by Jonah Lehrer
But tracing with what? Warhol had a light bulb in his projector; Ingres was drawing in the early nineteenth century, the last age of candlelight. After researching the available technology, Hockney decided that Ingres was probably using a camera lucida, essentially a prism on the end of a metal stick. When positioned in front of a subject, the prism creates the illusion of a mirror image on the paper below. (The invention of photography relied on these same spare parts.) As Hockney notes, the camera lucida is not easy to use. If the subject moves, or if the paper slides, the traced marks are useless. However, once an artist masters the trick, he or she can sketch quickly and with a seemingly inexplicable accuracy.
And it wasn’t just Ingres. Hockney would eventually conclude that most of the Old Masters also relied on optical science to create mysteries within their art. Take Caravaggio. Hockney was drawn to Caravaggio due to his technical breakthroughs. Never before had someone painted real people doing real things—eating, talking, decapitating—with such astonishing accuracy. As the critic Robert Hughes observed, “There was art before [Caravaggio] and art after him, and they were not the same.”18
Hockney found the tells in Caravaggio’s details. The Supper at Emmaus (1601), for instance, is a painting with the clarity of a photographic print: the sheen of the grapes, the foreshortened arms of Christ and his apostles, the mottled shadows on the textured tablecloth. “To any draughtsman, eyeballing that is virtually impossible,” Hockney writes. “Perhaps you have to be a practitioner to know this. Art historians don’t draw anymore.”19
And yet, the painting is also riddled with basic errors of composition. Cleopas’s hands are of different sizes; Christ exists in a different plane from everyone else; the fruit is luscious but oddly flat. For Hockney, these mistakes are like the misdirections used by a magician—their existence hints at the real explanation, at least if you know what to look for.III
According to Hockney, these details suggest that Caravaggio was relying on some kind of optical device to create his paintings. The most likely tool was a crude camera obscura, in which a system of convex lenses and mirrors project an image onto the wall of a dark room. Caravaggio would have his models pose one at a time, assembling the scene from their individual snapshots. (“For the people of those times, such paintings were motion pictures,” Hockney said. “Their eyes were invited to move through the unfolding story.”)20 This helps explain why Caravaggio preferred to work in basements with a single source of illumination: the setup provided the ideal conditions for his primitive camera.
Hockney is adamant that the use of these tools doesn’t lessen the genius of these painters. “Optics do not make marks, only the artist’s hand can do that,” he writes.21 What Hockney’s hypothesis does reveal, however, is the extent to which artists have always sought out new ways of representing the material world, creating paintings so realistic they become mysteries. And so we marvel at Ingres’s perfectly sketched mouth and the uncanny perspective of Caravaggio’s lute; the clarity of Vermeer’s light and the glittering eyes of Hal’s subjects. We keep staring at these fixed images because they feel like magic, works of impossible beauty hanging on the wall. Even when we know how they were done, we still can’t believe someone pulled it off.
The Clueless Code
Bill Tutte and his wife, Dorothea, spent most of their lives in a modest house by the Grand River in West Montrose, a small town in the Mennonite country of Ontario. Tutte enjoyed long walks in the woods with Digby, his boxer, and afternoon games of chess accompanied by a cup of hot cocoa.IV He was the kind of man who would apologize before winning: “I’m sorry, but I believe that is checkmate.” In the summer, he liked to weed the riverbank, clearing space for his favorite wildflowers.
But mostly what Bill Tutte did was work. A math professor at the University of Waterloo, he was an expert in graph theory, devoting much of his professional life to the four-color-map problem. In its simplest version, the problem refers to the minimum number of colors required so that every country has a color different from all of its bordering neighbors. Tutte spent decades trying to prove that the answer was four. (In the mid-1970s, a supercomputer confirmed that he was right.) Dorothea often lamented his strict work habits, but Tutte feared losing his mathematical talent. He was sure it would be gone by the time he turned forty.
What Tutte couldn’t tell anyone is that his greatest achievement might have come when he was only twenty-four years old. At the time the world was at war. Tutte was a code breaker, tasked with cracking the most difficult Nazi ciphers. In many respects, a code is like an act of magic, only there is no mystifying performance—just inexplicable information. (The strings of numbers on a lottery ticket are another code.) We know there must be a trick—the random symbols have to mean something—we just have no idea what it is. The job of the code breaker, in this sense, is to reverse engineer the ruse, to figure out the secret method of encryption.
Tutte had a gift for code breaking. His wartime work was so valuable that the British government classified it as Ultra, a higher classification than the standard highest, Most Secret, and retained this classification for decades. To have discussed this achievement, even in hushed tones to family and friends, could have earned Tutte a charge of treason. So, for his entire career as an academic, none of his colleagues knew that this quiet mathematician had changed the course of history.
Tutte was born in Newmarket, a small town about seventy miles north of London. His parents lived in a modest flint cottage at the end of a footpath; both worked at the local racing stable, his father as a gardener and his mother as a housekeeper. At an early age, Bill displayed a gift for numbers, which led to a scholarship to Trinity College of Cambridge University.22 Although he majored in chemistry—it seemed like the practical option—Tutte’s joy was mathematical puzzles.V
In 1940, Tutte’s reputation for solving these puzzles led to an interview with the Government Code and Cypher School (GC&CS), a division of the British intelligence services devoted to cracking the secret codes of other countries. Tutte aced the interview and was sent to Bletchley Park, a rambling estate that had become the center of the British government’s code-breaking efforts. At Bletchley, there were trained mathematicians, but also crossword puzzle experts, national chess champions, Egyptologists, classicists, dictionary editors, corporate managers, and even novelists. Winston Churchill, after meeting this ragtag assortment of code breakers, reportedly told the head of MI6, “I know I told you to leave no stone unturned to get staff, but I didn’t expect you to take me literally.”23 Tutte fit right in.
The young Tutte was assigned to Bletchley Park’s “Fish” team. Their job was to break a set of German codes that encrypted messages using a rotor machine that mapped each input letter, “plain text,” to an alternate output letter, “cipher text.” Different models of the same type of cipher machine differed in their complexity: the more rotors, the more difficult it was to reverse the trick and untangle the link between cipher text and plain text. Tutte was given the task of breaking the Lorenz cipher, reserved for use by only Hitler and the generals in his high command. Code breakers called it Tunny, or tuna fish.
The Germans believed this cipher to be unbreakable because the Lorenz SZ40 machine used twelve different rotors, thus generating more than 1.6 quadrillion cipher texts for the same plain text. (As a point of comparison, the Enigma cipher—the one cracked by Alan Turing—relied on a machine using three rotors.) In the early 1940s, there was no conceivable way to break the Lorenz cipher because the initial settings of the rotors would change with every message. It was like watching the same magic trick again and again, except the magician used a different method with every performance.VI
The Tunny messages arrived at Bletchley Park in their encrypted form, as pages of printouts with row after row of dots and Xs. Each of those “five-bit” rows corresponded to a particular letter. The letter A, for example, was two Xs followed by three dots; the Cyrillic Ж was one dot followed by four Xs. What made Loren
z so difficult to figure out was that in its cipher, two lines were used for each letter, meaning code breakers had to determine the method for combining each already encrypted letter to spit out the actual letter that was being transmitted.
Unfortunately for the Bletchley Park code breakers, they had no way of figuring out the cryptographic key, or how the rotors changed as the message unfolded, or even how many rotors the machine contained. The method of Tunny remained a mystery, Hitler’s secrets hidden behind near-infinite layers of orchestrated mechanical confusion.
On August 30, 1941, the Allies received a crucial break. A German army radio operator in Athens sent a Tunny message of approximately four thousand characters to an operator in Salzburg, Austria. However, something went wrong in that transmission; Salzburg asked for a resend. The Athens operator now made two catastrophic mistakes. First, he used the same twelve-letter key—HQIBPEXEZMUG—to send the second message. By itself, this wouldn’t have given the Bletchley code breakers any clues, since they would have just received the same impenetrable message twice. But the Athens messenger made slight edits to his resend, abbreviating certain words and adjusting the punctuation. (He almost certainly did so to save a few keystrokes.) These changes created a “depth,” an essential tool for the code breakers since it allowed them to compare the messages and test out various substitution ciphers. After ten days of painstaking decryption, Colonel John Tiltman deciphered the Athens message. It was the first Tunny code ever deciphered.
The triumph was temporary. Although Tiltman had cracked this message, the Bletchley Research Section still had no idea how the Lorenz machine worked. They could only crack the Tunny code when the German operators made a careless error, giving away a depth by accident. Roy Jenkins, a young Bletchley analyst who would go on to become a prominent British politician, describes what it was like to work on these codes:
“I went to a dismal breakfast having played with a dozen or more messages and completely failed with all of them. It was the most frustrating mental experience I have ever had, particularly as that act of trying almost physically hurt one’s brain.”24
In late November 1941, Bill Tutte received a series of Tunny messages to work on. His Bletchley bosses were not optimistic about his chances. Tutte would later say that his bosses gave him the messages as a “gesture of despair.”25 After all, how could one possibly solve a cipher with more than a quadrillion possibilities? Tunny remained a perfect magic trick.
Tutte’s ignorance was his advantage. He was a twenty-four-year-old chemistry student; he didn’t know enough about code breaking to realize the hopelessness of his situation. So Tutte treated Tunny like any other mathematical puzzle, trying out various strategies and looking for signs of progress. “I can’t say that I had much faith in this procedure,” Tutte said, in a 1998 speech, “but I thought it best to seem busy.”26
Tutte shared a small office with several other members of the Fish team. Their room was spare and simple, just a series of wooden desks set against bare walls. There were no luxuries at Bletchley: if the analysts wanted tea, they had to use their own rationed supply.27 Jerry Roberts was one of the code breakers who shared the room with Tutte. He marveled at the way Tutte worked: “I saw him staring into the middle distance for extended periods, twiddling his pencil and making endless counts on reams of paper for nearly three months. And I used to wonder whether he was getting anything done.”
Tutte was getting a lot done. His most important break came early in 1942, as he tested out countless periods, or numerical frequencies, on the Tunny messages. The tests went this way: Tutte would write out a stretch of Tunny code in rows of various lengths, filling reams of paper with charcoal Xs and dots. Tutte would then search for short repetitions in these vast spreadsheets, as frequent repetitions might signal frequently used words. When Tutte tried out a period of 575—itself a wild hunch, based on multiplying the number of possible keys in the Tunny cipher—he came up mostly empty. However, he noticed a few more repeats on the diagonal, which suggested he might have gotten better results by using a period of 574 instead. After rewriting his spreadsheet from scratch, he “found pleasingly many repeats of dot-cross patterns” that indicated words that were five or six letters long. This was his first insight into the methods of the Tunny code.
But now Tutte had another problem. A period of 574 suggested that the Lorenz rotors had 574 different starting positions. That seemed unlikely, even for an intricate German machine. So Tutte decided (again for reasons he couldn’t explain) to look at prime factors of 574, which led him to the number 41. When Tutte analyzed the Tunny code based on that, the improvement was clear: these repeats were real.
Tutte concluded that one of the primary rotors used to determine the Tunny cipher had forty-one starting positions. He then used the same technique—searching for repeats that didn’t feel random—to figure out the number of positions on the second rotor. Along with other members of the Research Section, Tutte went on to decipher the details on the ten remaining rotors. In essence, he found a way to solve the Lorenz machine mechanisms without ever seeing one. It was one of the greatest intellectual feats of World War II.28
How did Tutte do it? His colleagues at Bletchley used to say he had the “knack,” that uncanny ability to solve codes by intuition. While most Bletchley code breakers used the conventional tactics of their craft—they exploited depths and other transmission mistakes, which are akin to waiting for a magician to screw up—Tutte took advantage of his mind’s exquisite sensitivity to patterns. Tutte realized that, due to a quirk in the rotor settings of the Lorenz machine, guessing the first two wheels correctly would raise the odds of a dot to 55 percent. You could try out different key settings and eyeball the Tunny output for feedback. The asymmetry was slight, but Tutte could usually sense a preponderance of dots.
As Tutte would later note, some at Bletchley Park concluded that his triumph was just a “stroke of undeserved good luck.” But Tutte rejected those assertions, pointing out that the ability to notice patterns is itself a kind of “analytic reasoning.”29 He might not have been able to explain the patterns—it took him months to solve the entire machine—but he knew they were there. He could feel them, those subtle clues revealing the tricks of the machine.
The Nazis were convinced that the Lorenz SZ40 was the ultimate cipher, generating an impenetrable code. They assumed the Allied cryptographers would gain no traction, that even those with the knack would fail when faced with the twelve-rotor cipher. As a result, they never worried about a chemistry graduate student using the machine’s output to reveal its inner workings. But they failed to account for two factors. The first was the two slightly different messages sent from Athens using the same key, which gave Tiltman the toehold for a near-complete decryption of a long message, and Tutte the starting point he needed to search for patterns in the key. The second was the stubborn genius of Tutte himself. Almost all of his code-breaking attempts led nowhere, failure followed by failure. But he occasionally found a strategy that reduced the mystery, revealing a fragile pattern amid the deluge of Xs and dots. Magic works the same way: if the magician gives the audience the slightest foothold—if they get a glimpse of a wire or see a card vanish up a sleeve—the spell is broken. The wonder ends. The investigation begins.
The breaking of Tunny was a turning point in the war, giving the Allies an invaluable window into German strategy. It shaped the planning of D-Day—Tunny messages confirmed that the Nazis expected a landing much farther north—and allowed the Allies to avoid the best-defended beaches. “This was absolute intelligence gold dust,” writes Jerry Roberts. “The information had not come from spies”—which had proven problematic and untrustworthy—“but directly from Hitler via Lorenz decrypts, in the German’s own words.”30
Perhaps the most valuable use of Tunny came on the Eastern Front. In the summer of 1943, roughly 775,000 Nazi soldiers, along with three thousand panzer tanks, began gathering near Kursk, about three hundred miles southwest of Mo
scow. The goal of the German army was to split the Russian flank, cutting off their main salient, or bulge. But disagreements among German generals led to a spike in messages sent using the Lorenz machines, giving Tutte and his team of code breakers a detailed map of German positions.
The British recognized the importance of Kursk: if the Russians lost the battle, Moscow might fall. As a result, they passed their decrypted intelligence on to the Russian generals. (The Russian commander, General Zhukov, was so impressed by the British memos that he assumed they had a secret agent at the highest levels of the German army.)VII This intelligence on the movements of the German army allowed the Russians to establish a fierce line of defense at the points of expected attack. The Russians sent more than three hundred thousand civilians to work. They strung rows of mines, dug antitank ditches, and built concrete machine-gun bunkers. Around Kursk, these “defensive belts” were more than twenty miles deep.
On July 4, 1943, the Germans began their offensive, sending every panzer division on the Eastern Front to the Russian lines. They expected scattered resistance. Instead, they ran straight into a death trap. On July 13, nine days after the battle began, Hitler stopped the German attack. It was the first failed Nazi offensive of the war and set up a Russian counterattack that ended, two years later, in Berlin. The Russians referred to the Battle of Kursk as the Turning of the Tide.