Fearful symmetry

Tyger! Tyger! burning bright
In the forests of the night,
What immortal hand or eye
Could frame thy fearful symmetry?
—William Blake, The Tyger

Evariste Galois is perhaps one of the most romantic figures in mathematics. While still in school, he sent his great breakthrough in geometry to established Parisian mathematicians; unfortunately, the breakthrough was written out in such an ungodly scrawl that the wise men had no idea what to make of it. By the age of twenty, he was languishing in prison for his revolutionary acts (political, this time); with cholera threatening, he and other prisoners were sent to a clinic where he fell in unrequited love with a doctor’s daughter. Then, on May 30, 1832, he died of a wound from a gunshot fired in a duel that arose under murky circumstances.

The night before, realizing that he might not have another chance, Galois did some major cramming. He gave his best shot at explaining his ideas about geometry in the clearest language he could muster. (The name of his beloved, Stephanie, dotted the margins.)

“Maybe the fact that he stayed up all night doing mathematics was the [reason] why he was such a bad shot the next morning and got killed”, said Marcus du Sautoy in his TED talk on Galois and symmetry at the 2009 TED global conference. But du Sautoy, an Oxford mathematician, owes Galois quite a bit. The young mathematician discovered the rules governing symmetric shapes, shapes that can be rotated and flipped and look unchanged. Du Sautoy calls these manipulations the “magic trick” changes. “For Galois symmetry was all about motion, what can you do to a symmetrical object so it can looks the same,” du Sautoy said.

Symmetry, Marcus du Sautoy says, is “nature’s language.” It arranges the atoms in a ruby, and the piles of molecules that form a virus. Humans consider symmetric faces to be beautiful, he says, because symmetry, being difficult to achieve, is a token of strong genes and the sign of a desirable mate.

When Spain was under Muslim rule in the mid 14th century, the rulers built themselves a splendid palace known as the Alhambra, or the red fort. Because Muslim artists were forbidden from depicting animals or people, they found beauty in patterns and symmetry. Using Galois rules, you can determine that the gorgeous, intricate mosaics on the walls of the Alhambra contain 17 different kinds of symmetry in all, making it a treasure-trove for mathematicians. (A paper on the geometry of Islamic art appeared in Science in 2007.)

Walls and ceiling of the Alhambra. (Justus Hayes/Shoes on Wires/shoesonwires.com)

Du Sautoy goes on to say that there’s no stopping mathematicians from using Galois rules to go beyond three dimensions.
His breakthrough, du Sautoy says, “allows us to create symmetrical objects in the unseen world”—four, five, six dimensions and more.

“That’s where I work,” he says. “I create mathematical objects, symmetric objects using Galois’ language in very high dimensional spaces.”

As a final treat, du Sautoy named a new mathematical object he’d created after the person who could get closest to estimating the number of symmetries in a Rubix cube. (Try it yourself – he gives the answer at the end of the talk.) Of course, you can’t see a symmetric object in twelve dimensions, so the winner had to be content with a picture drawn in Galois’ mathematical language.

If you fancy having your own multidimensional symmetrical object named after you, you can donate \$10 to a Guatemalan charity that du Sautoy supports. He will then “stay up all night” building a new intangible, symmetric toy to stick your name on. He’s raised about \$3,000 this way—that’s a lot of late nights!