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Symmetry: A ‘Key to Nature’s Secrets’

Mike King
The five regular polyhedra. Steven Weinberg writes that ‘they satisfy the symmetry requirement that every face, every edge, and every corner should be precisely the same as every other face, edge, or corner…. Plato argued in Timaeus that these were the shapes of the bodies making up the elements: earth consists of little cubes, while fire, 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.’

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 esoteric particles—muons and dozens of types of mesons and baryons—most existing nowhere in nature except in cosmic rays. And we had to deal with mysterious forces: strong nuclear forces that hold partiicles together inside atomic nuclei, and weak nuclear forces that can change the nature 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 infinite probabilities. Nature, like an 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. There were symmetries that dictated that certain distinct processes all go at the same rate, and that also dictated the existence of families of distinct particles that all have the same mass. Once we observed such equalities of rates or of masses, we could infer the existence of a symmetry, and this we thought would give us a clearer idea of the further observations that should be made, and of the sort of underlying theories that might or might not be possible. It was like having a spy in the enemy’s high command.1


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 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 flat planes for faces, which satisfy the symmetry requirement that every face, every edge, and every corner should be precisely the same as every other face, edge, or corner. Thus the regular polyhedron called a triangular pyramid has four faces, each an equilateral triangle of the same size; six edges, at each of which two faces meet at the same angle; and four corners, at each of which three faces come together at the same angles. (See illustration on this page.)

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 fire, 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 representing the elements 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 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, by rotating it or moving it. In addition to describing a cube by saying that it has six identical square faces, we can also say that its appearance does not change if we rotate it in certain ways—for instance by 90° around any direction parallel to the cube’s edges.

The set of all such transformations of point of view that will leave a particular object looking the same is called that object’s invariance group. This may seem like a 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: Symmetry: A Journey into the Patterns of Nature by Marcus du Sautoy and Why Beauty Is Truth: A History of Symmetry by Ian Stewart.


The symmetries that offered the way out of the problems of elementary particle physics in the 1950s were not the symmetries of objects, not even objects as important as atoms, but the symmetries of laws. A law of nature can be said to respect a certain symmetry if that law remains the same when we change the point of view from which we observe natural phenomena in certain definite ways. The particular set of ways that we can change our point of view without changing the law defines that symmetry.

Laws of nature, in the modern sense of mathematical equations that tell us precisely what will happen in various circumstances, first appeared as the laws of motion and gravitation that Newton developed as a basis for understanding Kepler’s description of the solar system. From the beginning, Newton’s laws incorporated symmetry: the laws that we observe to govern motion and gravitation do not change their form if we reset our clocks, or if we change the point from which distances are measured, or if we rotate our entire laboratory so it faces in a different direction.2

There is another less obvious symmetry, known today as Galilean invariance, that had been anticipated in the fourteenth century by Jean Buridan and Nicole Oresme: the laws of nature that we discover do not change their form if we observe nature within a moving laboratory, traveling at constant velocity. The fact that the earth is speeding around the sun, for instance, does not affect the laws of motion of material objects that we observe on the earth’s surface.3

Newton and his successors took these principles of invariance pretty much for granted, as an implicit basis for their theories, so it was quite a wrench when these principles themselves became a subject of serious physical investigation. The crux of Einstein’s 1905 Special Theory of Relativity was a modification of Galilean invariance. This was motivated in part by the persistent failure of physicists to find any effect of the earth’s motion on the measured speed of light, analogous to the effect of a boat’s motion on the observed speed of water waves.

It is still true in Special Relativity that making observations from a moving laboratory does not change the form of the observed laws of nature, but the effect of this motion on measured distances and times is different in Special Relativity from what Newton had thought. Motion causes lengths to shrink and clocks to slow down in such a way that the speed of light remains a constant, whatever the speed of the observer. This new symmetry, known as Lorentz invariance,4 required profound departures from Newtonian physics, including the convertibility of energy and mass.

The advent and success of Special Relativity alerted physicists in the twentieth century to the importance of symmetry principles. But by themselves, the symmetries of space and time that are incorporated in the Special Theory of Relativity could not take us very far. One can imagine a great variety of theories of particles and forces that would be consistent with these space-time symmetries. Fortunately it was already clear in the 1950s that the laws of nature, whatever they are, also respect symmetries of other kinds, having nothing directly to do with space and time.

There are four forces that allow particles to interact with one another: the familiar gravity and electromagnetism, and the less well-known weak nuclear force (which is responsible for certain types of radioactive decay) and strong nuclear force (which binds protons and neutrons in the nucleus of an atom). (I am writing of a time, during the 1950s, before the formulation of the modern Standard Model, in which the three known forces other than gravity are now united in a single theory.) It had been known since the 1930s that the unknown laws that govern the strong nuclear force respect a symmetry between protons and neutrons, the two particles that make up atomic nuclei.

Even though the equations governing the strong forces were not known, the observations of nuclear properties had revealed that whatever these equations are, they must not change if everywhere in these equations we replace the symbol representing protons with that representing neutrons, and vice versa. Not only that, but the equations are also unchanged if we replace the symbols representing protons and neutrons with algebraic combinations of these symbols that represent superpositions of protons and neutrons, superpositions that might for instance have a 40 percent chance of being a proton and a 60 percent chance of being a neutron. It is like replacing a photo of Alice or of Bob with a picture in which photos of both Alice and Bob are superimposed. One consequence of this symmetry is that the nuclear force between two protons is not only equal to the force between two neutrons—it is also related to the force between a proton and a neutron.

Then as more and more types of particles were discovered, it was found in the 1960s that this proton–neutron symmetry was part of a larger symmetry group: not only are the proton and neutron related by this symmetry to each other, they are also related to six other subatomic particles, known as hyperons. The symmetry among these eight particles came to be called “the eightfold way.” All the particles that feel the strong nuclear force fall into similar symmetrical families, with eight, ten, or more members.

  1. 1

    This article is based in part on a talk given at a conference devoted to symmetry at the Technical University of Budapest in August 2009. 

  2. 2

    For reasons that are difficult to explain without mathematics, these symmetries imply important conservation laws: the conservation of energy, momentum, and angular momentum (or spin). Some other symmetries imply the conservation of other quantities, such as electric charge. 

  3. 3

    Strictly speaking, Galilean invariance applies only approximately to the motion of the earth, since the earth is not moving in a straight line at constant speed. It is true that the earth’s motion in its orbit does not affect the laws we observe, but this is because gravity balances the effects of the centrifugal force caused by the earth’s curved motion. This too is dictated by a symmetry, but the symmetry here is Einstein’s principle of general covariance, the basis of the general theory of relativity. 

  4. 4

    Lorentz had tried to explain the constancy of the observed speed of light by studying the effect of motion on particles of matter. Einstein was instead explaining the same observation by a change in one of nature’s fundamental symmetries. 

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