Thought Experiments

29
Dec

Beautiful Losers: Kepler’s Harmonic Spheres

This essay is part of the series Beautiful Losers.

Like Plato, the German astronomer Johannes Kepler believed that the five Platonic solids provided an essential blueprint for our universe. Six planets were known to Kepler, and he believed that they were carried around on nested globes that he called the celestial spheres. Kepler reasoned that five solids could correspond to six planets, if the solids—or more precisely, their bounding surfaces—marked the spaces between planetary spheres. He described this elegant construction in his Mysterium Cosmographium in 1596.

Kepler proposed that Mercury’s sphere supports a circumscribed octahedron, which is inscribed within Venus’s sphere. Then we have icosahedron, dodecahedron, tetrahedron, cube interpolating respectively Venus-Earth, Earth-Mars, Mars-Jupiter and at last Jupiter-Saturn. This revelation of cosmic order was, for Kepler, rapturous:

I wanted to become a theologian; for a long time I was unhappy. Now, behold, God is praised by my work even in astronomy.

It immediately suggests a plan of construction that human artists can mimic—as Kepler did himself—in worthy, gorgeous models:

A model of Kepler's solar system, on display at the Technical Museum, Vienna.

Photo by Sam_Wise. Source.

Though equally (that is, completely) wrong, Kepler’s conception reaches a higher level, scientifically, than Plato’s speculations, for it makes concrete numerical predictions about the relative sizes of planetary orbits, which can be compared to their observed values. The agreement, while not precise, was close enough to convince Kepler he might be on the right track. Encouraged, he set out, courageously, to prove it.

Thus Kepler’s discovery set him on his storied career in astronomy. As his work developed, however, problems with his original model emerged. Late in the 16th century, the astronomer Tycho Brahe was making exquisitely accurate observations of the motions of the stars and planets. As Kepler strove to do justice to Tycho’s work, he discovered that the orbit of Mars is not circular, but follows an ellipse. This and other discoveries fatally undermined Kepler’s beautiful system of celestial spheres. (Eventually even the numerology collapsed, with the discovery of Uranus in 1781, though Kepler was spared that ignominy.) Through the arduous, devoted labor his vision inspired he found other regularities among the orbits of the planets—his famous three laws of planetary motion—whose accuracy could not be doubted. In the end he’d arrived at a different universe than the one he first envisioned, and hoped for. He reported back:

I write the book. Whether it is to be read by the people of the present or of the future makes no difference: let it await its reader for a hundred years, if God himself has stood ready for six thousand years for one to study him.

Kepler’s hoped-for reader emerged not quite a hundred years later: Isaac Newton. To me, the first illustration of Newton’s Principia (1687) worthily transmits the most consequential thought-experiment ever. With a few cartoonish strokes, it both presages a new, universal theory of gravity and embodies a new concept of scientific beauty: dynamic beauty.

Newton's dynamical thought-experiment for the universality of gravitation. Via Wikimedia Commons

Imagine standing atop a tall mountain on a spherical earth, throwing stones horizontally, harder and harder. To keep things clear, remove from thought the damping influence of the atmosphere. At first it is clear what will happen, based on everyday experience: The stones will travel further and further before landing. Eventually, when the initial velocity becomes large enough, the stones will pass over the horizon; then its landing point will circuit the planet. Visualizing the developing situation, as in Newton’s diagram, it is easy to imagine the progress of trajectories leading to a circle (duck!). In this way we begin to see how the same force that pulls bodies to Earth might also support orbital motion. We see that orbiting is a process of constantly falling, but toward a (relatively) moving target.

I like to think that the images in this diagram reveal the deep inspiration, pre-mathematical and even pre-verbal, behind young Newton’s program of research (parallel to how young Kepler’s harmonic spheres inspired his). It contains the germ of universal gravitation: With (imaginary) taller mountains as launching pads, we fill the sky with possible orbits. Might the Moon occupy one of them? And if Earth’s Moon, why not Jupiter’s moons, or the Sun’s planets? And this question too begs for an answer: Throw harder still—what are the shapes of the resulting trajectories? The answer, which Newton’s mathematics allowed him to answer, is that they make more and more eccentric ellipses—the very shape that Kepler had used to fit planetary orbits!

In Newton’s dynamical approach, the beauty of planetary motion is not embodied in any one orbit, or in the particular set of orbits realized in the Solar System. Rather it is in the totality of possible orbits, which also contains trajectories of falling bodies. Putting it another way: The deepest beauty lies in the equations themselves, not in their particular solutions. Classical physics, initiated by Newton’s brilliantly successful celestial mechanics, suggests that it is misguided to expect, as Kepler and Plato did, to find ideal symmetry embodied in any individual physical object, be it the Solar System or an elemental atom. Astronomers in recent years have identified dozens of extrasolar planetary systems, and found that they come in a wide variety of shapes and sizes. And yet…

Physical requirements can privilege, among the infinity of possible solutions to beautiful dynamical equations, special ones—often especially beautiful ones. Consider crystals: They are of course quite real and tangible natural objects; they can be grown in controlled, reproducible conditions; and their form is often highly symmetric. Kepler himself wrote a monograph featuring the six-fold symmetry of snow crystals.

We discover the same thing, spectacularly, in the quantum theory of atoms. An electron interacting with a proton obeys the same species of force law as a planet orbiting the Sun. Schrödinger’s equation, no less than Newton’s, allows an enormous infinity of complicated solutions. (In fact, much more so!) But if we focus on the solutions with the lowest energies—the solutions that coldish hydrogen atoms will settle into, after radiating—we pick out a special few. And those special solutions exhibit rich and intricate symmetry. They fulfill, as they transcend, the visions of Plato, Kepler, and Newton.

The quantum mechanical wave function for a typical stationary state of the hydrogen atom. Image from http://vqm.uni-graz.at, used with permission of Bernd Thaller.

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Frank Wilczek

    Frank Wilczek has received many prizes for his work in physics, including the Nobel Prize of 2004 for work he did as a graduate student at Princeton University, when he was only 21 years old. He is known, among other things, for the discovery of asymptotic freedom, the development of quantum chromodynamics, the invention of axions, and the exploration of new kinds of quantum statistics. Frank is currently the Herman Feshbach professor of physics at MIT. His latest book is The Lightness of Being.

    • Guest

      This whole series is terrific! I remember learning some of this in university but that was more than 20 years ago; as I finished my survey courses as an undergrad and moved into liberal arts, I (unfortunately!) left behind this whole other world YOU live in, work in, and teach about. Thanks so much for reaching out to those of us who work in fields far from yours – I am really enjoying this!