The Fractal Geometry of Benoit Mandelbrot
Fast on the heels of chaos theory and the butterfly effect, the whirling tendrils that compose the Mandelbrot Set cropped up on 1980s T-shirts and computer screens, record albums and magazines, posters taped to dorm room ceilings and the title sequences of sci-fi TV. A heart-shaped ink blot on the pages of textbooks, the form is an unlikely eminence. Scale its outline smaller, and it teems like the tropics. Ferns, leaves, seahorse tails, and strings of beads mass and recede in this swarming composition. Curves pirouette through the foreground and pull toward the distance. Linger on a spiral till it spins away into a branching tree. But study the math that creates these patterns and find a realm as deftly woven as the one the ancients dreamed.
On the way to the conference, as luck would have it, the car ground to a halt twenty miles north of the Hilton’s chandeliered ballrooms and slow-moving escalators. Groans and a few choice expletives must have sounded inside the cabin: that afternoon, all five of its occupants were giving papers. A smattering of snow pinpricked the dashboard and vanished before more ice crystals dimpled its transparency. Benoit Mandelbrot observed the pattern, anticipating the columns that would form a fine interlacing on the cooling glass. Impossible to observe the paths of sand grains the wind dashes to ground. But if he found a way to chart the dance of these close-to-weightless discs, wouldn’t he learn something new about turbulence, that coiled energy intransigent as yet to mathematical expression?
The blare of a trucker’s horn bore down on the car, recalling Mandelbrot to a more prosaic if no less pressing problem. As the sixteen-wheeler rushed past, the high tone of its alarm lowered to a deeper note in obedience to the Doppler effect. Less than a second later, the wake raised by the truck’s passage slammed against the car with a thud and a cracking of windows ostensibly sealed against such atmospheric disturbance. Maybe the geometer sat stymied for a moment in the driver’s seat. More likely he slid out from behind the wheel and transferred his attention from the fidgeting of his colleagues inside the car to the problem with its transmission. He was, his wife, Aliette, often supplied, an erratic and distracted driver. But World War II had turned him into a good mechanic as well as a proficient horse groomer. New York’s raw winter air stole through his blazer as he opened the hood and studied the clanking, cooling machine. So what if the sky was leaden and the temperature barely in the double digits? If nothing else, war had schooled him to meteorological as well as man-made pain. Leaning over the distributor (there went his clean shirtfront), he wiped away the grease smearing the ignition points. One was pitted, presumably by a failing condenser. He extricated the part, returned to the relative warmth of the cabin, dried the condenser’s metal foil, replaced the scored point with a spare he grabbed from the glove compartment, and started up the engine—which, to the astonishment of his colleagues, gasped and stammered back to life.
The whole procedure had cost them only twenty minutes discomfort, Mandelbrot reflected as the windshield wipers swept away the snow, a barely perceptible anxiety compared with wartime’s quotidian dread. He had spent his teenage years hiding in plain sight in the south of France, far from Paris and his parents. Only a handful of days passed without some new terror. Would the refugee neighbor who nodded as they left the Tulle tenement at dawn turn him in before night? Would the high school colleague he chanced across in Saint-Junien accept the fake name Mandelbrot repeated slowly to cue his non-Jewish compatriot to silence? The woman he clumsily jostled on his way to the tool shop where he mended train parts: would she frown and ask to see his papers?
A gust of wind shook the car and a fresh clot of snow fell on the window. They had suffered in their separate cities, he and Léon, who his parents sent elsewhere to increase their sons’ collective chance of survival. But what family had not? Alone of their circle in Warsaw, and aside from the mother and daughter who lived above their own flat, the four Mandelbrots fled to France and escaped oblivion. Detained by ailing relatives or their grand piano or their window fronting the park, the rest deferred flight. At war’s end, come to Paris from the old apartment, Mrs. Braude and her daughter recounted deaths and disappearances. Horrified though she was, his mother “listened stone-faced” to their stories, Mandelbrot recalled in the memoir The Fractalist published two years after his death.
Now, as he drove toward the Hilton, slush slid across the glass, blurring the dreary New York landscape and encouraging the dead from that other continent to creep closer. Their familiar silhouettes clamored to be seen as the limbs of leafless trees zoomed past the rearview mirror. There, the perpetually pleased expression of the classmate who had strutted around Lycée Edmond Perrier repeating his parents’ progressive ideas only to be hung from a lamp-post in Tulle’s main square a year after Benoit and Léon fled south. Here, Mr. Wigdorczyk’s harried face softening briefly as ten-year-old Benoit opened the door of the fourth-floor walk-up in Warsaw. Here, that same summer, Mrs. Goldberg’s meaty arm extending in greeting from the horse cart parked by Mołodeczno’s wide-gauge train track.
The Honda jolted over a pothole, jerking Mandelbrot back several decades. At ten he had clutched the side of the buggy swaying rhythmically as the old Friesian plodded across the cobbles. Snorting and snuffling, the horse pulled up short and tried to shake off its reins. Mrs. Goldberg replied by levering herself forward and coaxing the creature on, patting its rump with surprising gentleness. The little cottage Benoit slept in that summer was built into a hill wind scoured in winter. In the previous century, the remnants of Napoléon’s disbanding army had stumbled past houses in this area as they retreated, Mandelbrot learned as an adult. Shrouded in snow, the hut to which Mrs. Goldberg conducted him in July’s heat must have stayed snug the winter of 1812 as the French soldiers filed past its door in the -30°C temperatures, their frostbitten faces blistered black.
Dusty summers of cleared fields and deserted town squares. Winters of broken windows and lightless doorjambs. The forties were desolate, but having witnessed the maimed and the murdered, Benoit could not but thank his better fortune. Scrap iron, a metal file, poorly contrived false papers: this was Périgueux in 1943, he recalled in his memoir. Mud and manure, crude wooden clogs, a knee purpled by the falling yoke of an oxcart: these things spelled Saint-André-des-Eaux for him the following year. Villagers screaming inside a flaming church during the Christmas break at Saint-Junien. Vichy radio crackling its false cheer as Count Benoit droned the pedigree of his long-dead Lyon Derby winner at dinner. And 1944’s champion? Klaus Barbie, strutting into the city in boots that barked a strike-slip-strike everyone learned to fear.
But there was still the sky, bluer than that Panzerman’s eye. Also, the silvered rain that ran down the ash trees circling the estate by Pommiers-en-Forez and whose boughs, forking upward, resolved in a twiggy fretwork of slender shapes a venturesome composer might have evoked with a piccolo’s climbing scale. Outside the manor where Benoit groomed horses, snow ridged the fluted sides of yellow squash piled high in a wheelbarrow. Frost crawled up the dining room window, edging the glass with a wobbly horizon of mountain peaks. Sometimes scalloped, sometimes lacy, this ice was whorled and curlicued, its ridged edge composed from countless crystals.
Did Mandelbrot explore the patterns he named “fractal” to cleave closer to nature? Or was the geometry of chaos he championed a way to think through the political disorder he survived as a young man? The meandering arcs of the Mandelbrot Set mirror the shifting contours of coastlines and clouds even as they repeat the branching conduits that bring oxygen to the lungs and blood to the tissues. “Every portion of matter may be conceived as like a garden full of plants and like a pond full of fish,” Gottfried Wilhelm Liebniz announces in The Monadology (1714). “But every branch of a plant, every member of an animal, and every drop of fluids within it, is also such a garden or such a pond.” Voltaire directed his Candide to sneer at Liebniz’s rosy understanding of Earth’s orchard. But surely, Mandelbrot might have reasoned from the vantage of more than two centuries’ distance, the Frenchman and the German who so vehemently disagreed on metaphysics could agree upon how quickly a pond dimpled with rain restored the reflection of the branches arching over its surface once the shower stopped and how unceasingly a part of every plant inclined toward constantly shifting light. What were the petty factionalism of the academy or the small-mindedness of nation-states compared with the faith flowers hold for the sun?
And look, he told himself: the freeway had just given way to city streets. His own faith in thinking had again been rewarded: despite his inattention to the road, the car had found its way to the Hilton’s tower. His colleagues climbed out of the vehicle, joking and chuckling, and then divided to their separate sessions. Mandelbrot’s talk would come later. Walking through an open door upon a presentation on wave velocities already in progress, he bumped his way past knees to a front row seat—only to rise again to query the speaker before he had finished commenting upon his last slide.
The Cornell scientist behind the podium straightened his papers and shut off the projector. Coldly, he slid his eyes toward his interlocutor. Mandelbrot waited at military attention, his bulky outline conspicuous against the room’s blandness, his broad shoulders rounded under his rumpled brown blazer. Tract lighting haloed the froth of hair on his nearly bald head. Speaking an accented, deliberate English that seemed at odds with his enthusiasm for argument, he outlined what was new in the paper (very little), what was distinct about its methodology (almost nothing), and what remained to be done (virtually everything). More than one in the assembled confraternity must have sighed inwardly as this scientist from the private sector plowed straight into censure. Perhaps a few shuttled their gaze from his disorderly profile to the room’s featureless architecture and found the latecomer all the sloppier. Yet irresistibly, their eyes returned to the IBM interloper who had wasted no time cataloguing the weaknesses of the experiment and analysis just presented. Imagining the vulnerabilities in their own papers exposed with identical want of ceremony, was it not understandable that some steeled their hearts against his brash intelligence? As for Mandelbrot himself, who wished only to deepen his understanding: could he not sense the eyes boring into his back as he drove toward this end?
“And the exceptional variations in wave swells?” Mandelbrot raised his palms up and spread them outward like a seraphic Eastern deity, his eyes twinkling despite the oceanographer’s annoyance. “What do we do with these extraordinary outliers?” He prompted, referring to the currents (not the experiment designed to measure them) with wincingly childlike curiosity. “I’m referring to the long tails. The points floating at the greatest distance from the central distribution,” he supplied to aid the students in the room. There was the barest hint of drollery in his voice, as if he intended his question to speak for displaced people as well as far-flung data points. (Years later, working upon his memoir, he would be more direct, defining statistical aberrations like the outlying wave swells as examples of the “extreme inequality that is a familiar pattern in nature and in the works of humans.”)
Two faculty from Paris’s École Normale Supérieur exchanged glances as Mandelbrot turned toward the young people, exposing a greasy black smear on his Sears-issue shirt.
“Beer?” one hypothesized.
“Gravy or dirt?” the other quipped with a Gallic shrug.
“Stress velocities and cross-shore gradients are fine.” Mandelbrot proceeded, the exchange behind him no more of an impediment than the periodic whump of air the room’s heater expelled. (“Trivial,” he would have told his friends in the frank shorthand with which he dismissed the uninteresting experiment, the obvious solution, and the question not worth asking.) “The large disproportions: that’s where the real question is. What happens if we factor in the swells rather than dispensing with these gross variations?”
“Ce type could make a Brioni suit look cheap,” one of the Normale men remarked as Mandelbrot gargled the “r” in “gross” in his evasive accent—part Belleville slum, part Polish ghetto, part indefinable.
The heat cycled on as the lecturer began listing surf-zone data he had collected at Nantucket Sound. Blocking out this numeric roll call, Mandelbrot watched the Sound’s coastline advance and retreat in time-lapse sequence with no projector but the technology humming in his head.
The condescending physicists, the standoffish oceanographer, and the session in the Hilton are my invention. But the memories of Poland are drawn from Mandelbrot’s memoir, and the breakdown, the grease-streaked shirt, and the alacrity with which he routinely discomfited scholars appear in recollections friends from physics, mathematics, music, and medicine contributed to Fractal Geometry and Applications, the two-volume Jubilee published to honor him during his eightieth year. Any one of the researchers who offered their reminiscences could tell you that to practice science is to find methods of measuring what you observe: leaves spinning in the wind, a skiff rocking in the wake of underwater shear turbulence, a plant sending roots around a boulder to absorb the minerals that seep into the surrounding soil. For Mandelbrot, perception accomplished without preconception was itself experiment. “To see is to believe,” he insists in The Fractal Geometry of Nature. “Look, look, look,” he repeats in Fractals and Chaos: The Mandelbrot Set and Beyond. Unlike rarified calculations pertinent to specific fields, the eye “is a universal tool,” he recapped to art curator Nina Samuel in an interview published in The Islands of Benoît Mandelbrot: Fractals, Chaos, and the Materiality of Thinking.
The speed with which he could transform algebraic equation into geometric form looked like revelation, not computation. Whether you call it intuition (as his friends did), label it laziness (as his critics volunteered), or attribute it to an education disrupted by war (as he avowed in The Fractalist), Mandelbrot’s uncanny aptitude let him sprint through proofs his high school teachers failed to finish and zip past university gates. The irregular structures he preferred examining were as different from Euclid’s geometry as atonal music is from Mozartian harmony. On the eve of World War I, Henri Poincaré dismissed these shapes as a “gallery of monsters,” Mandelbrot points out in his memoir. Though he grew up during the next such war, he maintained an imagination inimical to Aristotelian categorizing and a curiosity unfazed by all manner of bristling nationalisms. Hostile to intellectual constraint by temperament and time, Mandelbrot adopted the “zoo,” as he fondly called the figures Poincaré disparaged, for his own. Three decades after Potsdam, this late-blooming maverick published Les Objets Fractals: Forme, Hasard et Dimension and gave pride of mathematical place to shapes ill-fitted for Euclid’s arid planes but aptly suited to Earth’s curved space. Locating complex patterns in what looked like disarray, Mandelbrot insisted fractals into acceptance by demonstrating how closely their designs followed nature’s tangled forms.
Da Vinci engineered the multiple arms and legs of his Vitruvian Man to fit inside the circle Euclid defined almost two millennia before. But Mandelbrot knew intuitively that the Greek’s geometry, perfect for hoisting marble into the curves of arches and domes, would not explain the organic world’s wobbly, winding contours. The brambles of blackberry bushes and the cliffs of coastlines are recursive without perceptual ratio. In animals as well as plants and the landscapes that support them, serpentine vessels thrive. God may have created the world from the word, but it was Mandelbrot who would most accurately size this globe’s contents. Ferns, shells, feathers, snowflakes: he “saw that fractals are the geometry of nature,” Nathan Cohen insists in the preface to Benoit Mandelbrot: A Life in Many Dimensions, a collection honoring Mandelbrot after his death. “Note the absence of ‘a’ or ‘one of.’” Mandelbrot “left no doubt of that interpretation.”
Flux—not fixity—engaged him. Zephyr-like, he breathed life into equations and puffed them up into forms. He had only to conceive a shape to rotate, reflect, and resize it. Some, resenting his breezy trespassing across their disciplines, accused him of grandstanding, but Mandelbrot’s facility for finding solutions was unconscious and instantaneous. Numbers cried out for notice no less than the parochial professors who scrawled them on chalkboards and typed them into textbooks. Ellipses rose into cones while vector analyses plunged into whirlpools. Peano’s curve heaved up its arc into tessellated crags. Ampere’s law inflated itself into doughnut-shaped toroids and polynomials rushed to repeat their sequences in patterns that expanded in Mandelbrot’s head faster than fireworks rush their contrails through the dark skies above fields and fairgrounds. “Bottomless wonders spring from simple rules . . . repeated without end,” he pronounced these strangely beautiful shapes in a TED talk a few months before his death, relying once again on the unadorned language that grated on so many scholar specialists.
“Can the projector be turned back on?” Mandelbrot interrupted the oceanographer droning out data.
Mutely, the Cornell scientist returned to his PowerPoint.
“There.” Mandelbrot pointed to a black point vibrating on the matte white screen. “And there”—he raised his arm to gesture toward a dot swimming far from the myriad clustered in a brightly lit quadrant. “And again, here,” he continued, waving his hand rapidly through the air as if to gather in three specks so widely scattered they might have been mistaken for dust on the projector lens. “What would happen,” asked this man whose thought could jump lightning-quick from geometry to physics and branch out into music and art, “if we came up with an equation that accounted for these wanderers?” He smiled at the rest, narrowing his eyes as if he were staring at the sun rather than the lesser lights of the Normale physicists. “If we could establish one, we could connect the orbits of stars with the eddies in whirlpools.”
His challenge was genial, but turbulence coursed from the first row to the back seat. A few, lulled by the hum of the ventilation system or the oceanographer’s monotone, straightened their spines against the hard metal backs of the chairs. A trio argued, a larger mass in the middle of the room revised, and two students on one end of a back row gesticulated as if to amplify. Four people in turn scrawled elaborate equations on the poster propped by the podium. A phalanx of statisticians from Harvard’s Earth and Planetary Sciences leaned forward in their seats as Mandelbrot lined out one numerical sequence and scribbled a corrective. Nodding vigorously, a marine geologist whose curly black hair was sprinkled with silver scrawled a note on the pad of paper at his elbow. “But—” began an oceanographer from Oxford without being heard. “What about the—” he interjected a second time, then slumped before the verbal onslaught with tired, pouchy eyes. A chemical oceanographer from Scripps took Mandelbrot’s cue and proposed a clarification that veered sharply away from the general line. Like a cloud of electrons, fine brown hair frizzed about her face. Mandelbrot listened to her suggestion, pondered a second, and smiled. The Normale professors raised their eyebrows into supercilious half-moons. “Galaxies” to their graduate students, Peter Jones recalls in Benoit Mandelbrot: A Life in Many Dimensions, they were accustomed to being acclaimed, not ignored.
People filed in for the next session, bringing the easygoing fracas to a close. The impromptu discussants dispersed, leaving a few who carried on without glancing up at the newcomers. No, he had not expected to arrive at the Hilton looking like a mechanic, Mandelbrot answered two biologists with whom he was arguing about cardiac vascular structure. The car he was driving had stopped en route, he explained cheerily, pronouncing the last two syllables in that French the Normale scholars found so peculiar. But why wait hours for a tow if you could fix the problem in a few minutes? He added, raising his palms up in the quizzical gesture characteristic of him.
Did he brush shoulders with his French detractors before he left the room? If so, rebutting their frosty smiles, he might have assured them that his long-ago decision to withdraw from their institution after a single day’s matriculation jumpstarted his career and catalyzed his theory of roughness. Ah, yes, they would have nodded with tight faces. Fractals. Pretty patterns. Elegant, even, the equation. But what, more precisely, could his computations do?
Others had let this question hang in the air at conferences in economics, in linguistics, and in astrophysics. More would repeat the query before Mandelbrot, blinked at by the academy, was finally ushered into its fold. Yale awarded him a Sterling Professorship in Mathematics in 1975, but he would wait another twelve years to be tenured. Given the glacially slow recognition of his brilliance, the verbal dart thrown by the Normale scholar must have sailed past him with no more ceremony than the buzz of a fly. An iconoclast only loosely rooted in his own era, Mandelbrot repeatedly refused to follow a conventional career trajectory in order to maintain what he calls in The Fractalist (and always with a romantic flourish) his “Keplerian dream.” Prodigious insights like the one Mandelbrot worked toward possess a reach so uncommon they are rarely acknowledged in the discoverer’s lifetime. How marvelous, then, that he enjoyed an annus mirabilis in his fifties after describing the repeating forms mathematician Adrien Douady honored as the Mandelbrot Sett. The celebrity status he attained after was merely icing on the cake.
Under the gray walls of the Hilton’s conference room, a group of students awaited his attention, the turquoise and canary-yellow reproduction of his Set swirling conspicuously on the cover of The Fractal Geometry of Nature they clutched under their arms or pressed close to their chests. Before detaching himself from the Normale elite to sign books, the geometer might have told these physicists that he had chosen to attend college at the Polytechnique, their foremost rival. Would he have recalled for them the trip back to Paris from the farm where he had labored with his head down to protect his counterfeit papers? Unlikely, since this posture was one he steadily refused to reprise. He might have announced that the Bourbaki cult’s dry-as-dust approach then favored by Normale mathematicians had prompted him to reject their institution. In those days the Normale mathematicians were austere as monks, adamant in refusing the aid of images in proof and equation, and entirely unable to anticipate the irregular shapes Mandelbrot was himself investigating. The Polytechnique had given him room to flourish, he might have insisted. There, no arid theoreticians had turned their backs on the amorphous and the strange. He was fortunate the Normale had let him go without a murmur, since choosing to leave and being turned away had produced equal possibilities.
What could fractals do? What couldn’t they do would have been the better question, Mandelbrot might have murmured inwardly. Biologists use fractals to understand how insects distribute themselves in trees, cardiologists exploit them to identify the heart-rate patterns that produce life-threatening arrhythmias, and cosmologists manipulate them to develop equations about the contours of space-time. He had named them after the verb frangere to honor the Latin of his long past youth, he explained in his TED talk. But “fractus” is a “break” that also unifies. Juxtapose a photograph of cigarette smoke and one of an interstellar gas cloud thousands of light-years across, he proposes in The Fractal Geometry of Nature, and look in vain for their difference. Armed with a conviction that bordered on evangelicalism and an energy that verged on the prophetic, he insisted to all who would listen (and many who would not) that his zoo of shapes knit natural phenomena into an arrangement as artful as the design the Old Testament attributes to the deity. The resistance to changes in scale that provides fractals their self-similarity would give him—the one who named them—leeway to generalize across experiment and hopscotch from field to field.
He began his career by fine-tuning linguist George Zipf’s law of word frequency distribution, jumped to geography to formulate an argument about the fractal nature of coastlines, then swerved into economics to propose a direly accurate model for stock market crashes. “What a joy to quote the Bible as a pure scientific reference!” he exclaims in The Fractalist, having gleefully labeled the 1929 and 1987 market crashes the “Noah effect” and named the sustained dependency of price shifts on prior conditions after Joseph, the analyst of dreams who had so accurately predicted lean and fat years for Pharaoh. “In mathematical terms,” economist Richard Hudson translates in Benoit Mandelbrot: A Life in Many Dimensions for those of us fazed by the lexicon of his field, “Mandelbrot was saying that the guideposts that all prior financial theory” depended on “were in fact illusory.”
The fractalist’s own shifts from city to city and job to job seem no less meandering than the ragged contours his math revealed. In exposing hidden relationships between what seem to be entirely isolated mechanisms, the succession of experiments he conducted appear themselves to possess a fractal character. As a wave distributes its energy across the water, so the Zipf-Mandelbrot law explains the abundance of species ecologists study and ear-pleasing metrics musicians explore. Twigs maintain the proportions their parent branches establish, while the endless curves contained within continents mirror the limitless twisting of lung tissue packed within the chest. In turn, both rock outcroppings and alveolar arcs gesture toward the infinite variety of price shifts in the stock market, Mandelbrot argued. If the never-ending budding and dividing of the Mandelbrot Set restructures perception, so the manifold applications of fractal theory remodel thought. And yet the equation that creates the ark that shelters this immense range of terrestrial and celestial objects is as dazzlingly simple as Einstein’s own. Take “z,” multiply it by itself, and add “c” to obtain the branching, whirling contours of the black blot Mandelbrot derived through the laborious calculations of IBM’s first behemoth computers.
Showy, the Normale academics would have sniffed had they heard the researcher from IBM crowing about the shape he called the most complicated in mathematics. Grandiose, they might have added had they known he claimed likeness with Einstein, that supernova of scientists. Yet if da Vinci’s Vitruvian Man glorified humanity as driver of the world’s wheels, Mandelbrot’s more irregular shapes reduce us to minute mechanisms in a vast natural drama. We can sweeten this fall, as did he, by recognizing the rhythms of our hearts, the pulses of our neurons and the cycling of oxygen through our lungs as no less fractal in structure than the gasses that stream across space.
From fracture to fractal and war to wonder is not, after all, so great a leap. In 1944, as Mandelbrot rubbed down gentlemen’s horses, himself skinny as a rail and terrified of capture, could he envision the marvelous traces of order in the mess? The most astonishing fact of his life is not that he survived war’s disaster but that he refused bitterness over its damage. His mother’s forsaken dental degree and his father’s forgotten aptitude for invention, his own relocations across countries and removals to habitations ever shabbier and draftier, the pepper of gunfire outside windows beneath which he slept and the whispered recitations of torture he intercepted for those caught with false papers: such traumas might have stopped a man of lesser energy. How could the young Mandelbrot not feel as “solo, perdido, and abandonado” as Puccini’s Manon, whose aria he so admired? To counter such difficulty, he lost himself filing metal shards into the smooth join of spare parts to coax dynamited trains back into motion. After work, he watched water boil over rocks knowing that the froth leaking away between boulders offered illustration of the moon’s forcing of tides. In the end, who better to approach turbulence (that writhing monster of modern mechanics) than a man grown up in war’s whirlwind? Accepting suffering as the catalyst for his mathematical second sight, Mandelbrot converted the liability of his disrupted education into the gift of fiercely independent thinking.
Did his heart beat a little faster when he rose to contradict the oceanographer in that hotel room full of scholars irritated by his question? Did a hint of the estrangement that was his constant companion during wartime trouble his mind? If so, was it also possible that as the seconds ticked by the geometrical shapes blooming in his inner eye were overtaken by the figures that had once crouched in crawl spaces and hurried past soldiers and shuffled past guards? Might he then have had to will himself to remain standing? At twenty, he recalls in his memoir, he thought himself “a strong-willed person with clearly defined tastes.” At eighty he insisted on the blessing of knowing precisely who he was even as “successive bureaucracies” remained endlessly perplexed. Was he “ungentlemanly,” as a rival scientist called him in a slur an April 1990 issue of the Scientific American repeated a bit too quickly? Certainly. But while Mandelbrot was raised in Belleville and only tardily made a French citizen, he remained as impervious to scholarly factions as to political divides. For decades—through the reshuffling of the postwar period and the long years of useful but uncelebrated work at IBM—he brought his arguments to academia only to see them scoffed at, sniffed over, and ignored. After each rejection, a lonely figure under the light, he rose to speak his next idea into appearance.