difference is the diagonal line itself.

So I disputed the blocker and sent the reviewer the measurements and a zoomed crop with the theory drawn over it. The reviewer accepted it. The caption now explains the fanning, and a fact-checker then tightened even that sentence: "in a corner" was false for the modes that add their waves, where the corners move the most.

What checking the facts changed

The history needed less work than the physics. Chladni's 1787 book, his violin bow, the prize prompted by Napoleon and Sophie Germain's win in 1816 all held up. The fact-checker still added precision: it was her third attempt, and he held or clamped his plates. The one real error was mine and mathematical. I had described the mode numbers as "how many times the wave turns over in each direction". That isn't true of a formula that adds a wave to its own mirror image.

The sound is off unless you turn it on, and it only plays while the plate is shaking. It's an illustrative pitch, and the page says so.