The Scientific Complexity in a Seemingly Simple Snowflake

By Daniel Tammet | January 21, 2014 12:19 pm

snowflakesOutside it is cold, cold — ten degrees below, give or take. I step out with my coat zipped up to my chin and my feet encased in heavy rubber boots. The glittering street is empty; the wool-gray sky is low. Under my scarf and gloves and thermals I can feel my pulse begin to make a racket. I do not care. I observe my breath. I wait.

A week before, not even a whole week, the roads showed black tire tracks and the trees’ bare branches stood clean against blue sky. Now Ottawa is buried in snow. My friends’ house is buried in snow. Chilling winds strafe the town. The sight of falling flakes makes me shiver; it fills the space in my head that is devoted to wonder. How beautiful they are, I think. How beautiful are all these sticky and shiny fragments. When will they stop? In an hour? A day? A week? A month? There is no telling. Nobody can second-guess the snow.

The neighbors have not seen its like in a generation, they tell me. Shovels in hand, they dig paths from their garage doors out to the road. The older men affect expressions both of nonchalance and annoyance, but their expressions soon come undone. Faint smiles form at the corners of their wind-chapped mouths. Granted, it is exhausting to trudge the snowy streets to the shops. Every leg muscle slips and tightens; every step forward seems to take an age.

When I return, my friends ask me to help them clear the roof. I wobble up a leaning ladder and lend a hand. A strangely cheerful sense of futility lightens our labor: in the morning, we know, the roof will shine bright white again. Hot under my onion layers of clothing, I carry a shirtful of perspiration back into the house. Wet socks unpeel like Band-Aids from my feet; the warm air smarts my skin. I wash and change my clothes.

The Science of Snowflakes

Later, around a table, in the dusk of a candlelit supper, my friends and I exchange favorite recollections of winters past. We talk sleds, and toboggans, and fierce snowball fights. I recall a childhood memory, a memory from London: the first time I heard the sound of falling snow. “What did it sound like?” the evening’s host asks me. “It sounded like someone slowly rubbing his hands together.” Frowns outline my friends’ concentration. Yes, they say, laughing. Yes, we can hear what you mean.

A man sporting a gray mustache laughs louder than the others. I do not catch his name; he is not a regular guest. I gather he is some kind of scientist, of indeterminate discipline. “Do you know why we see snow as white?” the scientist asks. We shake our heads. “It is all to do with how the sides of the snowflakes reflect light.” All the colors in the spectrum, he explains to us, scatter out from the snow in roughly equal proportions. This equal distribution of colors, we perceive as whiteness.

Now our host’s wife has a question. The ladle with which she has been serving bowls of hot soup idles in the pot. “Could the colors never come out in a different proportion?” she asks. “Sometimes, if the snow is very deep,” he answers. In which case, the light that comes back to us can appear tinged with blue. “And sometimes a snowflake’s structure will resemble that of a diamond,” he continues. Light entering these flakes becomes so mangled as to dispense a rainbow of multicolored sparkles.

“Is it true that no two snowflakes are alike?” This question comes from the host’s teenage daughter. It is true. Imagine, he says, the complexity of a snowflake (and enthusiasm italicizes his word “complexity”). Every snowflake has a basic six-sided structure, but its spiraling descent through the air sculpts each hexagon in a unique way: the minutest variations in air temperature or moisture can — and do — make all the difference. Like mathematicians who categorize every whole number into prime numbers or Fibonacci numbers or triangle numbers or square numbers (and so on) according to its properties, so researchers subdivide snowflakes into various groupings according to type.

They classify the snowflakes by size and shape and symmetry. The main ways in which each vaporous hexagon forms and changes, it turns out, amount to several dozen or several score (the precise total depending on the classification scheme). For example, some snowflakes are flat and have broad arms, resembling stars, so that meteorologists speak of “stellar plates,” while those with deep ridges are called “sectored plates.” Branchy flakes, like the ones seen in Christmas decorations, go by the term “stellar dendrites” (dendrite coming from the Greek word for a tree). When these treelike flakes grow so many side branches that they finish by resembling ferns, they fall under the classification of “fernlike stellar dendrites.”

Sometimes, snowflakes grow not thin but long, not flat but slender. They fall as columns of ice, the kinds that look like individual strands of a grandmother’s white hair (these flakes are called “needles”). Some, like conjoined twins, show twelve sides instead of the usual six, while others — viewed up close — resemble bullets (the precise terms for them are “isolated bullet,” “capped bullet,” and “bullet rosette”). Other possible shapes include the “cup,” the “sheath,” and those resembling arrowheads (technically, “arrowhead twins”). We listen wordlessly to the scientist’s explanations. Our rapt attention flatters him.

Complexity Revealed

As he speaks, his white hands draw the shape of every snowflake in the air. Complexity. But from it, patterns, forms, identities, that every culture can perceive and understand. I have read, for instance, that the ancient Chinese called snowflakes blossoms and that the Scythians compared them to feathers. In the Psalms (147:16), snow is a “white fleece,” while in parts of Africa it is likened to cotton. The Romans called snow nix, a homonym — the seventeenth-century mathematician and astronomer Johannes Kepler would later point out — of his Low German word for “nothing.”

Kepler was the first scientist to describe snow. Not as flowers or fleece or feathers, snowflakes were at last perceived as being the product of complexity. The reason behind the snowflake’s regular hexagonal shape was “not to be looked for in the material, for vapor is formless.” Instead, Kepler suggested some dynamic organizing process, by which frozen water “globules” packed themselves together methodically in the most efficient way. “Here he was indebted to the English mathematician Thomas Harriot,” reports the science writer Philip Ball, “who acted as a navigator for Walter Raleigh’s voyages to the New World in 1584–5.” Harriot had advised Raleigh concerning the “most efficient way to stack cannonballs on the ship’s deck,” prompting the mathematician “to theorize about the close packing of spheres.” Kepler’s conjecture that hexagonal packing “will be the tightest possible, so that in no other arrangement could more pellets be stuffed into the same container” would only be proven in 1998.

That night, the snow reached even into my dreams. My warm bed offered no protection from my childhood memories of the cold. I dreamed of a distant winter in my parents’ garden: the powdery snow, freshly fallen, was like sugar to my younger brothers and sisters, who hastened out to it with shrieks of delight. I hesitated to join them, preferred to watch them playing from the safety of my bedroom window.

But later, after they had all wound up their games and headed back in, I ventured out alone and started to pack the snow together. Like the Inuit (who call it igluksaq, “house-building material”), I wanted to surround myself with it, build myself a shelter. The crunching snow gradually encircled me, accumulating on all sides, the walls rising ever higher until at last they covered me completely. My boyish face and hands smeared with snow, I crouched deep inside feeling sad and feeling safe.

“On t’attend!” my friends call up in the morning to my room. “We are ready and waiting!” I am the English slowpoke, unaccustomed to this freezing climate, the lethargy it imposes on the body, and the dogged, unshakable feeling of being underwater. What little snow I have experienced all these years, I realize now, has been but a pale imitation of the snow of my childhood. London’s wet slush, quick to blacken, has muddied the memory. Yet here the Canadian snow is an irresistible, incandescent white — its glinting surfaces give me back my young days, and alongside them, a melancholic reminder of age. After my sweater, I pull on a kind of thermal vest, then a coat, knee- long. My neck is wrapped in a scarf; my ears vanish behind furry muffs. Mitten fingers tie bootlaces into knots.

Business as Usual

Fortunately, the Canadians have no fear of winter. The snow is well superintended here. Panic, of the kind that grips London or Washington, DC, is unfamiliar to them; stockpiling milk, bread, and canned food is unheard of. Traffic jams, canceled meetings, energy blackouts, are rare. The faces that greet me downstairs are all kempt and smiling. They know that the roads will have been salted, that their letters and parcels will arrive on time, that the shops and schools will be open as usual for business.

In the schools of Ottawa, children extract snowflakes from white sheets of paper. They fold the crisp sheet to an oblong, and the oblong to a square, and the square to a right-angled triangle. With scissors, they snip the triangle on all sides; the pupils all fold and snip the paper in their own way. When they unravel the paper, different snowflakes appear, as many as there are children in the class. But every one has something in common: they are all symmetrical. The paper snowflakes in the classroom resemble only partly those that fall outside the window. Shorn of nature’s imperfections, the children’s unfolded flakes represent an ideal. They are the pictures that we see whenever we close our eyes and think of a snowflake: equidistant arms identically pliable on six sides. We think of them as we think of stars, honeycombs, and flowers. We imagine snowflakes with the purity of a mathematician’s mind.

Computer-generated snowflake. Courtesy Janko Gravner and David Griffeath

Computer-generated snowflake. Courtesy Janko Gravner and David Griffeath

At the University of Wisconsin, the mathematician David Griffeath has improved on the children’s game by modeling snowflakes not with paper, but with a computer. In 2008, Griffeath and his colleague Janko Gravner, both specialists in “complex interacting systems with random dynamics,” produced an algorithm that mimics the many physical principles that underlie how snowflakes form. The project proved slow and painstaking. It can take up to a day for the algorithm to perform the hundreds of thousands of calculations necessary for a single flake. Parameters were set and reset to make the simulations as lifelike as possible.

But the end results were extraordinary. On the mathematicians’ computer screen shimmered a galaxy of three-dimensional snowflakes — elaborate, finely ridged stellar dendrites and twelve-branched stars, needles, prisms, every known configuration, and others, resembling butterfly wings, that no one had identified before.

Unique Numbers

My friends take me on a trek through the nearby forest. The flakes are falling intermittently now; above our heads, patches of the sky show blue. Sunlight glistens on the hillocks of snow. We tread slowly, rhythmically, across the deep and shifting surfaces, which squirm and squeak under our boots. Whenever snow falls, people look at things and suddenly see them. Lampposts and doorsteps and tree stumps and telephone lines take on a whole new character. We notice what they are, and not simply what they represent. Their curves, angles, repetitions, command our attention. Visitors to the forest stop and stare at the geometry of branches, of fences, of trisecting paths. They shake their heads in silent admiration.

A voice somewhere says the Hull River has frozen over. I disguise my excitement as a question. “Shall we go?” I ask my friends. For where there is ice, there will inevitably dance ice skaters, and where there are ice skaters, there will be laughter and lightheartedness, and stalls selling hot pastries and spiced wine. We go. The frozen river brims with action: parkas pirouette, wet dogs give chase, and customers line up at the concessions. The air smells of cinnamon. Everywhere, the snow is on people’s lips: it serves as the icebreaker for every conversation. Nobody stands still as they are talking: they shift their weight from leg to leg, and stamp their feet, wiggle their noses and exaggerate their blinks.

The flakes fall heavier now. They whirl and rustle in the wind. Everyone seems in thrall to the tumbling snowflakes. Human noises evaporate; nobody moves. Nothing is indifferent to its touch. New worlds appear and disappear, leaving their prints upon our imagination. Snow comes to earth and forms snow benches, snow trees, snow cars, snowmen. What would it be like, a world without snow? I cannot imagine such a place. It would be like a world devoid of numbers. Every snowflake, unique as every number, tells us something about complexity.

Perhaps that is why we will never tire of its wonder.


Daniel Tammet is a mathematical savant and bestselling author. Excerpted from the book Thinking in Numbers by Daniel Tammet.  Copyright (c) 2012 by Daniel Tammet.  Reprinted with permission of Little, Brown and Company.

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