This was the gloomy end of the most romantic of all mathematicians—a young man in whose mind the sweeping ideals of the French Revolution were inseparable from the revolutionary new branch of mathematics he had invented.  Galois is the originator of Group Theory — the mathematical language that describes symmetry.    

 

Symmetry and Group Theory

Often, to truly understand a subject, you need to know a specific language that best describes that subject. The language of the financial world, for instance, is the language of arithmetic operations. You can compare at a glance the strengths of two corporations by examining a few key figures. In order to fully explore his laws of motion, Newton needed to invent the language of calculus. Mathematicians, scientists, stock market analysts, and even artists light their way through the labyrinths of the world of symmetries by the language of group theory. The meaning of “symmetry” is immunity to a possible change. Symmetry represents the stubborn cores of forms, music, and mathematical objects that remain unchanged under transformations.  For instance, the phrase “A Man, a plan, a canal, Panama” is symmetric with respect to back-to-front reading letter by letter. Sentences with this property are called palindromes. Snowflakes have six-fold rotational symmetry—rotating them about an axis (through their centers) perpendicular to their plane by angles of 60, 120, 180, 240, 300, and 360 degrees leaves the shape unchanged.  Any transformation that leaves a certain system unchanged (such as the above rotations of the snow flake) is called a symmetry transformation of that system. Symmetry transformations have an interesting property called “closure”—if you apply one symmetry transformation followed by another one, the result is yet another symmetry transformation. This is easy to understand, if X is a symmetry transformation, that is, its application leaves the system unaltered, and Y is another symmetry transformation, then clearly X followed by Y is also a symmetry transformation. This property has led to the realization that symmetry transformations can be described by mathematical structures called “groups.” Like some exclusive clubs, a “group” is composed of members that have to obey very strict rules.

A mathematical group is a collection of entities that obey four rules with respect to some operation. A familiar group is the collection of all the integer numbers (posi-tive, negative, and zero) in conjunction with the arithmetic operation of addition. The four properties that define a group are:

1. Closure. The offspring of any two members of the group that are combined by the group operation is also a member.  For instance, the sum of any two integers is also an integer.
2. Associativity. When combining any three ordered members, the result is the same, irrespective of which pair you combine first. For instance, (3 + 2) + 4 = 3 + (2 + 4), where the brackets (sometimes called “the punctuation marks of mathematics”) indicate which pair you added up first.
3. Identity. The group has to contain an identity element—one that when combined to any member, leaves the member unchanged. Clearly, in the integers, the number zero plays this role with respect to addition.
4. Inverse. Every member of the group has an inverse, such that when the member is combined with its inverse, it gives the identity. In the group of integers, the inverse of any integer is the integer that has the same absolute value, but the op-posite sign (e.g., 2 is the inverse of –2).
   

Hard to believe, but this simple definition has led to a theory that embraces all the symmetries in our world. Even seasoned mathematicians continue to be amazed, as the English geometer Henry Frederick Baker (1866–1956) has once put it: “What a wealth, what a grandeur of thought may spring from what slight beginnings.” Group theory simply manages to identify those unchanging cores of shapes, art forms, or mathemati-cal concepts, even if they are masquerading under dis¬guises of different disciplines.  Isn’t it absolutely mind boggling that this awe-inspiring theory was born in the mind of a teenager?

A Mysterious Death

You might have expected that every intimate detail in the life of such a prominent mathematician (as the person who originated group theory) would be widely known.  After all, historian of mathematics Eric Temple Bell (1883–1960) once commented that: “Whenever groups dis¬closed themselves, or could be introduced, simplicity crystallized out of comparative chaos.” Yet, Galois’s death has remained veiled in mystery for almost two centuries.  What is known is that Galois was killed in a duel with pistols on that fateful morning in 1832, but the questions of who killed him and why have been the subject of conspiracy theories galore.  Biographers have further been perplexed by the fact that the wounded young man appeared to have been aban-doned in the field.

Following three years of intensive research, I propose that the fog surrounding Galois’s mysterious death may have finally lifted.

The known facts concerning Galois’s activities in the last week of his life are precious few.  Even Galois’s own letters, indicating that “two patriots” (meaning active republicans in post-Napoleonic France) provoked the duel over  “something so contemptible” involving an “infamous coquette,” did not shed sufficient light on the identity of his opponents or their motives.  Galois ended his letter that was addressed “to all republicans,” with the sentence:  “Forgive those who kill me, they are of good faith.”  There is little doubt that these words, reminiscent of those of Christ on the cross, reflect traces of the religious education Galois had received from his mother.

The fact that Galois was a revolutionary firebrand inspired many of his early biographers to speculate that political enemies killed Galois.  A few have allowed their imaginative plots to take off and include even more intrigue, suggesting that the “coquette” was in fact a police agent masquerading as a prostitute.
The first clue pointing to an unrequited love as the potential cause for the duel came from the work of an unlikely “detective”—a Uruguayan priest.  Using a magnify-ing glass and special lighting to examine Galois’s papers, Carlos Alberto Infantozzi discovered in 1968 the identity of the “infamous coquette”—Stephanie Potterin du Motel.  This young woman lived in a building that housed a convalescent home where the troubled Galois was placed on parole upon being released from prison.  Stephanie was certainly neither a prostitute nor a police provocateur.  Based on Infantozzi's “forensic” work, as well as on a newspaper article from the period reproduced by the French author Andre Dalmas in 1956, Dalmas and the American physicist and author Tony Rothman have started to painstakingly put together the pieces of the puzzle.  They suggested that the person who shot Galois was Stephanie’s lover (and a personal friend of Galois), since the young seductress was playing a double game with the hearts of the two young men.

This was more or less the accepted consensus until 1996, when a new biography by an Italian historian of mathematics turned the entire story on its ear.  Laura Toti Rigatelli suggested that the famous duel wasn’t even a duel at all!  Rather, this some-what Machiavellian theory proposed that Galois sacrificed himself for the Republican cause—the republicans needed a corpse to stir up rebellion, and he offered his.  While many accepted Toti Rigatelli’s story, not all did.  The French researcher and author Jean-Paul Auffray, who conducted an extensive study of documents related to Galois, concluded that the duel was real.  Auffray reintroduced the theory that the duel was provoked by the unfortunate love affair with Stephanie, and suggested that one of the opponents was none other than Stephanie’s father.

I have always been fascinated by Galois.  How can you not be, when you realize that this flamboyant romantic brought about one of the greatest breakthroughs in mathematics, and that he achieved this feat before the age of twenty!  When I started to research the life, and especially the death, of this visionary genius, I decided to embark on this task with no prejudices, and to leave no stone unturned.  Having had the added advantage of being able to examine critically all the evidence collected by numerous researchers and their conclusions, in three years I was able to develop what at least appears to be an entirely self-consistent picture.  While the new theory clearly contains elements of previous scenarios, it combines these elements with new insights that give them, in my humble opinion, an enhanced credibility.  I therefore strongly believe that I have come closer to the truth than was ever possible before.

So, who killed Galois and why?  A key point ignored by many biographers is that Galois always talked about two people who provoked the duel.  One could, there-fore, not expect to find a complete answer without an identification of both opponents.  My conclusion is that these two people were Denis Faultrier and Ernest Duchatelet.  The former was a close friend of Stephanie’s family, and he later married her widowed mother.  The latter was Galois’s republican friend (and Stephanie’s presumed lover), and it was he who shot the fatal bullet. The entire affair was a classical case of cherchez la femme.  From the surviving pieces of Stephanie’s two letters to Galois we learn that either by some careless words, or by too impetuous a behavior, the inexperienced Galois offended Stephanie, who immediately informed her lover and “so-called uncle.”  When the two men confronted Evariste, the hot-blooded young man added insult to injury, referring to the entire incident as a “miserable piece of gossip.”  At a time when invitations to duels were issued at the drop of a hat, this was more than enough for the two men to challenge Galois.  A seventeen-year-old young woman who did not return his love sealed the fate of one of the most brilliant mathematicians to have ever lived.  Why did Galois appear to have been left wounded on the ground by most, if not all, of the seconds?  Galois’s autopsy report describes a large bruise on his head that was probably caused when he fell.  He might have been knocked unconscious and presumed dead.  One of the reports from the period notes that a “former officer” brought Galois to the hospital.  This fits Denis Faultrier, the second opponent in my scenario, like a glove.  Can, therefore, the 173-year-old “cold case” be closed?  Hopefully, yes.  But with so many gaps in the hard evidence, uncertainties are likely to always remain.  What is certain, is the fact that Evariste Galois will always be remembered as one of the most creative individuals to have ever lived.  The new branch of mathematics that he estab-lished has expanded far beyond the boundaries of pure mathematics, into the realms of music, visual arts, physics, and wherever symmetries can be found.  His romantic spirit makes him the “James Dean” of mathematics.