The Fourier Kinetic Series

Grenoble, February of 1807, and the prefect's study on the second floor was kept at a temperature his visitors described, depending on their charity, as tropical or as punitive. Jean-Baptiste Joseph Fourier sat in it wrapped to the chin in wool, a brazier at either elbow, and did not sweat.¹ Egypt had spoiled him. He had come back from the Nile convinced that heat was the one honest thing in the world — that it never lied about where it had been, that it always moved from more to less, that given long enough it would render every distinction in the universe into one lukewarm agreement. He intended to write down its law. He had, in fact, already written it down. The trouble was arithmetic.

His clerk that winter was a prefecture man named Berthier, who had come to the post expecting land surveys and marriage registers and who now spent his mornings multiplying sines. The method was straightforward and it was murder. Fourier would hand down a curve — the temperature along an iron ring, measured at thirty-six points — and Berthier would resolve it into its pure tones, and to find each tone he had to walk the whole ring, thirty-six multiplications, and there were thirty-six tones to find, and so the walk was made thirty-six times over. Twelve hundred and ninety-six products, each to five figures, each an opportunity to be wrong in a way that would not announce itself until the end.² Three weeks for one ring. Berthier had begun to dream in tables.

"It cannot be that God made the world this expensive," he said one morning, not quietly.

"God made it cheap," Fourier said. "We are the ones charging."

§

Downstairs, in the room where the prefecture kept the crates from Egypt, a seventeen-year-old was losing an argument with a dead language.

Fourier had found the boy six years earlier, in a classroom in Grenoble, and had done the thing he did best, which was to recognize a furnace when he saw one. He had shown him the plates — the cartouches, the endless patient signs — and the boy had asked whether anyone could read them, and Fourier had said no, and the boy had said I will, in the flat way children say things they have not yet learned to be embarrassed about. His name was Jean-François Champollion. He would be right. It would take him another fifteen years.³

That February he was working on a cartouche with a repeated group in it, and he was working on it with a pair of scissors, which is what brought Fourier to the doorway.

"You are cutting my plates."

"Copies." Champollion did not look up. "Look at it, though. Look." He had the signs written out in a long strip, and he had folded the strip in half, and in half again, so that the repeated group lay over its own recurrence. "Sign by sign it's hopeless — every sign might be a word or a sound or a nothing, and there are hundreds, so there are hundreds of hundreds of guesses. But it repeats. And if it repeats, I do not have to ask what each sign is. I ask only whether this half is the other half. One question instead of a hundred. And then I fold what's left and ask again."

"And if it is not?"

"Then I have still halved the pile." He grinned up at the prefect. "Twice in an afternoon and there is nothing left to be stupid about."

Fourier stood in the doorway for what Champollion later described, uncharitably, as an unpleasantly long time.

Then he went upstairs and shut the door and did not come out for two days, and the brazier went cold, and nobody dared mention it.

§

What he had seen was this.

The ring had thirty-six points on it. Walk the ring once and you get one tone, and you have paid thirty-six multiplications for it — but you have paid them in a particular order, and the order is where the waste hides. Take instead every other point. Eighteen of them, an eighteen-point ring, its own smaller problem. Take the ones you skipped: another eighteen-point ring, the same problem again, only shifted — turned slightly in its phase, like the second of two dancers who are doing the identical figure a half-beat apart. Solve the two small rings. Then twist the second by its half-beat and add, and subtract, and the thirty-six tones of the whole ring fall out of the eighteen tones of each half, in pairs, two at a time, for the price of an addition.

And then, of course, you do it again. Eighteen is two nines. Nine is three threes. You keep folding until there is nothing left to be stupid about.⁴

Twelve hundred and ninety-six products became a hundred and some. Berthier did in a morning what had eaten his January, checked it twice out of disbelief, and then wept a little, privately, over the second page.

Fourier called it the kinetic series,⁵ and he was not being fanciful. Heat had taught him the whole of it. No particle of iron moves the ring; each one only jostles its neighbor, and the great smooth cooling that looks like a single fact is nothing but an enormous number of small motions, summed. The tones were the same. You did not find them one at a time, grandly, by interrogating the whole. You found them two at a time, cheaply, by asking each half about the other.

He wrote it into the memoir. Thirty-one pages of theory, and at the back, in the small type reserved for the practical, a note on the ordering of the computation.

§

The memoir went to Paris in December and Paris hated it.

Not the arithmetic — nobody read the arithmetic. Lagrange, seventy-one and immovable, objected to the series itself: it was not decent that a corner, a jump, a discontinuity, could be built out of nothing but smooth eternal sines. Laplace was cool. Poisson was Poisson. Fourier was told his rigor was wanting and his physics was interesting and would he care to submit again, and he did, in 1811, and won the prize with the reservations attached to it like a tin can to a dog.⁶

By then he had a prefecture to run: a marsh at Bourgoin to drain, a road to Turin to survey, and the seven great volumes of the Description de l'Égypte to see through the press, for which he wrote the historical preface twice because the Emperor disliked the first one.⁷ The note on ordering the computation went into a crate with the Egypt proofs. It was clerk's work. It saved a clerk three weeks. It was not the sort of thing a man puts his name on when he is trying to convince Lagrange about the nature of the continuum.

He published Théorie analytique de la chaleur in 1822, and it is one of the great books, and it does not contain the fold.⁸

He was elected to everything. He worked out, in 1824, why the earth is warmer than the plain sunlight on it should allow, which was a better guess than anyone would improve on for sixty years.⁹ He got Champollion out of the draft, twice, and lived to see him read the names of kings aloud. And in May of 1830 he fell down a flight of stairs in Paris and died not long after, still cold, still wrapped in wool, sixty-two years old.¹⁰

What none of them knew: it was already lost twice. In Göttingen, around 1805, Gauss had wanted the orbit of an asteroid in a hurry and had folded the same fold, in Latin, in a notebook, and had not thought it worth printing either. Those pages did get published — in 1866, in volume three of the collected works, in a dead language, where they sat for another century being not read.¹¹

§

In the spring of 1965 two men named James Cooley and John Tukey published four pages in a journal of computational mathematics, and the thing came back.¹²

The reason it came back is worth saying plainly. The government wished to know whether the Soviets were testing bombs underground, and the only witness to an underground bomb is the ground itself, which shudders, and a shudder is a curve, and to ask a curve what it is made of you must resolve it into tones. The seismometers gave them curves by the thousand. The machines could not keep up, because the machines were doing it the way Berthier did it in January — walking the whole ring, once for every tone, paying the square.

Cooley and Tukey folded it. Halves, and halves, and halves, until there was nothing left to be stupid about.

The saving is not a saving; it is a change of kind. For a thousand points the old way costs a million and the fold costs ten thousand. For a million points the old way costs a trillion and would still be running, and the fold is done before you have looked up. Every telephone call you have made, every song you have streamed, every photograph in your pocket, every MRI, every radio astronomer's quasar, every one of them is that fold, running some enormous number of times a second, on a machine that does not know it is keeping a promise made in an overheated room in Grenoble by a man who could not get warm.

They called it the Fast Fourier Transform, which is fair, and a little sad, because it credits him with the wrong half. He was always going to get the series. Anyone who stared long enough at heat was going to get the series.

The fold he got from a boy with a pair of scissors, who had learned it the only way anyone learns it: by having a pile too large to carry, and discovering that you are permitted to ask it one question instead of a hundred, and that the question is always the same one.

§

And then the other thing happened, the one that always happens.

A transform built to take a thing apart became the best tool ever made for throwing parts away. Every song in your pocket is smaller than it should be because a perceptual coder ran the fold, sorted the components, and deleted the ones it calculated you would not miss.¹³ Fourier resolved a curve in order to see it. We resolve it in order to discard most of it and ship the rest cheaply. The method did not change. Only the verb did.

It went further. The austerity that made his work portable — his refusal to say what heat was, his insistence on describing only how it moved — meant the transform would work on anything that varied. That is why it is now a single line in a library, called by people who have never heard of aliasing, who sample too slowly and get back a clean, confident, entirely fictitious tone and publish it.¹⁴ The refusal to speculate, which was his discipline, is now the user's ignorance. Same austerity. Opposite result.

And the listening. The four pages of 1965 were written so that a nation could tell an earthquake from a bomb, which is a decent thing to want. But the fold does not care what it is pointed at, and what it is pointed at now is everything that varies: every channel, every cell, every wire, every room with a window in it.¹⁵ The algorithm that let us hear through the earth is the algorithm that lets anyone hear through anything.

Which returns the question to where it started, with a different weight on it. The boy with the scissors asked it about a cartouche. The prefect asked it about an iron ring. The machines ask it several billion times a second, about you.

Is this half like the other?