The Clockwork Universe

by Edward Dolnick

  Feverishly, Kepler put his brainstorm to the test. He drew a circle and inside it he drew the one triangle that stood out from all the other possibilities—the simplest triangle of all, the only one that fit perfectly inside the circle and was completely symmetrical, with all three sides identical. Inside the triangle he drew another circle. Again, he could have chosen any of countless circles; again, he made the only “natural” choice, the one circle that fit the triangle perfectly. He looked again at his drawing. The inner circle nestled snugly inside the triangle, and the triangle fit neatly and naturally into the outer circle. In Kepler’s mind, the outer circle represented Saturn’s orbit, the inner circle Jupiter’s. The triangle that tied the two together was the first shape in geometry. Kepler stared at that geometric emblem.

  He performed a quick calculation—the outer circle in his diagram was twice the circumference of the inner circle. And Saturn’s orbit was twice Jupiter’s. He had broken God’s code. Now Kepler set to work in a frenzy. If the orbits of the first two planets depended on the simplest geometric shape, a triangle, then the orbits of the next two planets must depend on the next simplest shape, a square.

  Kepler drew a circle, representing Jupiter’s orbit. The question was what circle would represent the orbit of the next planet toward the sun, Mars. In Kepler’s mind, the answer nearly shouted aloud. Inside the Jupiter circle, he drew a square. Inside that square he drew the one, special, God-designated circle that fit perfectly. That inner circle depicted Mars’s orbit.

  And Kepler could continue in this way for all the planets, working his way in toward the sun, arraying the planets just as Copernicus had shown them. The orbits were nested one within the other, and the size of one automatically dictated the size of the next. The first two orbits were built around a triangle, which has three sides; the next two around a square, with four sides; the next two around a pentagon, with five sides; and so on. Kepler set to work drawing squares, pentagons, hexagons, septagons, with circles in between.

  Kepler believed that God had arranged the planets’ orbits according to this geometric scheme. (For clarity, the diagram shows only the four outermost planets, not all six planets known in Kepler’s day.)

  Johannes Kepler had discovered the architecture of the solar system. Or so he believed, and in his fever dream he filled sheet after sheet with ever more elaborate geometric diagrams. For the young, unknown astronomer, this was dizzyingly exciting. Without looking out the window, he had not only assigned each planet to its proper place but shown why it had to occupy that place.

  It was perfect, it was elegant, and it was wrong. As Kepler took more time to compare the actual sizes of the planets’ orbits with the sizes his model predicted, he found mismatches he couldn’t explain away. He tried endless fixes. Nothing. How could God have led him astray?

  Chapter Twenty-Five

  Tears of Joy

  At last the light dawned. Kepler had been thinking in two dimensions, in the flat world of circles and triangles and squares. But the universe has three dimensions. How much time had he wasted? “And now I pressed forward again. Why look for two-dimensional forms to fit orbits in space? One has to look for three-dimensional forms—and, behold dear reader, now you have my discovery in your hands!”

  The switch to three dimensions represented far more than a chance to salvage a pet theory. One of the riddles that tormented Kepler had to do with the number of planets—there were exactly six. (Uranus, Neptune, and Pluto were not yet known.33) Why had God picked six, Kepler asked, “instead of twenty or one hundred”? He had no idea, and all his fussing with squares and pentagons and hexagons had brought him no nearer to an answer.

  But now he realized that he had overlooked a glaring clue. Euclid had proved, two thousand years before, that in three dimensions the story of symmetrical shapes has an extraordinary twist. Working in two dimensions, you can draw an endless succession of perfectly symmetrical, many-sided figures—triangles, squares, pentagons, hexagons, and so on, forever. If you had enough patience, you could draw a hundred-sided polygon or a thousand-sided one. (All you’d have to do is draw a circle, mark equally spaced dots on it, and then connect each one to its next-door neighbors.)

  In three dimensions, where there is more room, you might expect the same story—a handful of simple shapes like pyramids and cubes and then a cascade of increasingly complicated ones. Just as a pyramid is made of triangles pasted together, and a cube is made of squares, so you might guess that you could glue fifty-sided shapes together, or thousand-sided ones, and make infinitely many new objects.

  But you can’t. Euclid proved that there are exactly five “Platonic solids”—three-dimensional objects where each face is symmetrical and all the faces are identical. (If you needed dice to play a game, the mathematician Marcus du Sautoy points out, these five shapes are the only possible ones.) Here is the complete array. There are no others:

  Only five. And there are six planets. Now Kepler had it. He still had to work out the details, but at last he’d seen the big picture. Each planet traveled around the sun, its orbit confined to a particular sphere. The spheres sat one inside the other. But what determined the sizes of the spheres? God, the greatest of all geometers, surely had a plan. After a false start, Kepler had seen it. Each sphere fit snugly and symmetrically inside a Platonic solid. Each Platonic solid, in turn, fit snugly and symmetrically inside a larger sphere. In a flash, Kepler saw why God had designed the cosmos to have six planets and why those orbits have the sizes they do. He burst into tears of joy.

  “Now I no longer regretted the lost time,” he cried. “I no longer tired of my work; I shied from no computation, however difficult.” On and on he calculated, computing orbits, contemplating octahedrons and dodecahedrons, working without rest in the hope that at last he had it right but always terrified that once again his “joy would be carried away by the winds.”

  Kepler devised a new, more elaborate scheme to explain the planets’ orbits. God had built the solar system around the five “Platonic solids.” The diagram at right, with the sun at its center, is a detail of the drawing at left. The sun sits inside a nested cage; the inmost shape is an octahedron.

  But it wasn’t. “Within a few days everything fell into its place. I saw one symmetrical solid after the other fit in so precisely between the appropriate orbits, that if a peasant were to ask you on what kind of hook the heavens are fastened so that they don’t fall down, it will be easy for you to answer him.”

  * * *

  Kepler rejoiced in his success. “For a long time I wanted to become a theologian,” he told an old mentor. “For a long time I was restless. Now, however, behold how through my effort God is being celebrated through astronomy.”

  In 1596 he presented his theory to the world in a book called The Mystery of the Universe. Even with his book completed, Kepler fretted about whether his model fit the actual data about the planets’ orbits quite well enough. For the time being, he managed to fight down his doubts. He happily devoted long hours to constructing models of his solar system from colored paper and drawing plans for a version made of silver and adorned with diamonds and pearls. “No one,” he boasted, “ever produced a first work more deserving of admiration, more auspicious and, as far as its subject is concerned, more worthy.”

  In the decades to come Kepler would make colossal discoveries, but his pride in his elaborate geometric model never faded. Centuries later the biologist James Watson would proclaim his double helix model of DNA “too pretty not to be true.” Kepler had felt the same joy and the same certainty, but eventually the data left him no choice but to acknowledge that he had gone wrong, again.

  His perfect theory was only a fantasy, but it proved enormously fruitful even so. For one thing, Mystery of the Universe transformed Kepler’s career. He sent a copy of the book to Tycho Brahe, the leading astronomer of the day, who found it impressive. In time Kepler would gain access to Tycho’s immense and meticulous trove of astronomical data. He would pore
over those figures incessantly, over the course of decades, trying to make his model work and uncovering other patterns concealed in the night sky. Later scientists would rummage through Kepler’s collection of numerical discoveries and find genuine treasure among the dross.

  Kepler valued Mystery of the Universe so highly because it was there that he had unveiled his great breakthrough. But in the course of discussing his model of the heavens, he had scored another history-making coup. Kepler had followed Copernicus in placing the sun at the center of his model, but then Kepler had moved a crucial step beyond all his predecessors. Not only did all the planets circle the sun, he noted, but the farther a planet was from the sun, the slower it traveled in its orbit. Somehow the sun must propel the planets, and whatever force it employed plainly grew weaker with distance.

  Kepler had not yet found the law that described that force—that would take him another seventeen grueling years—but this was a breakthrough even so. Astrologers and astronomers had always focused their attention on mapping the stars and charting the planets’ journeys across the sky. The goal had been description and prediction, not explanation. No one before Kepler had ever focused on asking what it was that moved the planets on their way. From now on, scientists looking at the heavens would picture the stars and planets as actual, physical objects pushed and tugged by some cosmic engine and not simply as dots on a chart.

  “Never in history,” marvels the historian of science Owen Gingerich, “has a book so wrong been so seminal in directing the future course of science.”

  Chapter Twenty-Six

  Walrus with a Golden Nose

  From the start Kepler’s faith that God was a mathematician both impeded him and spurred him. First his faith lured him into devoting years to his Platonic pipe dream; when that dream dissolved it motivated him to search elsewhere, in the certain knowledge that there had to be some mathematical pattern that explained the solar system. Through all his years of searching, Kepler’s fascination was less with the objects in the sky—the sun, stars, and planets—than with the relationships among them. Not the things but the patterns. “Would that God deliver me from astronomy,” Kepler once wrote, “so I can devote all my time to my work on harmonies.”

  In time that work would yield many patterns, a few of them among the highest achievements of human thought but most of them nearly inexplicable to modern readers. When Kepler finally abandoned his elaborate geometric model of the planets, for instance, he replaced it with an equally arcane model based on music. This new search for “harmonies” built on Pythagoras’s age-old insight about strings of different lengths producing notes of different pitch. Kepler’s notion was that the planets in their various orbits, traveling at different speeds, corresponded to different musical notes, and “the heavenly motions are nothing but a continuous song for several voices (perceived by the intellect, not by the ear).”34

  Kepler’s new system, with its sopranos and tenors and basses, was as farfetched as its predecessor, with its cubes and pyramids and dodecahedrons. As it turned out, neither model had anything to do with reality. But in the course of his obsessive, misguided quest to prove the truth of his theories, Kepler did make genuine, epochal discoveries. Scientists would eventually dub three of these “Kepler’s laws,” though Kepler never gave them that name nor deemed them any more praiseworthy than his other finds.35

  Late in his life, when he looked back over his career, Kepler himself could scarcely pick out his breakthroughs from the mathematical fantasies that surrounded them. “My brain gets tired when I try to understand what I wrote,” he said later, “and I find it hard to rediscover the connection between the figures and the text, that I established myself.”

  Kepler was one of the most daring, insightful thinkers who ever lived, but his career only took off when he joined forces with an astronomer who was his opposite in almost every respect. Kepler was poor and lean, a creature of ribs and patches. Tycho Brahe was rich beyond measure. Kepler was shy and ascetic, Tycho hard-drinking and rowdy. Kepler was imaginative and creative, sometimes alarmingly so, Tycho a brilliant observer but a run-of-the-mill theoretician. But the two great astronomers needed one another.

  Tycho36 was a Danish nobleman with a private observatory on a private island. The most eminent astronomer of the generation before Kepler and Galileo, it was Tycho who had startled the world in 1572 by proving that the new star that had somehow materialized in the sky truly was a star. Nothing about Tycho was run-of-the-mill. Round, bald, sumptuously dressed, he looked like Humpty Dumpty with a walrus mustache and a velvet cloak. He ruled his mini-kingdom like a mini-king, presiding over lavish banquets and cackling over the antics of his court jester, a dwarf named Jepp.

  In his student days, Tycho had lost part of his nose in a swordfight. In one version of the story, the trouble started at a wedding celebration when another wealthy young Dane reminded everyone of some odd events from a few months before. Tycho had announced with great fanfare, in a poem written in elegant Latin flourishes, that a recent eclipse of the moon foretold the death of the Turkish sultan. But, it turned out, the sultan had died six months before the eclipse. Tycho’s rival told the story with gusto, and nearly all his listeners enjoyed it. Not Tycho. The retelling of the story led to bad blood and, soon after, a duel. Tycho nearly lost his life and did lose a chunk of his nose. For the rest of his life he sported a replacement made of gold and silver.

  Despite the bluster and showmanship, Tycho was a genuine scholar. His observatory was the best in Europe, outfitted with a dazzling array of precision-made sextants, quadrants, and other devices for pinpointing the positions of stars. The observatory stood in a grand, turreted castle that boasted fourteen fireplaces and an astonishing luxury, running water. In Tycho’s library stood a celestial globe five feet in diameter and made of brass; when a star’s position was established beyond a doubt, a new dot was carefully added to the globe. Tycho boasted that his observatory had cost a ton of gold, and Kepler complained that “any single instrument cost more than my and my whole family’s fortune put together.”

  Kepler had sent his Mystery of the Universe to Tycho and all the other eminent scientists he could think of. Many could not fathom what he was up to. Tycho, more mystically minded than Galileo and some of the other skeptics, replied enthusiastically and soon took Kepler on as his assistant. It was an arrangement with obvious benefits for both men. Tycho had devised a hybrid model of the solar system, partway between the ancient Earth-centered model and Copernicus’ sun-centered version. In this picture, the sun and moon orbited the Earth, and the five other planets orbited the sun. Tycho had compiled reams of scrupulously accurate observations, but without Kepler’s mathematical help he could not demonstrate the truth of his hybrid model. Kepler had no interest in Tycho’s model, but in order to make progress on his own theories he desperately needed Tycho’s records.

  But Tycho hoarded them. Torn between hope that the younger man could find patterns hidden within twenty years’ worth of figures and fear that he was giving away his treasure, Tycho clung to his numbers with a miser’s grip. Kepler snarled helplessly. And then, out of the blue, Tycho died. (He died of a bladder infection brought on, according to Kepler, by drinking too much at a banquet and refusing to leave the table to pee.) Kepler had been with Tycho only eighteen months, but now he had what he needed. “I was in possession of the observations,” Kepler noted contentedly, “and refused to hand them over to the heirs.”

  Chapter Twenty-Seven

  Cracking the Cosmic Safe

  For nearly twenty years Kepler would stare at Tycho’s figures, certain that they concealed a hidden message but for months and years at a time unable to make any headway in deciphering it. He had long since abandoned his geometric models, on the grounds that they simply did not fit the data. The problem was that nothing else did, either.

  Kepler knew, for example, how long it took each planet to orbit the sun—Mercury, 3 months; Venus, 7 months; Earth, 1 year; Mars, 2 years; J
upiter, 12 years; Saturn, 30 years—but try as he might he could not find a rule to connect those numbers. This was a task with some resemblance to making sense of the numbers on a football scoreboard if you had never heard of football. The number 3 appears sometimes and so do 7 and 14, but never 4 or 5. What could be going on?

  Even armed with Tycho’s astronomical data, Kepler took six years to find the first two of the three laws now named for him. The story of Kepler’s discovery of his laws is a saga of false starts and dead ends piled excruciatingly one upon another, while poor Kepler despaired of ever finding his way.

  Kepler’s first law has to do with the paths the planets travel as they orbit the sun. Kepler shocked his fellow astronomers—he shocked himself—by banishing astronomy’s ancient emblem of perfection, the circle. But Tycho’s data were twice as accurate as any that had been known before him, and Kepler, who had indulged himself in endless speculative daydreams, now turned the world upside down because of a barely discernible difference between theory and reality. “For us, who by divine kindness were given an accurate observer such as Tycho Brahe,” Kepler wrote, “for us it is fitting that we should acknowledge this divine gift and put it to use.” To take Tycho’s measurements seriously meant to acknowledge, albeit slowly and reluctantly, that the planets simply did not travel in circles (or in circles attached to circles or any such variant).

  Worn down by endless gruesome calculations, Kepler nearly despaired of ever finding the patterns hidden inside the astronomical records. (He referred wearily to his hundreds of pages of calculations as his “warfare” with the unyielding data). Finally he found that each planet orbits the sun not in a circle but in an ellipse, a kind of squeezed-in circle. This meant, among other things, that the distance from the sun to a planet was not constant, as it would be if the planet traveled in a circle, but rather always changing.