The good people at the US Postal service do produce some splendid sets of stamps on quite a regular basis. Today, to my delight, I was offered this set as one of my choices at the post office:
There’s so much physics behind the microscopic lattice of H2O molecules underlying such a lovely shape, with the pleasant C6 symmetry of the macroscopic result – the snowflake. I found a lovely “morphology diagram” showing the sorts of shapes you get depending upon temperature and supersaturation of the water vapour that condenses to form the snowflake. It is from the lab website of Kenneth G. Libbrecht, of Caltech. Go there (particularly to the snowflake primer) for much more information.
You just have to sit back and admire sometimes:- Nature’s just fantastic, you know.
-cvj
[Update: Bee has a post about patterns and symmetry on Backreaction, where she talks about snowflakes too, among other wonderful things.]
A cavil: if when you said C6, you meant the cyclic group of order six, that’s not right: surely you ought to have said D6 (also called D12), the dihedral group of order twelve.
Webern and symmetry. Basically, Webern’s mature compositions were constructed using the twelve tone system. (There’s a really good article on Wikipedia about serialism which explains this compositional technique.) The music is constructed from from a row of twelve notes which involves taking all twelve notes of the chomatic scale and arranging them into a particular order without repeating them. This prime form of the row can then be transposed, inverted, played backwards and the inversion can also be put into retrograde. The pitches that form the basis of the composition can be arranged into a 12 by 12 matrix by representing each pitch with a particular number which make it possible to observe any symmetry. Webern tended to use ‘derived rows’ (again a good explanation of this can be found on Wikipedia), which involve palindromes and hence the symmetry. Op 24 (Concerto) and Op 28 are really good examples as both note rows result in high degress of invariance and combinatoriality. Downloads can be found on http://www.webern.com
It isn’t easy listening, but it’s wonderfully constructed!
Dr Johnson,
I didn’t know they were connected either and probably would not have made the connection but the catalog description of the book had mentioned that the pictures had been used by the post office.
TheGraduate: – Really? That’s an amazing coincidennce indeed, since I’d no idea those sources were connected.
Thanks!
Mary:- Hi! What did Webern do with symmetry in his compositions? Any examples to point out to us to go and listen to? Thanks.
-cvj
A delightful post with interesting links! Webern (one of my favourite composers) was fascinated by snowflakes (and indeed other crystals), but given his preoccupation with symmetry in his music, I suppose this is not so surprising.
How does this long range order show up in a snowflake? Why does the water molecule in the northeast corner align itself to be symmetrical to the water molecule that is attaching itself to the lattice on the southwest corner?
The only answer I have is that these two latecoming molecules encounter identical surfaces to latch on to, because all the previous molecules that attached to the snowflake did so too. But if the symmetry of the snowflake is set by how the core condenses, then how are there virtually infinitely many patterns?
(Maybe I should be more clear. It’s the book from which the pictures on the postal stamps above were taken.)
Hi Dr. Johnson,
I was thumbing through the Barnes and Noble catalog that I got from in the mail today and I stumbled on the following:
“The Little Book of Snowflakes” by Kenneth Libbrecht
It’s the book from which the snowflake pictures were taken! It’s weird how coincidences like that work.
http://www.amazon.com/Little-Book-Snowflakes-Kenneth-Libbrecht/dp/0896586529
Cheers.
Clifford, I didn’t mean to imply that there’s a contradiction between strings and chaos. However, I do see the two as being on opposite ends of Nature’s spectrum, so to speak.
While the aim of string theory is to reveal simpliciy in Nature, the objective of chaos theory – on the other hand – is to uncover complexity in Nature.
Perhaps you see an overlap between strings and chaos. Yet on a fundamental level, I envision the two as being separate and distinct. In fact, string theory seems more attuned to the primordial Universe. By contrast, chaos theory appears more pertinent to the evolutionary Universe, especially the Universe following “the Dark Ages”.
But then, I could very well have it all wrong, not even wrong.;) Strange though it may sound, simplicity and complexity might have been joined together since the beginning…
Have a most enjoyable evening!
Cynthia
“Although I absolutely adore strings, I’ve acquired a closet passion for chaos/complexity.;)”
I don’t see a contradiction here….
-cvj
Clifford, thanks for sharing a most thought-provoking post on snowflakes!
Without surprise though, when I see a snowflake not only does the Renaissance fingerprint of Kepler springs to mind, but the primordial fingerprint of chaos/complexity suddenly emerges as well. Johannes Kepler (1611) formulated his conjecture after pondering the question of why snow crystals always exhibit a six-fold symmetry. Then in 1998, a proof of Kepler’s conjecture was derived via a computer program. But because this 1998 proof severely lacks the fundamental component of human thought, many mathematicians still regard Kepler’s conjecture as an enigma.
As far as chaos/complexity and snowflakes are concerned, in 1904, Helge von Koch – a pioneer in the field – calculated his famous fractal dimension (the dimension of 1.2618) called The Koch Snowflake.
Although I absolutely adore strings, I’ve acquired a closet passion for chaos/complexity.;)
Pingback: Snowflakes: how they get that way « Later On
Bee! That’s excellent! Great minds think alike, they used to say. Will update…
-cvj
Hi Clifford,
What a lovely post! And what a coincidence: I just used snowflakes as one of my favourite examples for beauty in symmetry, see
The Beauty of it All
Best,
B.
Fantastic images, and nice to see the morphology diagram too. Gives some further insight into sun haloes, par-helia and suchlike as the varous displays are controlled by crystal shape, orientation etc.
I’d love to know how they photograph snowflakes, it’s hard i know (actually there’s a section detailing how to do it) : http://journals.aol.co.uk/bobmcnaught/astromcnaughts_home_planet/entries/821 from my late un-lamented (except by me) blog, shows my miserable attempts at frost. It insisted on melting all too quickly! There’s also a link to the pioneer of snowflake photography, one Wilson “Snowflake” Bentley.
Incidentally Clifford, my blogging attempts were directly inspired by yourself. Sometimes i wish i had persisted, but nobody ever bothered to read it, so i didn’t see the point.
Hmm. trying again.This is my favorite Libbrecht page.
Kenneth Libbrecht’s site is glorious, isn’t it? I was enamoured with this particular page in the past; passing the link around to my friends.
A few years ago while watching the first snow with the big puffy snowflakes fall in Heidelberg and missing some things, including the fluid ability to speak and understand speech in my native language, I wrote the following snow poetry. Enjoy…