Snow · ice · a cellular model · what “alike” means

Six Arms, One Weather

Nothing tells a snowflake's six arms to match. The hexagonal lattice of ice explains why there are six of them; it does not explain why they agree. They agree because they fell through the same cloud at the same moments, so every change in the weather was written into all six at once. Grow crystals here with Clifford Reiter's 2005 model, shake every growing cell with noise, and measure: an arm turns out as like its five sisters as it is like an arm of a separate crystal that met the same weather. That is also the honest meaning of “no two alike”.

Six is the easy half

Why six? That part was answered long ago. A snow crystal is a single crystal of ice, and Kenneth Libbrecht, a physicist at Caltech who grows snow crystals in his laboratory, puts it in one line:

“The six-fold symmetry you see in a snow crystal arises from the arrangement of water molecules in the ice crystal lattice.”

K. G. Libbrecht, Snowflake Science, SnowCrystals.com

A small crystal that grows slowly is a plain hexagonal prism, a slender column or a thin plate: the lattice, made big. The hard half is the other one. A star-shaped crystal grows six arms, and in a good specimen they are not just six: they carry matching ornaments at matching distances out, although the tips are far apart (two lab-grown crystals of Libbrecht's measured 2.0 millimetres from tip to tip). Some people, he reports, are so struck by this that they suspect “some acoustical or quantum mechanical oscillations are enforcing symmetrical growth.” His answer:

“What synchronizes the growth of the arms? Nothing. The six arms of a snow crystal all grow independently, as described in the previous section. But since they grow under the same randomly changing conditions, all six end up with similar shapes.”

K. G. Libbrecht, Snowflake Science, SnowCrystals.com

The previous section he refers to says why the conditions are shared: a crystal tumbles through the cloud meeting ever-changing temperatures and humidities, and “the six arms all took the same path, and so each experienced the same changes at the same times.” Each arm is a record of the same journey, written six times by six writers who never compare notes.

That is easy to say and hard to feel. So below, grow one.

Grow one

Clifford Reiter, of the mathematics department at Lafayette College, published a model of snow crystal growth in 2005 that is simple enough to state in a paragraph. The plane is tiled with hexagonal cells, and each cell holds a number that, in his words, very informally measures “the amount of water at that cellular location”. A cell at 1 or more is ice. A cell is receptive if it is ice or touches ice. Each step, receptive cells keep what they hold and gain a small constant, γ; every other cell's water is free to move, and moves toward the average of its neighbours. Far away, at the edge of the board, the water is held at a background level, β. Start with one cell of ice in the middle and run it.

That is the whole rule, and it treats all six directions alike. Reiter is plain that it is not physics: “while we use general physical notions to design our algorithm, we are not trying to fit equations of the physics.” Neither number is a temperature or a humidity. What it does have is the thing that matters here: a crystal that grows out of its own neighbourhood, with no cell able to see further than the cells beside it.

Reiter also tried what the page does next. To mimic a crystal falling “through air with different temperature and humidity”, he changed γ partway through a run. Here the change is yours to make: the strip below is the crystal's history, forty spells of fifty steps each. A gold spell gives the receptive cells a lot of extra water (γ = 0.01), and the edges fill in; a blue spell gives them very little (γ = 0.0001), and the tips run ahead and branch. Click a spell to flip it. The crystal is painted in the colour of the spell in which each cell froze, so you can read the history back off the ice.

With the noise at zero the stacked arms are pure white, and the readout says every frozen cell has all five of its rotated copies frozen too, on the very same step. Of course it does: the rule and the starting cell are symmetric, so the crystal cannot be anything else. (The verifier checks one subtlety. The computer adds the six neighbours in a fixed order, so a cell and its rotated copy can round differently in the last binary place; they do, by under a millionth of a millionth, and for history A over 1,949 steps it never once moves a cell across the line into ice.)

That perfect symmetry is a property of perfect arithmetic. To see what survives without it, turn the noise up. The noise is this page's addition, not Reiter's: every step, every receptive cell gets its own random share of γ, anywhere from none to double at the top setting.

Now the arms stop being identical. At ±50% noise, across eight noisy crystals with history A, only 54 per cent of the frozen cells, on average, have all five rotated copies frozen as well. Cell for cell, the arms disagree almost half the time. Yet look at the crystal: it is plainly six arms of one design, and the second gold spell in the history has coated all six of them out to the same distance, 45 cells from the centre on a board of radius 100. The stack shows where they agree: the spine and the band hold white, the fine side branches go red-brown. The noise rewrites the details. The history writes the shape.

Twins and strangers

Here is the measurement that turns Libbrecht's sentence into something you can test. If the arms are not coordinating, then an arm should be no more like its five sisters than it is like an arm of a different crystal grown through the same history, with its own noise. Call that crystal a twin. And if history is what makes arms alike, then a crystal grown through a different history, a stranger, should be less alike.

To compare two arms, rotate one onto the other and lay a grid of blocks over them. In each block take the share of cells that are ice, and add up how much the two arms differ, block by block. Likeness is 1 when the shares agree everywhere and 0 when the arms share nothing. Blocks one cell wide compare the arms cell for cell; blocks sixteen cells wide are a weak lens that sees only the broad shape.

The recorded run, eight crystals of each history at ±50% noise, gives the same answer at every lens. Cell for cell, an arm's likeness to its own five sisters averages 0.754, and its likeness to the arms of a twin averages 0.755. Through 16-cell blocks the two are 0.937 and 0.935. The difference is never more than 0.002 at any of the five block sizes. In this model the arms of one crystal share the same water, drawn from one field, and could in principle crowd or feed each other; the measurement says whatever they do to each other adds nothing you can see. They are as alike, to within 0.002, as two crystals that never met.

History B, which spends its first spells starved and gets one later burst, gives strangers: 0.607 cell for cell, 0.820 through 16-cell blocks. History C, a near neighbour of A with its bursts shifted by a spell or two, sits between, and every twin pair in the run beats every C pair at every block size. A weak lens narrows that gap without closing it, from about 0.10 cell for cell to 0.029 through 16-cell blocks.

One more control makes the point from the other side. Grow crystals under weather that never changes, all blue, and every crystal is a twin of every other: through 16-cell blocks two of them are 0.970 alike on average, as alike as one crystal's own arms. Remove the history and there is nothing left to tell them apart.

So, no two alike?

The phrase has an author of sorts. Wilson Bentley, a Vermont farmer who took his first photographs of snow crystals through a microscope in December 1884, wrote at the end of 1923, in an article published in 1924, that his collection numbered 4,200, “no two crystals being alike.” Libbrecht credits Bentley's images with being “largely responsible for the widespread notion that no two snowflakes are alike.” And Libbrecht's own answer follows from the same sentence as the symmetry: “since no two snow crystals follow the exact same path through the clouds as they fall, no two look exactly alike.” The model agrees with that, and adds the qualification that the word “alike” was hiding. Whether two crystals look alike depends on the lens. Cell for cell, even twins grown through one history score only 0.755; through a weaker lens, a crystal from a nearby history closes most of the gap on a twin.

The real world has run the experiment both ways. In the laboratory Libbrecht grows what he calls identical-twin snowflakes, two seed crystals side by side under one controlled, changing environment, and they come out “clearly very similar to one another, although they are not precisely identical.” His own summary of what that shows is the whole of this page: “Snow crystal symmetry requires symmetrical growth conditions. If the six arms experience different environments, they will grow differently.”

And in the sky, on 1 November 1986, a research flight over Wausau, Wisconsin exposed an oiled glass plate to a cloud for 11 seconds, and Nancy Knight of the National Center for Atmospheric Research found two crystals on it that were, if not identical, very much alike. Her letter appeared in the Bulletin of the American Meteorological Society in May 1988 under the title its editor gave it, “No Two Alike?”, and produced, as her husband and colleague Charles Knight later put it, “the deluge of publicity”. The crystals were not stars. They were hollow columns, about a quarter of a millimetre long, a far simpler shape with far fewer details in which to differ. Charles Knight said afterwards what had been lost in the coverage: “Lots and lots of snow crystals are ‘alike,’ meaning similar,” but the phrase no two alike “seemed to translate in people’s minds to ‘no two identical’.”

This page does not claim two identical crystals either. What it shows is narrower and, I think, more interesting: the likeness you see between snowflakes, and between the arms of one snowflake, is the same likeness, made by the same thing. A shared history.

What the model is not

It is flat. Real crystals have thickness, and a whole class of them (columns, needles, the capped columns that grow a plate on each end after drifting into colder air) cannot happen in a plane. It has no temperature: which shapes grow where in a real cloud is the Nakaya diagram, after Ukichiro Nakaya, who with his collaborators mapped it in a cold laboratory in the 1930s. Libbrecht's summary: thin plates near −2 °C, slender columns near −6 °C, large thin plates near −15 °C, and more branching when the humidity is high. Why the shape flips with temperature is, in Libbrecht's words, still something science “cannot explain”. Reiter's two numbers stand in for the humidity side of that only loosely, and the colours on this page are γ, not weather.

And most real snow is not symmetric at all. Libbrecht is emphatic: “the vast majority of snow crystals are not very symmetrical”, and the irregular ones are by far the most common. The six-armed beauties in the photographs are picked: he typically glances over thousands of crystals on his collection board before selecting one to photograph. The model makes that easy to believe. A crystal stays symmetric only while its arms share their conditions: in one pair of his lab twins, once the branches of one began interfering with the other's, the inner branches had no room and became stunted. Give one side of a crystal different conditions from the others, a neighbour close by, say, and the six records stop being copies.

Bentley had seen this a century earlier. Showing a journalist a photograph of crystals that each reached out further on the side facing its neighbour, he explained that in a thick snowstorm the crystals form close together and are distorted by their neighbours: “To be perfectly symmetrical, they must be outside the range of influence of other crystals.”

Sources

1. C. A. Reiter, “A local cellular model for snow crystal growth”, Chaos, Solitons & Fractals 23 (2005) 1111 to 1119, sections 2 to 5 and Figs. 6 and 7. doi:10.1016/j.chaos.2004.06.071. His update is printed as u + (α/12)(−6u + the sum of the six neighbours), the same arithmetic as the page's u + (α/2)(neighbour average − u). His boards were about 400 cells across; this page's are about 200, so that a crystal grows in a browser in seconds.
2. K. G. Libbrecht, “Snowflake Science”, snowcrystals.com/science, fetched 8 October 2026.
3. K. G. Libbrecht, “Identical-Twin Snowflakes”, snowcrystals.com/identicaltwins, fetched 8 October 2026.
4. K. G. Libbrecht, “Guide to Snowflakes: the Snow Crystal Morphology Diagram”, snowcrystals.com/morphology, fetched 8 October 2026.
5. K. G. Libbrecht, “The physics of snow crystals”, Reports on Progress in Physics 68 (2005) 855 to 895, p. 858 (author's copy). doi:10.1088/0034-4885/68/4/R03.
6. W. A. Bentley, “Forty Years' Study of Snow Crystals”, Monthly Weather Review 52 (1924) 530 to 532, p. 530 (Internet Archive scan).
7. M. B. Mullett, “The Snowflake Man”, The American Magazine (1925), an interview with Bentley, read here in the Jericho Historical Society's transcription (snowflakebentley.com), not an original scan; and the society's biography of Bentley (snowflakebentley.com/biography), both fetched 8 October 2026.
8. N. C. Knight, “No Two Alike?”, Bulletin of the American Meteorological Society 69 (1988) 496. doi:10.1175/1520-0477-69.5.496. Not read here: the publisher's page refused every request. The details of the collection are from the Associated Press report of the letter (University Daily Kansan, 15 June 1988, p. 7, KU Libraries), and Charles Knight's words from “Nancy Knight: The life of a singular scientist”, NCAR & UCAR News, July 2011.