## Shape of snowflakes

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Is there a mathematical theory that explains the shape of a snowflake? Why is it not round?

Update Tree-like metric spaces appear often as limits of sequences of metric spaces (say, asymptotic cones or boundaries of metric spaces). I wonder if similar objects can be obtained as shapes minimizing some kind of energy functional. This may lead to new constructions in geometric group theory.

I just saw Igor Rivin's answer which may be what is needed. Perhaps somebody can give a more detailed answer?

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Water crystallizes in a hexagonal lattice, so small snowflakes are just hexagons. For reasons of surface chemistry that I don't really understand, water molecules are more likely to attach at a vertex than in the middle of an edge, so as the hexagons get large the vertices grow faster than the edges, creating a non-convex figure with 12 edges. Iterating this procedure (growing faster at vertices than at edges) gives a snowflake--for a picture of this, see pg. 884 here: its.caltech.edu/~atomic/publist/rpp5_4_R03.pdf (the whole article is pretty interesting. So really, the Koch... – Daniel Litt Dec 23 2011 at 20:12
...snowflake with appropriately chosen parameters is not a bad model at all. – Daniel Litt Dec 23 2011 at 20:12
@Daniel: This may not be what I want since I wanted an explanation, not a description. An example of an explanation could be something like this: "consider the following `energy' functional ..., the shapes that minimize that functional are snowflakes." – Mark Sapir Dec 23 2011 at 20:27

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The review that Daniel Litt linked to explains that the shapes of snowflakes are the outcome of a complicated growth process, involving the competition of between the rates of diffusion and attachment of water molecules and heat transport, among other things. In particular, as snowflakes are formed very far from the equilibrium state of water and ice, their overall shapes won't be the minimizers of any natural thermodynamic energy functional, or even randomly perturbed versions of such, in the way that large facets of crystals might be. – jc Dec 24 2011 at 5:24

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I think their shapes are more akin to the random fractals arising from diffusion-limited aggregation, where molecules diffusing in "from infinity" attach to a seed. I think this paper psoup.math.wisc.edu/papers/h2l.pdf by the group that created the pictures that Joseph O'Rourke linked to below is probably the best current explanation for snowflakes, and has a similar philosophy starting from growth rather than minimization - create a random growth model (on a lattice, even) with certain physically motivated properties and one finds shapes that match well the real life snowflakes. – jc Dec 24 2011 at 5:35
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## 3 Answers

Yes, there is a quite active theory of crystal formation, in which the late Fred Almgren and the very much with us Jean Taylor did groundbreaking work. If you google "ALmgren Taylor dendrites" you will be enlightened. You can read the papers (and papers referring to the papers) -- I think the theory is not so simple.

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Janko Gravner at UC Davis and David Griffeath at the University of Wisconsin-Madison have modeled snowflake growth, as reported on this web page:

the researchers were able to recreate a wide range of natural snowflake shapes. Rather than trying to model every water molecule, it divides the space into three-dimensional chunks one micrometer across. The program takes about 24 hours to produce one "snowfake" on a modern desktop computer.

Paper, code, and movies on this modeling are available here. Here is a 1min17sec YouTube video of growth simulations following this model; and here is a 26sec, more colorful set of simulations.

Added. Some added detail from the G-G papers:

The building blocks for snowflakes are hexagonally arranged molecules of natural ice (Ih). Just how the elaborate designs emerge as water vapor freezes is still poorly understood.... The solidification process involves complex physical chemistry of diffusion limited aggregation and attachment kinetics....Our basic set-up features solidification Cellular Automata on the triangular lattice $\mathbb{T}$ (to reflect the arrangement of water molecules in ice crystals).

To echo Igor, it's "not so simple"!

A more physically based, 3D model is explored in the paper "Monte Carlo Simulation of the Formation of Snowflakes," by Maruyman and Fujiyoshi Journal of the Atmospheric Sciences, 2005. Comparisons of the shape to "observed snowflakes" are made:

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@Joseph: Thanks! But right now I see lots of snowflakes appearing in real time outside my window.:) – Mark Sapir Dec 23 2011 at 20:49

I was curious about the OP's second question, which I now think is actually rather difficult. Namely

Why is it [the shape of a snowflake] not round?

Various physics-y sources (e.g. this paper) suggest the following "explanation" for the shape of a snowflake, which I mention in my comment on the original question--namely, that water crystallizes in a hexagonal lattice, so small snowflakes are just hexagons; new water molecules are more likely to attach at corners than edges or faces (for complicated reasons I don't really understand) so vertices grow faster than edges. Thus the hexagon will become a non-convex 6-pointed star; then the edges of this figure will split similarly, and so on. This interpretation is born out by e.g. the picture on pg. 884 of the paper above.

This inspired the following simple model, which comes in both deterministic and random flavors. We'll build a snowflake on the standard hexagonal lattice in $\mathbb{R}^2$, spanned by e.g. $(1, 0)$ and $(1/2, \sqrt{3}/2)$. Start with a single regular hexagon of side length $1$ centered at the origin, with vertices the six shortest lattice vectors.

In the deterministic version of the model, at each positive integer time $t$ we add a regular lattice hexagon with side length $1$ centered at each lattice point which is the vertex of exactly one hexagon. In the random version, at each positive integer time $t$, we add a hexagon centered at a random lattice point which is the vertex of exactly one hexagon with uniform probability over such lattice points.

I had some time this morning, so I coded up both models in the language "Processing." Here is a typical pair of snowflakes from the deterministic model:

This model has the following interesting properties, none of which are particularly difficult to prove.

1) By the envelope of a snowflake I mean the smallest simply-connected polygon containing it. Let $S_n$ be the envelope of the snowflake at time $n$. Consider the sequence $S_n$ in the space of plane polygons metrized by Hausdorff distance, modulo homothety (two polygons are homothetic if one is congruent to a rescaling of the other). Then $S_n$ is recurrent (that is, any homothety class visited by $S_n$ is approached arbitrarily closely infinitely many times). However, the only homothety class taken infinitely many times by the $S_n$ is that of a regular hexagon. (Thus in this setting the adage that no two snowflakes are alike is pretty far off.)

2) Let $H_n$ be the smallest regular hexagon containing $S_n$. Then $$\frac{\text{area}(S_n)}{\text{area}(H_n)}$$ is bounded above by $1$ and below by, say $1/2$ (though one can do better). By virtue of the recurrence of the $S_n$, however, this ratio does not attain a limit.

3) Certain interior triangles are never filled in, and as is visible in the pictures above these follow a beautiful regular pattern which I haven't bothered to work out.

Now let's look at the random model. Here are two typical snowflakes:

As you can see, these are quite round, so they might be better called snowballs. I understand this model much less well than the deterministic one above. However, the following conjectures are natural given the pictures.

4) (Conjecture) In the space of homothety classes of plane polygons, metrized by Hausdorff distance, as in (1) above, the envelopes of these shapes tend towards the homothety class of a circle with probability $1$.

5) (Conjecture) The ratio $$\frac{\text{perimiter}(S_n)}{\sqrt{\text{area}(S_n)}}$$ tends to infinity with probability $1$.

In other words--the random model I implicitly suggested in my comment on the original question seems to give round snowflakes! So I at least think the physics question as to why snowflakes aren't round is still pretty interesting.

In the comments, Rebecca Bellovin suggests another random model--namely, fix a probability $0\leq p\leq 1$ and at each time $t$, and each valid lattice point (namely, each lattice point which is the vertex of exactly one hexagon) add a hexagon centered at that point with probability $p$. At least for small $t$ (e.g. $t<10000$), this seems to interpolate between the two models I give here, and certainly if one scales $p$ in proportion to the number of valid lattice points (so that for example the probability is negligible that more than one hexagon will be added, or that no valid points will be missed), these models will behave exactly like the ones I give. On the other hand, for middling $p$, something interesting happens--namely, the snowflakes look like rounded hexagons. At Rebecca's request, I am posting a picture for $p=0.7$, below:

I have no real explanation for this phenomenon; only unconvincing heuristics.

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I'd be happy to send the Processing code/java applet to anyone who'd like to play with these models--my email is in my MO profile. – Daniel Litt Dec 24 2011 at 22:47

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I don't like your random model: it seems much more reasonable to fix a probability p and at time t, attach a new hexagon at every available lattice point with probability p. Assuming a fair amount of water in the air, water crystallizing at one vertex should be independent of water crystallizing at other vertices (to a first approximation). – Rebecca Bellovin Dec 24 2011 at 22:58

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Nice! Is it similar to percolation clusters, self-avoiding random walks, etc.? – Mark Sapir Dec 24 2011 at 23:50

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@Daniel: I know some people who know these subjects. I will ask when I see them. – Mark Sapir Dec 26 2011 at 1:19

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The random model looks very close to the Richardson growth process; my web page shows 3D images www-fourier.ujf-grenoble.fr/~bkloeckn/images.html and Olivier Garet's one has many images of related random processes iecn.u-nancy.fr/~garet/images.php – Benoît Kloeckner Mar 14 2012 at 13:52
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