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dream_weaver

The Snowflake Conjecture

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It snowed last night.
It snowed last night.
The sky bears had a pillow fight.
They tore up every cloud in sight,
and tossed down all the feathers white.
Oh, it snowed last night.
It snowed last night…

Every snowflake is different. No two snowflakes are alike. Looking out over snow covered fields, there are a lot of snowflakes. They melt. More snow falls. They don't melt. More snow falls, increasing the depth of coverage.

That's a lot of snowflakes.

Isolate a snowflake. Catch one falling from the sky. Look at its intricate detail. Catch another one. Compare it. Every snowflake is different. No two snowflakes are alike. There are too many to compare. There are too many winters from which the snowflakes are no longer available for such a comparison.

Their icy, crystalline structure have a clear prism appearance to them. Yet the field of snow appears white. The crystalline structure has an appearance of symmetry, often across multiple planes of symmetry. Still, with such a vast number of snowflakes that fall . . . how many snowflakes have fallen? Break the problem down into smaller, soluble considerations. How wide is one snowflake? How long is it? How thick is it? Measure a few more. Identify a range for each axis of measurement. Wait. These snowflakes are not thin square prisms. They are more like thin hexagons. Hexagons can be nested, except . . . they don't all fall laying flat. And there are more than can be seen from one geographic location. And then there's this snowfall, and snow has fallen before. How can it be that every snowflake is different and that no two snowflakes are alike? Snow falls in both the northern and southern hemispheres. Is the planet Earth the only place the in the universe that snow forms?

Look at the shape of those snowflakes again. How many permutations can there possibly be. Can there be more permutations than there are snowflakes? Is this the axis down which every snowflake is different and that no two snowflakes are alike?

snowtypes4.jpg

All S is P. All snowflakes are different.

Wherein lies the logical leap? In the finiteness of quantity? In the finiteness of permutations? Should a variation in the specific temperature of the particular snowflakes be tossed into the mix to augment the permutation angle? In the grand scheme of things, this may be a trivial inquiry. But is it? Does this challenge the notion for the basis of accepting the truth or falsehood of a proposition?

Another consideration may be a case of two snowflakes found to which no measurable difference of any kind can be found. (I would consider this a hypothetical scenario.) In one of Pat Corvini's talks, she approaches this angle using Achilles and the Tortoise. The short of her conclusion, with which I agree, is that if two lengths or distances are indiscernible by available measurement means . . . they are the same.

The onus of proof lay with the asserter of the positive. If mom or dad said that all snowflakes are different, being challenged, could invoke the response "Look for yourself, and see." Does the limited sampling of comparisons justify the logical leap?

From my examination of Collatz Conjecture, I would have to conclude that the truth or falsehood of "Every snowflake is different" and "No two snowflakes are alike" can be reached. In the examination of Collatz Conjecture, all sorts of patterns emerge that can be identified which repeat themselves, yet each pattern is distinctly different. Permutations manifest that repeat in a similar manner without being the same, other than in their form. I'm not so sure I can do the same with this snowflake conjecture . . . before they melt.

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As someone who has hiked in various types of snowstorms, I know that snowflakes are very much alike, in size, shape, and pattern. But if your definition of "alike" means "the same entity," then obviously every snowflake is different, i.e., not the same entity. Each snowflake is its own individual entity.

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Point taken. That is not what is being pursued though. Consider two stamps that are indistinguishable from one another. Or two coins struck from the same die. Put the coins or stamps out of sight. Switch their positions or leave them in the same location. Upon seeing them again, can you tell if they are in the same location or have their locations been switched? This is the type of alikeness being alluded to in the OP.

Of course, if mom and dad simply meant that each snowflake is its own individual entity, then this exercise is making a big mountain out of a little mole hill.

In a similar conclusion drawn from an article looking at Snowstorms or Snowflakes thru the "prism" of collectivism or individualism:

No two snowstorms are alike, but a far more amazing fact is that no two snowflakes are identical either—at least so far as painstaking research has indicated. Wilson Alwyn Bentley of Jericho, Vermont, one of the first known snowflake photographers, developed a process in 1885 for capturing them on black velvet before they melted. He snapped pictures of about 5,000 of them and never found two that were the same—nor has anyone else ever since.

Is 5000 a large enough sample to make the logical leap? Or could it be that "alikeness" is being conflated with "individuality" here too?

 

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8 hours ago, dream_weaver said:

From my examination of Collatz Conjecture, I would have to conclude that the truth or falsehood of "Every snowflake is different" and "No two snowflakes are alike" can be reached.

Just in case clarification need be made here:

I would have to conclude the the truth or falsehood of the two propositions can be reached. My question is are the propositions demonstrably true, or can it  be be shown to be demonstrably false?

Here are two citations attesting to the falsehood. Using a controlled environment: Who Ever Said No Two Snowflakes Were Alike? And from Guinness World Records: First identical snow crystals.

This article is found at Forbes: Ask Ethan: Could You Have Two Perfectly Identical Snowflakes? drawing these two particular attested falsehoods into question.

 

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A snowflake's form, is its identity as a type of (pun intended) frozen history.  

Starting from a similar nucleus, which in fact are often themselves different, each snowflake as it falls and swirls about encounters different temperatures, pressures, humidities, in different orders and  for different times at various magnitudes and combinations and with varying asymmetry or gradients versus homogeneities, and this happens as it makes it entire journey.  What it encounters directly affects and is frozen in what it becomes.  As such a single flake's identity in form corresponds to its identity in past history.  

To be sure, some snowflakes will be nondescript enough and have seen boring enough of a history such that many will look very similar (degenerate cases) but given how varied and chaotic each flakes history will normally have been (given standard chaotic and active weather systems) it is no surprise they will normally appear quite different from each other.

To my mind how a flake even appears is not as interesting as pondering how unique and different each ones journey must have been, even if we cannot see it in their faces, so to speak.

To be sure in actuality no two snowflakes are exactly the same.  The sheer number of molecules involved and the different historical permutations are simply staggering.  

Edit:  read the science article after my post... interesting stats!

Edited by StrictlyLogical

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20 hours ago, StrictlyLogical said:

To my mind how a flake even appears is not as interesting as pondering how unique and different each ones journey must have been, even if we cannot see it in their faces, so to speak.

To be sure in actuality no two snowflakes are exactly the same.  The sheer number of molecules involved and the different historical permutations are simply staggering.  

I hadn't thought of the snowflake's journey that way. Between influence a snowflake's unique journey has (different journeys, different results), and the lab produced snowflakes (same journey, essentially same result), I have what I need to be satisfied that the two propositions are indeed true.

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