Nakaya Ukichirō, who grew the first artificial snow crystal, called snow crystals letters sent from the sky: read a crystal's shape and you read the weather it grew in. I took him at his word. I wrote messages into the air a growing crystal falls through, then tried to read them back from the crystal alone.
Type a word. Each letter becomes one snow crystal. Its five-bit code is written into the first five layers of air the crystal falls through, dry for 0 and humid for 1, followed by seven layers of whatever weather comes next. The crystal grows here in your browser, in the same model and with the same faint noise as every experiment below. Then a reader looks at the finished crystal, and nothing else, and says what it thinks the letter was. It learned to read from 3,600 crystals and has never seen these.
Click a layer to change it and regrow this crystal. The reader was trained on faintly turbulent air; in still or rough air it is reading a hand it has not practised on.
On 12 March 1936, in a cold room at Hokkaido University, Nakaya grew the first artificial snow crystal on the tip of a rabbit hair. Over the next years he mapped how a crystal's shape depends on the temperature and humidity it grows in: plates here, needles there, ferny stars where the air is wettest. A crystal that lands on your sleeve has carried that record down from the cloud. Hence his sentence, 雪は天から送られた手紙である: snow is a letter sent from the sky.
A letter carries information, so it's fair to ask how much. I can't fly through a cloud, but I can grow crystals in one of the best models there is: Janko Gravner and David Griffeath's 2008 "mesoscopic" lattice model. On a honeycomb of cells it tracks three kinds of mass (vapour, a quasi-liquid skin and ice) and whether each cell has joined the crystal, and from seven numbers it grows startlingly real snowflakes. Inside it, I can be the weather.
Its authors, interestingly, are sceptics. Because a perfectly steady sky already grows ornate crystals, they write that they are "wary of interpreting specific structural features as signaling environmental changes." So here is a direct test. Write a known message into the sky, grow the crystal, photograph it, and see how much of the message a reader can get back.
Two kinds of air: dry, with vapour density ρ = 0.42, and humid, with ρ = 0.65. Everything else is the base weather of the paper's Figure 11. A message is a sequence of layers, each lasting the same number of steps. When the crystal passes from one layer to the next, all the air around it changes at once, as when a falling crystal drops into a new layer of cloud. Gravner and Griffeath put a cell at about 5 µm and a step at a few hundredths of a second, so these crystals are about 1.5 mm across and grow for a few minutes. With layers of 450 steps, twelve layers fill a crystal.
The noise matters. Gravner and Griffeath let the vapour in every cell flicker by a fraction σ each step, and suggest σ ≈ 10⁻⁵ as plausible. I call that faintly turbulent, and also try σ = 10⁻³ (rough) and σ = 0 (still).
The reader gets the photograph and nothing else: which cells are ice, and how thick. It never sees when any cell froze.
Here is one crystal two ways. On the left is the photograph. On the right, each cell of ice is coloured by the layer of sky it froze in: dry or humid, alternate layers a shade darker. The message is written in chevrons, like tree rings, because every layer adds a band of ice down the whole length of each arm, not just at the tip.
Growing thousands of these and reading them back taught me three things about how a crystal keeps its letters.
In steady air, a tip grows about 6 times faster in humid air than in dry. Switch the air every few hundred steps, though, and the tip barely notices. Both kinds of layer move it about 0.027 cells per step, the speed of air with the average humidity. A tip carries its own shape forward: a blunt tip from a dry spell stays slow for a while in humid air, and a sharp one stays fast when the air dries. The two speeds only pull apart for layers of several hundred steps or more.
A layer writes its band, then every later layer keeps adding ice beside and around it. By the time the crystal lands, only 17–24% of the ice in a middle layer's ring froze during that layer. The rest is later weather.
Without noise, the model is a perfect recorder. Swap the weather for a single step in about five thousand, and 96 to 5,874 lattice cells of the finished crystal come out differently, in all six trials. It isn't fragile, though. Nudge the vapour by one part in a million for one step and nothing changes at all: small one-off disturbances heal. Continuous noise is different. Two crystals grown through the same sky at σ = 10⁻³, with different noise, differ in 2,646 to 9,492 cells. That smudge is larger than many single-step marks.
For each layer length and noise level I grew 800 crystals with random messages and trained readers on 600 of them. The bits a reader recovers are counted honestly: the mutual information between each layer and the reader's call on 200 crystals it never trained on, summed over layers, with a small-sample correction so that guessing scores zero.
The best reader I built is also the plainest: a weighted sum of the photograph's pixels for each layer (a ridge regression on the folded photograph). It reads 6.6 bits at best, from 12 layers of 450 steps in faintly turbulent air. It reads the first five or six layers almost perfectly, and then the record dissolves. The reader in the telegram above learned from 3,600 crystals and does better: 8.1 bits, and the letter right 99.5% of the time on 400 crystals it never saw.
| layers × steps | read, still | read, faint | read, rough | genie, faint | genie, rough |
|---|---|---|---|---|---|
| 104 × 50 | 2.4 | 2.5 | 1.3 | – | – |
| 69 × 75 | 3.4 | 3.2 | 2.8 | 49.0 | 9.4 |
| 52 × 100 | 3.3 | 3.8 | 2.8 | – | – |
| 35 × 150 | 4.4 | 4.5 | 3.1 | 25.3 | 8.6 |
| 26 × 200 | 5.2 | 5.2 | 2.8 | 20.8 | 9.5 |
| 17 × 300 | 6.3 | 6.0 | 3.2 | 13.8 | 8.8 |
| 12 × 450 | 6.9 | 6.6 | 3.3 | 8.5 | 5.7 |
| 9 × 600 | 6.8 | 6.2 | 3.1 | 7.4 | 4.4 |
| 6 × 900 | 5.0 | 4.8 | 3.5 | 4.2 | 3.0 |
At the page's setting, twelve layers of 450 steps, the learned reader gets 6.9 bits in perfectly still air. The faint turbulence Gravner and Griffeath think plausible costs almost nothing (6.6 bits). Ten times rougher already costs two bits (4.6), and the roughest air I tried leaves 1.6 bits at σ = 3·10⁻². Even there, the first layer is still read right 97% of the time: the centre of a crystal is the last part of the letter to go.
Is the rest of the message lost, or just unreadable? To find out, I gave a reader a genie that tells it every layer except one, then asked it to decide that one. It compares the photograph with the average crystal grown under each answer, and picks the nearer. In faintly turbulent air it gets most layers right even when they are short. With 69 layers of 75 steps, it calls the median layer right 94% of the time (the outermost layers are harder). That adds up to an estimated 49 bits, where the learned reader gets 3.2.
So the letter is mostly there. What defeats the reader is that each layer's mark depends on every layer before it, which sets where and how it grew, and on every layer after, which grows over it. A reader that knows everything else can see one layer clearly. A reader that knows nothing, which is the only kind the sky ever gets, has to untangle them all at once.
The genie's total is evidence that the marks are there, not a count of what could be extracted. Its simple comparison misses things a trained reader sees: with the longest layers it scores below the learned reader. And when layers depend on one another, a sum of one-layer-at-a-time answers can overstate what the whole crystal holds. Somewhere between the two panels is the true size of the letter; I could only reach the left one.
| layer | read | with a genie |
|---|---|---|
| 1 | 100.0% | 100% |
| 2 | 100.0% | 100% |
| 3 | 100.0% | 100% |
| 4 | 100.0% | 99% |
| 5 | 99.5% | 99% |
| 6 | 99.8% | 100% |
| 7 | 96.8% | 98% |
| 8 | 93.8% | 94% |
| 9 | 85.8% | 85% |
| 10 | 75.8% | 85% |
| 11 | 65.2% | 77% |
| 12 | 58.2% | 63% |
I expected cleverer readers to do better. Only one beat the plain reader, once, and none did reliably. Each failed for a reason worth knowing:
The plain reader is still learning, too. At 300 steps per layer in rough air it climbs from 1.9 bits with 150 training crystals to 4.5 with 4,400, about half a bit per doubling and slowing. At the page's setting the same doublings took it from 4.4 bits to 8.3. More practice helps, but slowly: on that trend, the reader in rough air would need hundreds of times more crystals to reach the genie's 8.8 bits, if it got there at all.
A snowflake has six arms, each with two mirror-image halves, all growing in the same air. In most of these experiments the noise is shared by all twelve, so they stay identical. That keeps the computing cheap: only a twelfth of the crystal has to be grown. Real arms each meet their own noise. Growing whole crystals with independent noise in every cell, and folding the twelve half-arms on top of each other before reading, gives the reader twelve slightly different copies of the letter.
In rough air, with 17 layers of 300 steps, that is worth a lot: trained and tested on the same numbers of crystals (900 and 300), the reader gets 5.4 bits from twelve independent copies against 3.5 from twelve identical ones. In faint turbulence, with the page's 12 layers of 450 steps, the gain is +0.7 bits (7.5 against 6.8). The rougher the air, the more the twelve copies differ, and the more there is to gain by comparing them. They are not fully independent witnesses, though, since they share the vapour between them.
I wrote these down before growing a single crystal.
In still air, a snow crystal remembers nearly everything: change the air for one step in five thousand and thousands of cells come out differently. In the faint turbulence the model's authors think plausible, a reader who is told all the rest can still pick out most of the layers, an estimated 49 bits in a 69-layer sky. So Nakaya was right that the weather is written down.
But the sky never sends the rest of the message along with the letter. A reader who starts from nothing gets about 8.1 bits: the first six layers almost perfectly, a few more in part, and then the writing disappears into the weather that came after. That's enough for one letter of the alphabet, with a few bits to spare. Taken literally, a letter sent from the sky is exactly that: a single letter.