Six Arms, One Story -- balasubramanian.org

What you are watchingNobody drew this snowflake. There is no picture of one anywhere in the code -- only a grid of cells, each trading water vapor with its six neighbors under one small rule. Freeze when you are full; feed the vapor to whoever is already frozen. The arms, the side branches, the six-fold symmetry: none of it is placed. It all falls out of the rule.

Two forces, arguing on every arm

Facetingthe flat, patient force

order, pulling toward a plain hexagonWater molecules settle into ice along a six-sided lattice, and that lattice prefers flat faces. Left undisturbed, a crystal would just thicken into a plain hexagonal plate -- tidy, blunt, and boring. Faceting is the force that keeps the edges straight.

Branchingthe greedy, unstable force

a bump that reaches into richer airGrowth eats the vapor near the crystal, so a face sits in air it has already thinned out. Any tiny bump reaches past that shadow into fuller air and grows faster -- which makes it stick out more, which makes it grow faster still. A flat edge cannot stay flat. This runaway is the Mullins–Sekerka instability, and it is where the arms come from.

Every snowflake is these two forces fighting -- faceting flattening the arms, branching splitting them -- and the temperature and humidity of each passing moment decide who is winning.

The mathematics

How the sky writes a snowflake

the question01

Why always six?

In 1611, Johannes Kepler had no New Year's gift for his patron, so he wrote one: a short essay asking why snowflakes always fall with six corners, never five, never seven. Nobody had thought to ask before. He guessed the answer lay in how the smallest units of water pack together -- the same way cannonballs or seeds stack into hexagons -- and he was, in spirit, right. But he had no way to prove it. The real reason, the six-fold shape of the ice crystal lattice itself, could not be confirmed for another three hundred years, until we could see that far down.

the diary02

A letter from the sky

In the 1930s, the Japanese physicist Ukichiro Nakaya became the first person to grow snow crystals on demand -- nucleating them on a single rabbit hair in a cold chamber -- and so the first to change the weather around one and watch the shape answer. He found that the form is set almost entirely by two dials: temperature, and how much moisture is in the air. Near −2°C, thin plates. Near −5°C, slender needles and columns. Near −15°C in humid air, the grand six-armed stars. A crystal that falls through changing layers records each one in turn. "A snow crystal," Nakaya wrote, "is a letter from the sky."

The same water, a different letter at every temperature

the arms03

Why a flat edge cannot stay flat

Start with a simple hexagonal plate and let it grow. As it pulls water out of the air, it leaves a thinned shell of vapor hugging its faces. Now suppose one point on an edge pushes out a hair further than its neighbors. That point sticks past the thinned shell into fuller air, so it grows faster -- which pushes it further out, into even fuller air, faster still. The edge sharpens into a tip; the tip throws side branches by the same logic; the branches throw their own. That is the whole engine of a dendrite. Faceting fights back, trying to flatten every tip, and the standoff between the two sets how far apart the branches sit.

The flat face is unstable · any bump wins, and keeps winning

the agreement04

Why the six arms match

Here is the quiet marvel. Each arm grows entirely on its own -- there is no signal passing from one to another, no plan holding the flake to a template. And yet the six come out nearly identical. The reason is almost embarrassingly simple: a snow crystal is tiny, a few millimetres across, so all six arms sit in the same little parcel of air and fall the same path through the sky. They meet the same warmth, the same moisture, in the same order, at the same instants. Same conditions, same growth. When the flake drifts into colder air and starts to branch, all six branch at once. The symmetry isn't cooperation. It is six arms reading the same weather.

the claim05

No two alike -- and why

The famous line is truer than it has any right to be. An elaborate stellar crystal is the record of a long, wandering fall, and the number of ways to assemble its roughly 10¹⁸ molecules along a unique path is so vast that no two ornate flakes have ever matched, or ever will. But be honest about the fine print: simple crystals -- plain little hexagonal prisms, barely fallen -- can and do look identical. In 1988 a scientist named Nancy Knight photographed two that were essentially the same, plucked from one Wisconsin storm. "No two alike" holds for the fancy ones -- and it holds because each of them took a different walk through the sky.

the reckoning06

And then it lands

It is the most intricate thing most people will ever hold, and it is assembled by nobody -- molecule onto molecule, over a fall of maybe half an hour, out of nothing but cold air and the rules that cold air obeys. Then it touches a warm palm, or a windshield, or the ground, and in a second or two it is water again, the whole diary erased. A hundred million fall for every one a person notices, and each of those is a letter from the sky that no one will ever open. The most careful writing in the world, addressed to no one, and gone before it can be read.

“There must be some definite cause why, whenever snow begins to fall, its initial formation is invariably in the shape of a six-cornered starlet. For if it happens by chance, why do they not fall with five corners, or with seven?”

Johannes Kepler · On the Six-Cornered Snowflake, 1611

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Six arms, one story