Further, what constrains snowflakes to be so thin and so planar? When we do see a snowflake that has a sharp “bend”, why are the two portions of the snowflake individually almost planar?

Concerning the SA article, I think it is reasonable for it to state that temperature and humidity are the main factors in determining the pattern of a snowflake (although the best known explanation does omit humidity). But, concerning the global perfection of snowflake symmetry, this article is just as vague as all the other explanations I’ve seen.

It’s as though these professors want to prove that physics has a simple explanation for every phenomenon in nature, so they make up something that sounds plausible. On close inspection, all the explanations fall apart due to “hand waving” (a lack of rigor). They focus on supposedly relevant principles but omit a sufficient amount of actual explanation.

The part of this article that I believe is the statement that hydrogen bonds (proton-electron cloud attraction) cause a sixfold symmetry. The part I don’t believe is that they also explain the fact that each of the six arms are identical (except for minor dislocations, which are always corrected).

The question remains: why is each arm virtually identical?
The answer to this specific question at http://www.scientificamerican.com/article.cfm?id=why-are-snowflakes-symmet isn’t science. It basically says “during formation of a snowflake, each adsorbed molecule of water must go in one and only one place due to electrical attraction and repulsion.” This begs the question entirely. I’m shocked to see such a pseudoscientific statement published by SA.

I admit that crystalline structures have been well understood for many years. But the familiar case of solid crystals growing in liquid don’t make “arms” like snowflakes. Can all crystals growing by adsorption in a cold fog make arm patterns, or is this restricted to water?

And what causes the corrections of dislocations?

Maybe it’s time for physicists to admit that they don’t really know and are unwilling to experiment sufficiently to discover the reason?

I’m intrigued by the commutative/associative comment but I’m not quite sure what you mean by it. What aspect are you thinking of [I can think of non-associative ideas, but they're also not commutative, unless you mean that with two items it's always clear unless it's a tie, but with 3 it matters which pair goes first.]

I thought you were going in a completely different direction with this, namely, that RPS makes a good example of commutative but not associative!

The Bourbaki symbol seems very similar but slightly different in orientation, see e.g. this page or this page. (Both are translations, so there is a small probability that the French originals were typeset differently.)

Is it the case that flowers display the same symmetry as their fruits? I looked up banana flowers, but they look more like artichokes so that wasn’t revealing.