Conference rusure::math

If they weren't, they would be aerodynamically unstable and would fall too
fast to grow. Many of the smallest snowflakes are NOT symmetrical.

    
    The real reason they are symmetrical is that there's no bias to cause
    them to be otherwise.
    
    

Ah, but there is randomness. The overall shape is determined by the 
crystalline structure of ice (which leads to another child's question, of 
course) but the rate of growth is determined by very local conditions that 
are maintained in free fall. Look at real snowflakes; they are not all
really all that symettrical. 

What do you mean by "symmetrical"?
If there's NO directional bias, why aren't they spherical?
As pointed out before, the shape of the ice crystal lattice
must have something to do with it.

    Take a small number of water molecules, arrange them into a
    three-dimensional pattern and imagine that pattern repeated
    indefinitely in space.  Now measure the electrostatic energy of the
    arrangement.  Vary the number, position and orientation of the 
    water molecules until the energy is minimized.  You will find that the
    resulting lattice has hexagonal symmetry.  Note that I know of nobody
    actually doing this calculation for water, but similar calculations for
    simpler systems do confirm that observed crystal structures are minimum
    energy.
    
    Because the lattice has hexagonal symmetry, the macroscopic crystal
    will also have hexagonal symmetry.
    
    But there a lot of macroscopic shapes with hexagonal symmetry: prisms,
    plates and branches among other things.  Which of these forms depends
    on what faces of the crystal grow fastest, which is in turn determined
    by the temperature and humidity of the air from which it is growing.
    Because the crystal is so small, all faces are exposed to the same
    conditions, so they grow the same way.
    
    This is just theory, of course.  Real snow crystals are not exactly
    symmetrical, and some have beautiful and striking asymmetries.  I
    suspect that this is often the result of dust landing on a growing
    face.
    

	Thanks for all your suggestions.   Here are my initial thoughts:

> If they weren't, they would be aerodynamically unstable and would fall too
> fast to grow. Many of the smallest snowflakes are NOT symmetrical.

	Interesting idea, but I don't buy it.   It doesn't by itself account
for the *hexagonal* symmetry, nor do the shapes *look* 'aerodynamic'.   They
just look hexagonal.   Is the final sentence experimentally observed?   
Could it be that the smallest snowflakes *were* symmetrical, then melted
asymmetrically?   Or if not, then this result has consequences for some of
the later replies, which would expect the larger snowflakes to be *more*
assymetrical than the smaller ones.

>    The real reason they are symmetrical is that there's no bias to cause
>    them to be otherwise.
    
But why hexagonal?   Nah, don't buy it.

> Ah, but there is randomness. The overall shape is determined by the 
> crystalline structure of ice (which leads to another child's question, of 
> course) but the rate of growth is determined by very local conditions that 
> are maintained in free fall. Look at real snowflakes; they are not all
> really all that symettrical.

Well, how random effects influence or fail to influence the growing crystal
is clearly the nub of the question, once the 'overall shape' is decided.
I agree that the crystalline structure of ice must determine the
*hexagonality* of the structure.   Rate of growth is hard to determine given
the final snowflake, but what I'm asking is something different, the *form*
of the snowflake.   Another experimental remark, I can't judge.

> What do you mean by "symmetrical"?
> If there's NO directional bias, why aren't they spherical?
> As pointed out before, the shape of the ice crystal lattice
> must have something to do with it.

	Yes.

>    Take a small number of water molecules, arrange them into a
>    three-dimensional pattern and imagine that pattern repeated
>    indefinitely in space.  Now measure the electrostatic energy of the
>    arrangement.  Vary the number, position and orientation of the 
>    water molecules until the energy is minimized.  You will find that the
>    resulting lattice has hexagonal symmetry.  Note that I know of nobody
>    actually doing this calculation for water, but similar calculations for
>    simpler systems do confirm that observed crystal structures are minimum
>    energy.

	Very plausible.   Sorta reminds me of those freefall guys who link hands
to form cycles.   I wonder if the water molecules *do* link up in a chain
which then bends round to meet itself, at the beginning.	
    
>    Because the lattice has hexagonal symmetry, the macroscopic crystal
>    will also have hexagonal symmetry.

	Perhaps.
    
>    But there a lot of macroscopic shapes with hexagonal symmetry: prisms,
>    plates and branches among other things.  Which of these forms depends
>    on what faces of the crystal grow fastest, which is in turn determined
>    by the temperature and humidity of the air from which it is growing.
>    Because the crystal is so small, all faces are exposed to the same
>    conditions, so they grow the same way.

	Now this isn't obvious to me for two reasons.

(1) Non-isotropy.   Gravity and wind direction would seem to me to be both
directional forces, which will impact the crystal asymmetrically, whatever
the size of the crystal.

(2) Chaos theory.   Growing a 'small number of molecules' up to a crystal
whose symmetries can be oberved by the naked eye is a process of many
stages.   Given the complicated nature of each individual arm of the
crystal, it would seem to me likely that the crystal growth process is
chaotic, and the separate arms would diverge in shape.

I would expect the second to be especially true if temperature and humidity
were significant factors, rather than the shape of the static seed.   Even
if temperature and humidity could be assumed to be constant over the
small seed, sure this could not be the case at the macro-level.

And if temperature and humidity are not significant, and initial seed
determines everything, then how many possible snowflakes are there?

>    This is just theory, of course.  Real snow crystals are not exactly
>    symmetrical, and some have beautiful and striking asymmetries.  I
>    suspect that this is often the result of dust landing on a growing
>    face.
    
	Another experimental result quoted.   It would be nice to have
access to a population of typical snowflakes, to support or refute the
allegations.

	Basically, the Neilsen line is most plausible to explain the
symmetry at the small crystal level.   I think he has more work to do if
he wants me to believe that this causes symmetry at the macro level.

Andrew.

    re:         <<< Note 1133.6 by HERON::BUCHANAN "Andrew @vbo DTN 828-5805" >>>

>for the *hexagonal* symmetry, nor do the shapes *look* 'aerodynamic'.   They
>just look hexagonal.   Is the final sentence experimentally observed?   
>Could it be that the smallest snowflakes *were* symmetrical, then melted
>asymmetrically?   
    
    This was not my idea, but I like it, as applied to the plate form.  The
    largest single flakes you see are all plates (footnote 1: journal of
    casual looking).  Plates are well-known to be the slowest falling
    shapes (footnote 2: annals of dimly-remembered areodynamics).  I think
    there's a connection here.  
    
    The small single flakes I have looked at did not appear to be melted at
    all.  They generally look like tiny prisms.  I have seen remelted snow, 
    but this usually takes the form of agglomerations of many small blobs.
    
>	Very plausible.   Sorta reminds me of those freefall guys who link hands
>to form cycles.   I wonder if the water molecules *do* link up in a chain
>which then bends round to meet itself, at the beginning.	
    
    I don't know if the study has been done for ice, but most crystals
    studied grow by individual atoms or molecules approaching the surface,
    then either locking into the right place (with the right orientation)
    or sliding around on the surface until they find the right, that is,
    low energy place.  It is well known that ice forms very slowly without
    a seed of some sort.
    
>>    Because the lattice has hexagonal symmetry, the macroscopic crystal
>>    will also have hexagonal symmetry.
>
>	Perhaps.
    
    This is a well-established rule of crystallography.  The macro symmetry
    of the crystal must reflect the micro symmetry of the lattice.  There
    are, of course, some complications, but the agreement is good enough to
    let us expect that snowflakes should have hexagonal macro symmetry.
    
>(1) Non-isotropy.   Gravity and wind direction would seem to me to be both
>directional forces, which will impact the crystal asymmetrically, whatever
>the size of the crystal.
    
    Wind by itself does not affect an ice crystal.  It is simply carried along
    in a moving air mass.  It is too small to feel any wind-shear, and too
    light to resist the effect of gusts.
    
    An ice crystal will feel gravity.  On average, it will be falling with
    its terminal velocity (very slow, during most of its growth) through
    the cold, moist air.  But this still leads to hexagonal symmetry, in
    most cases.  The plates and flakes fall in a horizontal orientation, so 
    gravitation is parallel to their hexagonal symmetry axis.  The prisms
    mostly fall with hexagonal symmetry axis horizontal, and rapidly
    tumbling around this axis, so the effect of gravity averages out. 
    Prisms may also fall with hexagonal axis vertical, in which case they
    may show a significant difference between one end and the other.

    Deviations are interesting, and I have seen them reported.  For
    example, the downward facing surface of a flake will sweep up smaller,
    slower-falling ice and dust.  If the flake does not flip over too
    often, the two faces will have a different appearance.
    
>(2) Chaos theory.   Growing a 'small number of molecules' up to a crystal
>whose symmetries can be oberved by the naked eye is a process of many
>stages.   Given the complicated nature of each individual arm of the
>crystal, it would seem to me likely that the crystal growth process is
>chaotic, and the separate arms would diverge in shape.
    
    You are right, in the sense that a typical flake will have six arms,
    they will be generally similar (for example, all six finely branched),
    but the individual arms will differ in the details of their branching.
    And this is exactly what we observe.  The hexagonal symmetry holds only
    at the overall level: all six arms are usually plate-like or
    branch-like or whatever.

>I would expect the second to be especially true if temperature and humidity
>were significant factors, rather than the shape of the static seed.   Even
>if temperature and humidity could be assumed to be constant over the
>small seed, sure this could not be the case at the macro-level.
    
    Several degrees, or several percent humidity mark the transitions
    between different forms of ice crystal.  These changes typically
    correspond to hundreds of feet of atmosphere.  Adjacent cells of
    atmosphere do a good job of staying in equilibrium with each other.
    So even a fairly large snowflake, say an eighth of an inch across, 
    will see the same temperature and humidity, closely enough to grow 
    the same form of crystal.

>access to a population of typical snowflakes, to support or refute the
>allegations.
    
    I have seen books with many snowflake photographs in them.  Sorry I
    can't give you any detailed references.  I may try when I get home.
    _Field Guide to the Atmosphere_ (one of the Peterson Series) is
    probably the source for a lot of this information.  I've also spent
    some time looking at snowflakes myself, naked eye, hand lens, and small
    microscope.

    I believe good old Dover publishes a paperback of snowflake
    photographs.
    
    I recall reading someplace that the reason snowflakes differ from
    one another is that they *do* experience significant macroscopic
    changes in temperature and humidity, as they move through the
    atmosphere.  While they're growing, they apparently move up and
    down quite a bit, and don't just "fall" continuously. 
                  
    len.
    

    re:                      <<< Note 1133.8 by DRUMS::FEHSKENS >>>
    
>    I recall reading someplace that the reason snowflakes differ from
>    one another is that they *do* experience significant macroscopic
>    changes in temperature and humidity, as they move through the
>    atmosphere.  While they're growing, they apparently move up and
>    down quite a bit, and don't just "fall" continuously. 
    
    This is correct, and explains both why snowflakes differ from each
    other and why single snowflakes often contain multiple forms: stellars
    terminated by plates or prisms sandwiched between plates.  These
    movements, up and down and lateral, may cover miles during the growth
    of a snowflake.
    
    The previous discussion was whether there was enough uniformity across
    the dimension of the snowflake to explain the general hexagonal
    symmetry.  Variation on the scale of miles is not relevant to variation
    on the scale of tenths of an inch.