When mathematicians talk about symmetry, they mean immunity to possible change. In the words of the great mathematician Hermann Weyl: “A thing is symmetrical if there is something you can do to it so that after you have finished doing it, it looks the same as before.” For instance, the phrase: “Madam I’m Adam” reads the same backward or forward. In this case, we say that the sentence is symmetric under the operation of back-to-front reading. One of the most familiar of all symmetric patterns is that of a repeating, recurring motif. From the friezes of classical temples to carpets, the symmetry of repeating patterns has always produced a comforting familiarity and a reassuring effect. The symmetry transformation in this case is called translation, meaning a displacement by a certain distance along a line. The pattern is considered symmetric if it looks the same after we have displaced our view. The Victorian artist, poet, and printer William Morris, for instance, has produced many sumptuous wallpaper designs that are the embodiment of translational symmetry (Figure 1).
Symmetry under translation is not limited to the visual arts. Music, with its spirit of rhythm, and the periodic repetition of identical parts, is the temporal equivalent of Morris’s designs. Examine, for instance, the opening bars of Mozart’s famous Symphony no. 40 in G Minor (Figure 2). You don’t have to know how to read music to realize that the notes repeat, not only within each line of the score, but also between the first and within the second lines (compare “a” and “b”).
The most dramatic manifestation of symmetry under translation, however, is neither in the symmetry of shapes nor in scores of music. It is in the symmetry of the laws of nature. Not until the seventeenth century did humans even dream of the possibility that a body of laws exists that would explain the entire cosmos. Through the works of the likes of Galileo, Descartes, Newton, Maxwell, and Einstein, however, physicists now believe that they can formulate such laws. Here is where nature has been kind to us. The laws of nature are symmetric under translation! Unlike in the real estate business—where everything is location, location, location—our location in space does not make any difference to the laws of nature we deduce. If not for this symmetry, scientific experiments would have to be repeated in very new lab across the globe, and any hope of ever understanding the remote parts of the universe would be forever lost. Luckily the cosmos is governed by universal laws, rather than parochial bylaws—a hydrogen atom on Earth behaves precisely the same as a hydrogen atom in galaxies that are billions of light-years away (Figure 3).
Why do the laws of nature obey certain symmetries? We don’t really know, except that perhaps in a universe that does not possess such symmetries, complexity and life could not have emerged.