Varieties of Symmetry
In Third Thoughts, Nobel Prize-winning author, Steven Weinberg offers a wise, personal, and wide-ranging meditation of science and society. Here is a brief excerpt looking at Weinberg’s perspective on symmetry.
When I first started doing research in the late 1950s physics seemed to me to be in a dismal state. There had been a great success a decade earlier in quantum electrodynamics, the theory of electrons and light and their interactions. Physicists then had learned how to calculate things like the strength of the electron’s magnetic field with a precision unprecedented in all of science. But now we were confronted with newly discovered exotic particles, some existing nowhere in nature except in cosmic rays. And we had to deal with mysterious forces: strong nuclear forces that hold particles together inside atomic nuclei, and weak nuclear forces that can change the species of these particles. We did not have a theory that would describe these particles and forces, and when we took a stab at a possible theory, we found that either we could not calculate its consequences, or when we could, we would come up with nonsensical results, like infinite energies or in finite probabilities. Nature, like a resourceful enemy, seemed intent on concealing from us its master plan.
At the same time, we did have a valuable key to nature’s secrets. The laws of nature evidently obeyed certain principles of symmetry whose consequences we could work out and compare with observation, even without a detailed theory of particles and forces. It was like having a spy in the enemy’s high command.
I had better pause to say something about what physicists mean by principles of symmetry. In conversations with friends who are not physicists or mathematicians, I find that they often take symmetry to mean the identity of the two sides of something symmetrical, like the human face or a butterfly. That is indeed a kind of symmetry, but it is only one simple example of a huge variety of possible symmetries.
The Oxford English Dictionary tells us that a symmetry is “the quality of being made up of exactly similar parts.” A cube gives a good example. Every face, every edge, and every corner is just the same as every other face, edge, or corner. This is why cubes make good dice; if a cubical die is honestly made, when it is cast it has an equal chance of landing on any of its six faces.
The cube is one example of a small group of regular polyhedra — solid bodies with at polygons for faces — that satisfy the symmetry requirement that every face, every edge, and every corner should be precisely the same as every other face, edge, or corner.
These regular polyhedra fascinated Plato. He learned (probably from the mathematician Theaetetus) that regular polyhedra come in only five possible shapes, and he argued in Timaeus that these were the shapes of the bodies making up the elements: earth consists of little cubes, while re, air, and water are made of polyhedra with four, eight, and twenty identical faces, respectively. The fifth regular polyhedron, with twelve identical faces, was supposed by Plato to symbolize the cosmos. Plato offered no evidence for all this — he wrote in Timaeus more as a poet than as a scientist, and the symmetries of these five bodies evidently had a powerful hold on his poetic imagination.
The regular polyhedra in fact have nothing to do with the atoms that make up the material world, but they provide useful examples of a way of looking at symmetries, a way that is particularly congenial to physicists. A symmetry is at the same time a principle of invariance. That is, it tells us that something does not change its appearance when we make certain changes in our point of view. For instance, instead of describing a cube by saying that it has six identical square faces, we can say that its appearance does not change if we rotate our frame of reference in special ways, for instance by 90 degrees around directions parallel to the cube’s edges.
The set of all the transformations of points of view that will leave something looking the same is called its invariance group. This may seem like an awfully fancy way of talking about things like cubes, but often in physics we make guesses about invariance groups, and test them experimentally, even when we know nothing else about the thing that is supposed to have the conjectured symmetry. There is a large and elegant branch of mathematics known as group theory, which catalogs and explores all possible invariance groups, and is described for general readers in two recently published books.
Each of Plato’s five regular polyhedra has its own invariance group. Each group is finite, in the sense that there are only a nite number of distinct changes in point of view that leave the polyhedron looking the same. All these different finite invariance groups are contained in an in finite group, the group of all rotations in three dimensions. This is the invariance group of the sphere, which of course looks the same from all directions.
For aesthetic and philosophical reasons, spheres also figured in early speculations about nature — as a model not of atoms, but of planetary orbits. The seven known planets (including the Sun and Moon) were supposed to be bright spots on spheres that revolve around the spherical Earth, carrying planets on perfect circular orbits. But it was hard to reconcile this with the observed motions of planets, which at times seem even to reverse their direction of motion against the background of stars. According to the neo-Platonist Simplicius, writing in the sixth century ad, Plato had put this problem to mathematicians at the Academy, almost as if assigning a bit of homework. “Plato lays down the principle,” says Simplicius, “that the heavenly bodies’ motion is circular, uniform, and constantly regular. Therefore he sets the mathematicians the following problem: What circular motions, uniform and perfectly regular, are to be admitted as hypotheses so that it might be possible to save the appearances presented by the planets?”
“Save the appearances” is the traditional translation, but what Plato meant by this is that a combination of circular motions must precisely account for the apparent motions of the planets across the sky.
This problem was addressed in Athens by Eudoxus, Calippus, and Aristotle, and then more successfully, with the introduction of epicycles, at Alexandria by Hipparchus and Ptolemy. The problem of planetary motions continued to vex astronomers and philosophers in the Islamic and Christian worlds, up to and beyond the time of Copernicus. Of course, much of the difficulty in solving Plato’s problem arose from the fact that the Earth and what we now call the planets go around the Sun, not the Sun and planets around the Earth. The Earth’s motion explained in a natural way why planets seem sometimes to jog backward in their paths through the zodiac. But even when this had been understood by Copernicus, he still had trouble making his theory agree with observation, because he shared Plato’s conviction that planetary orbits had to be composed of circles.
No really satisfactory solution to Plato’s homework problem could be found, because planetary orbits are actually ellipses. This was the discovery of Kepler, who incidentally as a young man had like Plato also been fascinated by the five regular polyhedra. Astronomers and philosophers for two millennia had been too much impressed with the beautiful symmetry of the circle and sphere.