Saturday, December 26, 2009

On the Five-Pointed Snowflake

There’s a fun letter in the latest issue of Nature from Thomas Koop at the University of Bielefeld pointing out how often snowflakes in festive decorations, greetings and wrapping paper are misdrawn with fewer or more than six points. Above is a particularly fine example that I discovered this year, in which some of my presents were (ironically) wrapped (available at Sainsbury’s, no doubt now at a reduced price). It never ceases to amaze me that designers have failed to assimilate this very basic fact about the six-pointed snowflake, though it’s been generally known for centuries. Indeed, the knowledge goes much further back than Kepler’s seventeenth-century treatise, as I pointed out here and in the ‘Branches’ volume of my recent trilogy Nature’s Patterns.

(I’m amused too to see that Nature’s marketing folk are still managing to embarrass the scientific staff. I could tell other tales, but it would be cruel.)

Yet these pentagonal snowflakes set me thinking. As is widely known, the only way ‘crystals’ can display growth habits with fivefold (or indeed eightfold) symmetry is if they are in fact quasicrystals. But could water form quasicrystals? Certainly, in the liquid state it is much more congenial for water molecules to form fivefold rings than the sixfold ones present in ice, because the bond angles are then much closer to that preferred in the tetrahedral coordination geometry. And these pentagonal rings are a general feature of the crystal structures of clathrate hydrates, in which water is frozen around nonpolar solutes such as methane. Now, I’m no crystallographer but I have the impression that it would be naïve to imagine that a local pentagonal packing symmetry is all it takes to make quasicrystallinity feasible. But on the other hand, it’s a good start; our current understanding of quasicrystals grew partly out of Charles Frank’s early work on icosahedral clustering in simple liquids. And large icosahedral structures for water have certainly been postulated. In fact, I’m very puzzled that I can seem to find no discussion in the literature of the possibility of quasicrystallinity in water – either I’m failing badly to understand something (quite possible) or I’m somehow looking in the wrong places (also possible). But I will ask my water structure gurus about this, and if anything comes of that, watch this space.

Incidentally, strict twelvefold symmetry is also forbidden in true crystals but known in quasicrystals. Yet a sort of pseudo-twelvefold symmetry has been seen in snowflakes, due to the coalescence of two snowflakes.


omnivorist said...

While you are in contact with your 'water structure gurus' (wish I had one of my own) I wonder whether you might also ask them if they have an answer to this other puzzler re snowflakes:

We are all familiar with the various portrayals of snow flake crystals - all showing 6-fold symmetry, but nevertheless exhibiting a rich variety of forms.

What intrigues me is why (it appears) the geometry of a single snowflake is consistent. To put it another way, there would seem to be no reason why a particular real snowflake shouldn't look like a collage or hybrid of the idealise patterns we see in books. Once the crystal has grown outwards beyond it's seed, it would seem reasonable to expect each of the arms to develop differently.

I can see that it might be the case that, in the seed itself and the initial crystal layers there is some sort of unique, yet symmetric shape that serves as a pattern for everything that follows. Nevertheless, given the seeming variety of forms, it is difficult to see how there is sufficient 'information space' in the starting crystal to encode all the different variants.

The only other hypothesis that seems plausible is that, at each stage of growth, there is some sort of delicate electrical field that is maintained consistently around all 6 axes and created by what has gone before, that determines the optimum sites for further attachment of water molecules.

To summarise: growth along each axis of the snowflake must follow some sort of pattern - different in different snowflakes but consistent in a single flake. Is this 'pattern' information communicated to the axes from the central seed or between the separate arms as they grow outwards.

It could be of course that truly symmetrical snowflakes only appear in books and that real snowflakes are hybrids as described. I've tried looking, of course, but it's not as easy as one might imagine (as I am sure you are aware).

Rocki said...

The five points -- as in pentagram -- was not a scientific gaff, but a totally intentional pagan/witch holiday design.