Sand that knows the note

Take a flat plate of brass, hold it at the center, sprinkle it with sand and draw a violin bow down one edge. The plate sings one note, and as it sings the sand jumps. It leaps off the parts of the plate that are moving and lands on the parts that are still, and in a second or two it has drawn a figure: a cross, a star, a set of nested rings, a lattice of curves nobody put there. Bow a different note and the sand gets up and draws a different figure. Ernst Chladni, a law professor's son from Wittenberg who had given up law for sound, did this in 1787 and published the plates, and for the next thirty years he drove around Europe in a carriage doing it in drawing rooms for anyone who would pay.

Napoleon's question

In 1809 the audience was Napoleon. Chladni played the figures for him at the Tuileries, and the Emperor, who had been a good student of mathematics, asked the right question: nobody could write down why the sand went where it went. He put up a prize of three thousand francs for a mathematical theory of the vibrating plate. The Academy ran the contest three times. The only entrant all three times was Sophie Germain, who had taught herself mathematics from her father's library because the Ecole Polytechnique did not admit women, and who had been corresponding with Gauss under a man's name. Her third attempt won, in 1816, with the equation that still carries her name, and she was the first woman to win a prize from the Paris Academy. She did not attend the ceremony.

Why the sand goes there

Here is the why, and it belongs on this shelf more than anything else here. A plate can only vibrate in certain ways, the way a string can only sound certain notes. Each way of vibrating, each mode, has a shape: some parts of the plate swing up and down and some lines through it never move at all. Those lines are called nodes. Sand bounces off the moving parts until it lands on a node, where there is nothing to bounce it, and stays. The figure in the sand is a drawing of the mode. The note and the shape are the same fact, heard and seen.

That is what Pythagoras was reaching for with the monochord, at the start of this shelf. He found that the notes that sound right together are the ones whose string lengths are small whole numbers. Chladni found the other half: a note is not just a number, it is a shape, and the shape is in the metal before the bow ever touches it, waiting for the right pitch to call it out. Kepler would have taken the plate as proof of his whole book.

The instrument

The plate below is square and brass, like the one in Chladni's drawing room. Its notes come in pairs of whole numbers, so many half-waves across the plate and so many down it, and the figure for each pair is one wave laid across another, the construction Charles Wheatstone worked out in 1833 to explain Chladni's drawings. Three thousand eight hundred grains of sand random-walk, pushed hardest where the plate swings most, and settle on the still lines in a few seconds. The two dials pick the pair. Each change bows the plate and plays its tone, the pitch rising with the mode the way it does on the real brass.