Symmetry in Chaos: A Search for Pattern in Mathematics, Art and Nature

by Michael Field, by Martin Golubitsky
Oxford University Press, 218 pp., $35.00

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 …

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