The Clockwork Universe: Isaac Newton, the Royal Society, and the Birth of the Modern World - Edward Dolnick (2011)
Part II. Hope and Monsters
Chapter 25. 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.”