Symmetry in Chaos: A Search for Pattern in Mathematics, Art and Nature
When Blake wrote of the Tyger’s “fearful symmetry” he was using the noun as a synonym for beauty. Today the word usually means any kind of regular pattern. Geometers sharpen the definition by making symmetry the property of a figure that stays the same after a given operation is performed. A snow crystal, the Star of David, and patterns in a kaleidoscope, for examples, have hexagonal or six-fold symmetry because they look the same after a rotation through any multiple of sixty degrees. You and the tiger have bilateral or mirror reflection symmetry because you both seem unchanged after a mirror has exchanged left and right sides. A wallpaper pattern has translation symmetry, meaning it is unaltered when shifted in any direction. If every other unit of a periodic pattern is mirror reversed, such as RЯRЯRЯR…, the symmetry is called a glide reflection.
The letter A has bilateral symmetry with respect to its vertical axis. H is richer in symmetry because it also looks the same when turned upside down. The letter O, if shaped like a circle, is the most symmetrical of all. In addition to reflection symmetry it has circular symmetry—it stays the same after an infinity of rotations.
Palindromes, such as “Straw? No, too stupid a fad. I put soot on warts,” have reversal symmetry with respect to letters. “You can cage a swallow, can’t you, but you can’t swallow a cage, can you?” is reversible with respect to words. The following poem, by J.A. Lindon, is reversible with respect to lines:
As I was passing near the jail
I met a man, but hurried by.
His face was ghastly, grimly pale.
He had a gun. I wondered why
He had. A gun? I wondered…why,
His face was ghastly! Grimly pale,
I met a man, but hurried by,
As I was passing near the jail.
Sentences, even musical scores, can be written that have upside-down and/or reflection symmetry. NOW NO SWIMS ON MON, a sign by a swimming pool, is the same inverted. CHOICE QUALITY appears on the side of Camel cigarette packages. Turn the words upside down and view them in a mirror. QUALITY is reversed but CHOICE, having a horizontal axis of reflection symmetry, is not.
All these symmetries involve static forms. Physicists broaden the term to cover properties of dynamic systems and their equations. Electrical charge is symmetrical with respect to an interchange of positive and negative. Magnetic force is symmetrical with respect to an interchange of north and south poles. Protons are the same as neutrons after what physicists call a “rotation”—in this case meaning that the spin directions of their three constituent quarks are reversed.
Emmy Noether, an eminent mathematician forced out of Germany because she was Jewish (she died in the United States in 1935), was the first to show that associated with every symmetry is an algebraic structure called a group. The “elements” of a group are operations. A square, for example, stays the same after rotation in either direction through multiples of right angles. A 360-degree turn is the same as no turn at all. This is called the “identity” operation. Four other operations of the square’s group are turning it over or reflecting it with respect to its vertical or horizontal axis, or either diagonal. If combining two operations has the same effect as the identity, they are called “inverses” of each other. For example, turning a square 90 degrees in one direction is the inverse of turning it 270 degrees the other way.
It is impossible to understand modern physics without understanding symmetries and groups; above all, without grasping the concept of broken symmetry. Water has spherical symmetry. Like a crystal ball it looks the same no matter how you turn it. But when water freezes, under certain conditions this perfect symmetry shatters to produce the lower but more beautiful hexagonal patterns of snowflakes.
With a bit of glue stick a golf ball to the top of a Mexican sombrero. The structure has circular symmetry. Without the glue, however, it is unstable. The ball rolls to the brim, breaking circular symmetry but leaving a lower mirror-reflection symmetry. Hold a cork in the center of a bowl of soup and you preserve the system’s circular symmetry. Let go. Surface tension breaks the symmetry by floating the cork to one side.
Immediately after the big bang, an unimaginably hot universe had perfect spherical symmetry. As it cooled, this and other symmetries began to vanish. Some of the breaks are fairly well understood, but others remain conjectural. The symmetrical electroweak force, a union of weak and electromagnetic forces, is believed to have been broken into the two forces by what is called the Higgs field, with its quantized Higgs particle. Do Higgs particles exist? Probably, but they are yet to be detected. There are many rival TOEs (Theories of Everything), but all of them place us in a cold universe of broken symmetries.
Several nontechnical books have discussed symmetry, notably Hermann Weyl’s classic Symmetry (1952), but Fearful Symmetry, by Ian Stewart and Martin Golubitsky, is far and away the most informative and up-to-date book yet on the topic. Moreover, it is entertainingly written and lavishly illustrated. Stewart is a mathematician at England’s University of Warwick, author of some fifty books, and the current writer of Scientific American’s department of Mathematical Recreations. Golubitsky is a University of Houston mathematician. He and Stewart have collaborated on many papers about how symmetry is related to the fast-growing research on chaos theory.
In chaos theory extremely tiny fluctuations in forces that make up a complicated physical system, such as weather, are rapidly magnified. It is said that the flutter of a butterfly’s wings in South America could trigger a cascade of causes and effects that would produce a cyclone in Kansas—a storm no meteorologist could have anticipated. In still air the smoke from a cigarette may rise in a straight plume until slight air movements break the symmetry and the plume dissolves into chaotic swirls. Chaos theory studies the ways in which symmetrical structures, both in pure mathematics and in the physical world, can plunge into chaos.
It is one of the great surprises of recent mathematics that seemingly disheveled structures are far from patternless. Fearful Symmetry was written after Stewart’s popular introduction to chaos, Does God Play Dice? (1989) Because symmetry is simpler, and underlies chaos, he regards his new books as a “prequel” to the previous one. On symmetry generally he and Golubitsky write:
Some symmetries are imposed by human agency. We can make a spherical ping-pong ball or a cylindrical coke can. We usually do. Their symmetry is convenient for technological control and development; but we could put fizzy drinks into asymmetric cans if we wanted to. For some reason, we seem to prefer symmetrical things. Aeronautical engineers have done calculations showing that an aircraft with one wing swept back and the other swept forward actually has advantages over the conventional bilaterally symmetric configuration. However, no aircraft manufacturer has yet dared to make a jumbo-yet with a skewed wing; it’s unlikely that the public would trust such an ungainly design.
While we may not understand the reasons for this human preference for symmetric objects, it’s easy to see how that could lead to their widespread manufacture. Manufacturing processes themselves are conducive to symmetry: it’s easier to make lots of copies of the same thing. Human-inspired symmetry isn’t such a problem. But symmetries arise in nature, spontaneously, and pose much deeper questions. Raindrops and planets are spherical. Crystals have lattice symmetries. Galaxies are spiral. Waves in the ocean are spaced periodically. The polio virus is an icosahedron. Hornets, hamsters, harriers, herrings, and humans are bilaterally symmetric. The methane molecule is a tetrahedron, with a carbon atom at the center and a hydrogen atom at each of the four vertices. A new form of the element carbon, known as “Buckminsterfullerene” after the famous architect, has recently been discovered: it has 60 carbon atoms, arranged at the vertices of a truncated icosahedron. Although designed to human specification, it’s proving unusually stable, and is believed to exist naturally in interstellar space.
How does all this symmetry arise?
The common underlying cause seems to be the fact (itself an even deeper mystery) that the universe is “mass-produced”—it’s composed of large numbers of identical bits, rolling off the cosmic production line.
The authors introduce broken symmetry by reproducing a classic photograph of the splash produced by a spherical drop of milk falling into a bowl of milk. Because the bowl, drop, and milk all have circular symmetry, the form of the splash is totally unexpected. It is a jeweled crown with twenty-four spikes, each with a tiny bead at its end! As the authors point out, it is something of a paradox that forces of high symmetry can transform a circularly symmetric structure to one of twenty-four-fold symmetry.
The book’s examples of broken dynamic symmetry are taken from a vast array of both natural and human-made phenomena. You can learn from Fearful Symmetry how to construct, from a disk and elastic bands, a whimsical “catastrophe machine.” It has unstable bilateral symmetry which suddenly snaps to asymmetry. You will be introduced to the broken symmetry of plants with helical stalks, the helical patterns of bark on certain trees, and the spiral forms of snail shells. One might expect the dew that glistens along the strands of spider webs to coat each thread smoothly. But no, surface tension breaks the continuity into regular spaced droplets.
Cosmology, as the authors make clear, swarms with broken symmetry. The spherical shapes of stars are demolished by a variety of forces: rotations that bulge equators, vibrations that alter shapes, asymmetrically placed sunspots, and other phenomena. The spiral arms of certain galaxies are broken symmetries probably caused by gravity, though cosmologists are not sure how. Something also not understood has broken the circular symmetry of Saturn’s rings into mysterious “spokes.” The same planet’s north pole is surrounded by a strange hexagonal pattern that slowly rotates. It is called “Godfrey’s kinky current,” and there is a photograph in the book to prove it exists.
In recent years cosmologists have been shaken by the discovery that the universe is much lumpier than previously suspected. Galaxies clump into enormous bubble-like clusters that surround monstrous voids, and here and there form huge “sheets” or “walls.” Exactly what broke the spherical symmetry of the primeval universe into these vast asymmetries is a question now hotly debated.
Once you grasp the concept of symmetry breaking, the authors write, you see it everywhere. In living forms an obvious example is provided by the beautiful symmetry of a freshly laid egg. Its interior is quickly transformed into the complicated bilateral symmetry of a bird or reptile. You and I are broken symmetries of fertilized eggs. Even the bilateral symmetry of animals is broken in hundreds of ways. Examples include the irregular spots on leopards, dogs, and cats, the huge claw of the fiddler crab, flatfish with both eyes on the same side, a bird’s crossed bill. Our heart is on the left side.
The authors describe several easily performed experiments in symmetry shattering. Let a hose hang vertically, with water coming from its nozzle. As the flow speed increases, the system’s circular symmetry becomes unstable and the hose starts to wobble violently. Rapidly rotating cylinders of water produce surprising patterns of vortices, stripes, and helices. Squashing symmetrical cans and cartons are familiar examples of how our actions can both destroy symmetry and produce new patterns.
A delightful chapter is devoted to time asymmetries in the gaits of horses, elephants, cats, kangaroos, camels, centipedes, and other creatures. A camel’s gait has a peculiar way of breaking bilateral time symmetry. Its left legs follow exactly the same movements as its right legs, but half a period out of phase. Most birds flap both wings simultaneously. The authors don’t mention it, but the chimney swift, like a bat, flaps them alternately. The chapter is headed by the following stanza:
A centipede was happy quite,
Until a frog in fun
Said, “Pray, which leg comes after which?”
This raised her mind to such a pitch,
She lay distracted in a ditch
Considering how to run.
The highest time symmetry possible is possessed by such steady-state systems as rocks. They may weather over very long time periods, but for shorter periods a rock has greater temporal symmetry than you and I have. However, as the authors confess, we are not inclined to rhapsodize over a stone’s “magnificent temporal symmetry.”
In their final chapter the authors take up briefly a philosophical question raised by the physicist Eugene Wigner in his often cited paper, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” How is it that mathematical theorems, obviously invented by human minds, apply so accurately to the outside world? They offer a simple, reasonable, answer: “Mathematics is effective in describing the universe because that’s where we got it from.” The cosmos, of which we are part, has an astonishingly deep mathematical structure that we only dimly comprehend. “God,” declared Plato, “ever geometrizes.” Earlier the authors quoted Paul Dirac, one of the great pioneers of quantum mechanics: “One could perhaps describe the situation by saying that God is a mathematician of a very high order, and he used very advanced mathematics in constructing the universe.”
In that sense, “Yes,” the authors answer the question in their book’s subtitle, “God is a geometer.” Then they add: “But never forget. She’s much better at it than we are.”
Four color plates in Fearful Symmetry display striking patterns created by computer programs which mix chaos with symmetry. The astonishing fact that out of chaos can emerge beautifully symmetrical patterns is the topic of another impressive book, Symmetry in Chaos, on which Martin Golubitsky also collaborated. The book’s other author, Michael Field, is a mathematician at the University of Sidney. It is an oversize volume with fifty-four full-color plates and more than a hundred illustrations in all. The book overlaps Fearful Symmetry in covering basic concepts of symmetry and chaos, but there is greater emphasis on periodic patterns such as wallpaper designs and quilts, and on chaos and fractals.
Fractals are swirling patterns that look the same when portions of them are dilated. Blow up even a very small segment and you will see that it contained the elements of the large one. Chaos and fractals closely intertwine. So many marvelous books have been written about fractals, starting with The Fractal Geometry of Nature by Benoit Mandelbrot, the “father of fractal theory,” that almost everybody now knows vaguely about them. The authors reproduce a New Yorker cartoon by Sidney Harris that shows a living room in which one woman is saying to another, “We did the whole room over in fractals.”
Field and Golubitsky are very good in explaining how equations and natural forces can produce stunning chaotic patterns with their “bifurcations” and “strange attractors.” The colors, by the way, are arbitrary. Each point or “pixel” on a computer’s monitor screen may be visited many times as the pattern evolves in obedience to a program. Colors assigned to the pixels are based on how many times a pixel is visited. For computer buffs, one of the book’s appendices gives several programs in BASIC that generate symmetrical chaos patterns.
Symmetry in Chaos begins innocently enough with elementary mathematics, but by the time you get to the end you will have been introduced to nontrivial number theory, group theory, probability theory, and the complex plane. The complex plane is similar to the familiar Cartesian coordinate system except that the horizontal axis represents real numbers, and the vertical axis represents imaginary numbers (products of real numbers and the square root of minus one). Computer pictures of fractals and chaos are generated on the complex plane.
It is almost impossible to write a book about symmetry without reproducing one or more pictures by Maurits Escher, and both books under review are no exceptions. American art critics still ignore Escher, but he has acquired an enormous cult-like following among mathematicians, physicists, and college students in general. Scores of prints of his mathematically inspired graphic art are available and now wrapping paper, T-shirts, scarves, neckties, and even socks feature Escher tesselations.
A dozen books about Escher are in print, but none is as mathematically sophisticated as Doris Schattschneider’s M.C. Escher: Visions of Symmetry. The author is a mathematician at Moravian College, in Bethlehem, Pennsylvania, and former chief editor of Mathematics Magazine. Her book’s more than four hundred illustrations, many in color, include almost 180 Escher drawings never before published.
Mrs. Schattschneider devoted fifteen years of research to her book, including trips to the Netherlands to interview friends of the artist. The book was first published in 1990, but I mention it here because it has just been reissued in less expensive paper covers. The author has enlightening comments on individual pictures, and there is an extensive bibliography. It, too, is an excellent introduction to geometrical symmetry.
It may be surprising to learn that a triangle or a quadrilateral of any shape will “tile” the plane periodically—i.e., can be laid edge to edge on a plane, as with tiles on a floor, without overlaps or spaces in between. When polygons have more than four sides, or have curved edges, it is not always easy to decide whether a given shape will tile a surface. There are many fascinating unsolved problems in tiling theory. It is amazing that Escher, who had no formal training in mathematics, would have based his tilings on so many different kinds of planar symmetry, even including technically difficult tesselations of non-Euclidian planes.
On a more recreational level involving word play, several oddly talented persons discovered about two decades ago that it is possible to write names and even phrases in such a way that they remain identical when turned upside down or reflected in a mirror. The effect can seem magical. Scott Kim’s Inversions (1980) was the first book about such hard-to-believe calligraphy. It was followed by Douglas Hofstadter’s Ambigrammi, published in Italy in 1987, but not yet here. Wordplay by John Langdon (for which I wrote a foreword) is the latest collection of this curious symmetry art. They are called ambigrams because they exploit the mind’s surprising ability to recognize words even when their letters are distorted in ambiguous ways. The illustrations on this page show some of Langdon’s constructions. Invert them and try to see how they were made.