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What’s a more rigorous description of why snowflakes are so symmetric?

Symmetrical snowflakes with detailed, unique branches

The general explanations of why they’re symmetrical are:

  1. Theyre not. The branches actually vary.
  2. Snowflakes are somewhat symmetrical because the branches form in very similar environments (they are right next to each other after all).

This is unsatisfying. First of all, the ones I’ve looked at appear remarkably symmetrical. Occasionally they are lopsided, like the one in the lower right above, but usually not. Secondly, wouldn’t the formation of each branch be determined by chance and/or minute, almost micron-scale factors (factors right at the end of each particular branch)? Why do these formation dynamics match the other branches so well?

Furthermore, according to Pinterest creator Amy Dayton, on her page dedicated to radial symmetry in nature, the entire formation begins due a piece of dust or pollen. Usually life is behind the symmetry. It seems that smaller particulates at each branch would greatly affect their formation as well. But even if we assume otherwise, trillions of molecules are involved in every turn of the branch, so the crystalline structures should be free to go wherever. Seems like a lot of “memory” for the water of adjacent branches to continue in similar ways.

Snowflake images by Amy Dayton

For example, the one in the upper right. The existing explanations imply that just the local conditions (temperature, pressure, purity, etc) plus hexagonal crystals, plus that start to a branch.. will result in the unique palmetto tridents shown! (And these tridents appear after the branches go through differences in how they look!).

A correct answer will give us an intuition about why, after quadrillions of layers of molecules and ice crystals, opposite branches are still growing in the same way (beyond saying they are in similar environments).

Finally, a related question provides no help and just says ice makes hexagonal crystals at the molecular level.

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    $\begingroup$ This isn't an answer, but there's this extremely comprehensive textbook on snowflakes on arxiv here arxiv.org/abs/1910.06389 I haven't read it, though $\endgroup$
    – user1379857
    Commented Sep 9, 2021 at 19:47
  • $\begingroup$ Looks great eg:“Some might argue that creating an accurate.. simulation would not constitute a true understanding of the underlying phenomenon. Debating this point would require a precise definition of the word “understanding,” which is itself a nontrivial undertaking. Snow crystal formation involves a multitude of complex physical processes acting over a broad range of length and time scales. It may indeed be the case that our small brains cannot fully absorb all aspects of what is happening. If that is true, then we will have little choice but to let our machines do the heavy lifting for us” $\endgroup$
    – Al Brown
    Commented Sep 9, 2021 at 20:03
  • $\begingroup$ You are not asking about the fundamental hexagonal symmetry of the molecules, no? $\endgroup$
    – Cosmas Zachos
    Commented Sep 9, 2021 at 20:22
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    $\begingroup$ @CosmasZachos I just edited question because i see why youd say that. Yes I am aware of that. Didnt explain the larger phenomenon imo though. Baffling $\endgroup$
    – Al Brown
    Commented Sep 9, 2021 at 20:31
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    $\begingroup$ Related: physics.stackexchange.com/q/158532/247642, physics.stackexchange.com/q/3795/247642, physics.stackexchange.com/q/609410/247642 $\endgroup$
    – Roger V.
    Commented Sep 29, 2021 at 9:40

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I agree with the points you make about the common explanations appearing to be unsatisfactory. This is a guess, not an informed answer, but I belief it is logical to infer that since a) there is a high degree of symmetry, and b) the snow crystals grow through the accretion of water molecules, then the arrangement of the first few elements of the crystal- the foundation rings of molecules, if you like to think of it that way- must set a constraint that determines how subsequent layers can be slotted into the structure. If that were not the case, and if it was possible instead for the new layers of molecules to be fitted-in in a wide range of ways, then the pattern would break down as molecules were added in different ways at random.

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  • $\begingroup$ Oh that’s actually an interesting thought. $\endgroup$
    – Al Brown
    Commented Sep 9, 2021 at 20:39
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    $\begingroup$ I've just scanned the reference that was mentioned in the first comment attached to your question. It seems that the foundational core of the snowflake has a hexagonal symmetry and attracts molecules to each of its corners. Since the crystals are small compared with the distances over which atmospheric conditions vary, each corner of the initial hexagon experiences pretty much the same conditions as the other corners, and so they attract new molecules in the same way. That each snowflake encounters unique combination of local atmospheric conditions means that they are all highly individual. $\endgroup$
    – Marco Ocram
    Commented Sep 9, 2021 at 20:58
  • $\begingroup$ The book seems marvellous, and will certainly answer whatever detailed questions you might have about the process! $\endgroup$
    – Marco Ocram
    Commented Sep 9, 2021 at 20:58
  • $\begingroup$ I mentioned that explanation in my answer. Is not nearly enough though. Must be missing something basic. I mean in the snowflake I discuss, the symmetry happens after sections that differ between branches. Hundreds of thousands of crystals later being a match is not explained by initial symmetry and similar micro-environments $\endgroup$
    – Al Brown
    Commented Sep 9, 2021 at 21:05

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