weaselman
weaselman
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December 8th, 2011 at 8:00:21 AM permalink
Quote: FrGamble

Seriously I guess I should say thanks, did I tell you about the time a student was late to class because he said he licked a flagpole in the frezzing cold and his tounge stuck, oh well nevermind I'm sure that is not true either... :)


Priceless! Thank you, father!
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Face
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Face
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December 12th, 2011 at 9:03:28 PM permalink
TOPIC: "Random Snowflakes"

It's that time of the year when you're bound to hear some inspirational message conveyed with the point that "No two snowflakes are ever the same". Me being me, I had to challenge the idea and as a result now believe this cliche to be false.

Snowflakes can be very varied indeed. Sometimes, on a very cold, dry night, they come down almost as diamond dust. I have seen some I would judge to be fractions of a millimeter, yet big enough to see the detail with naked eye (these are the ones I find to look like the stereotypical representation we see in decorations, very symetrical and very geometric). I believe the record for largest fallen snowflake is fifteen inches, although I would estimate a "typical" big flake as two to three.

Now, snowflakes are just ice, which forms as a crystalline structure. That right there slashes the possible configurations of the water molecules by a great number as they can't join randomly or haphazardly. And in the history of Earth as we know it, I'd put the number of flakes fallen somewhere over the number 10 followed by, I dunno, one hundred thousand zeroes? (I'd be interested if anyone could ball park it better than a top-'o-the-head guess, although I assume it's probably impossible)

Doesn't it stand to reason that they've been duplicated possibly hundreds if not thousands of times? I haven't done any book work yet, just thought experiment, but I think if I can find the proper numbers and do a little crunching, I can prove mirror snowflakes are not the unicorns people believe them to be. Rather, I think they could be seen in a lifetime, maybe even within a decade or year.

But before I do I thought I'd open it to discussion, see what all the thinkers here feel on the matter. Random snowflakes - any truth to it whatsoever?
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weaselman
weaselman
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December 13th, 2011 at 5:41:49 AM permalink
I'll start with the first part for now:
Quote: Face

And in the history of Earth as we know it, I'd put the number of flakes fallen somewhere over the number 10 followed by, I dunno, one hundred thousand zeroes? (I'd be interested if anyone could ball park it better than a top-'o-the-head guess, although I assume it's probably impossible)



It is actually much less than that. The hardest part is to estimate how much snow actually falls on Earth every year. The record annual snowfall in US seems to be just under 1000 inches or 25.4 meters. Granted, some areas (South Pole?), probably, get much more snow, but others don't get any at all, and there is more of the latter. There were ice ages, but there were also some very hot periods (especially, towards the beginning of the history). Besides, this is the record number, not an average, so, I think, it is appropriate to be used as high estimate. You'll see in the end that it does not matter very much for the qualitative answer if I happen to underestimate it by a few orders of magnitude.

Now, the surface area of the Earth is about 500 million square kilometers, or 5*10^14 square meters. Multiply this by 24.5 to get 1.3*10^16 cubic meters of snow per year.
The age of the Earth is about 5 billion years, so 6.5*10^25 cubic meters total. Snow is about 12 times less heavy than water. Let's say 10 times for simplicity, which makes one cubic meter to be about 100 kilograms. This gets us to about 6.5*10^27 kilogram of snow since the beginning of time.

Now, a snowflake contains about 10^19 molecules, and a single water molecules weighs about (16+2)/6.022*10^23 ~ 3*10^-23, which makes one snowflake to be about 3*10^-4 grams, or 3*10^-7 kilograms.

Bringing it all together, we get about 2.2*10^34 snowflakes since the beginning of time. Only 34 zeroes, give or take, not at all close to thousands (1, followed by just a hunderd zeroes is an enormous number, greater than the number of atoms in the whole observable universe, way larger, than its age in nanoseconds or size in Planck lengths etc.).

Unfortunately, I don't know crystal science well enough to speculate on how many different configurations can 10^19 molecules form, but something tells me it's a lot :) I would guess, while not entirely impossible, it is extremely unlikely the a pattern ever repeats itself exactly. However, if you asked if two snowflakes are ever "alike", where "alike" is defined as not distinguishable with a naked eye, that would be a different story. I would think, that the later statement is, probably, false.
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Face
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Face
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December 13th, 2011 at 12:00:46 PM permalink
Jumpin Jehoshaphats! I feel gorged on information.

Thanks for bringing things into perspective, weasel. I'm rubbish at conceptualizing the very large, and this "very rough estimate" is quite what I was looking for.

I, too, am ignorant of any sort of crystal science, and using 10^19 units to make anything, even if it was simple triangles, would leave you with a ridiculously high number of possible combinations. That sucks =( lol

Because yeah, my original thought was pertaining to exact replicas down to the molecular base. I've often looked at those really small ones I mentioned in my OP, and given enough time, I'm sure I could find 2 that looked alike. I wanted to know the possibility that they were identical. It seems I have severely underestimated the smallness of water molecules.

From what I've read, ice crystals typically form in hexagons. Wouldn't 10^19 water molecules / 6 = "ice units"? And taking a leap of faith and assuming snowflakes are symetrical, couldn't we then take that number and divide it in half, and just figure how many different combinations of these remaining units there are? (since what changes on one side would have to change on another) Or did I just bastardize the entire concept of math, science and logic? ><

If I'm in any way correct, then it seems that I'd have to figure out the possible configurations of [(10^19 / 6) / 2] "ice units", then figure the probability of two of these configurations repeating within 2.2*10^34 times.

In the words of Jeremy Clarkson, "How hard could it be?" ;)
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P90
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December 13th, 2011 at 12:41:51 PM permalink
Remember what you said about numbers with 100,000 zeroes?

If you have just 10,000 elements, there will be something on the order of 10^35000 possible combinations.
For 10^19 elements, there would be something about 1.20247678*10^168091349662870733545 or over 10^^21 possible combinations. That's a number with a sextillion digits. If you gathered every single data storage device in the world, it would only just be enough to simply store that number in decimal notation, and it would take a few to send that number between the Old World and the New World using all communications methods combined.

These combinations hold for different elements, since molecules involved are the same, the number is much smaller. But "much" is still an order of magnitude somewhere around 10^1,000,000,000,000. So in absence of very strict formal structure the probability of identical crystals is zero.
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Nareed
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December 13th, 2011 at 1:31:17 PM permalink
I read somewhere that a Googleplex (10^10^100) is too big a number to actually measure anything in the universe.

Is this so? I can comprehend a Google (10^100), barely. But that's where I draw the line :)
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weaselman
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December 13th, 2011 at 2:23:04 PM permalink
It's actually spelled "googol" and "googolplex", nothing to do with google.com :)
Googleplex is actually what they call Google.com's HQ :)

As to what you can measure with it, yeah ... like I said earlier, it is hard to come close to even 10^100 (the googol) enumerating various "things" in the universe (there are far fewer atoms in the universe for example), leave alone 10^10^100.
There are some extravagant examples that could still invoke googolplex. Here is one.
If you pack the whole universe solid with dust, and number every particle, so that they are all different, the number of different ways to arrange those particles would be somewhere close (as in within a few googols, probably :)) to one googolplex.
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Garnabby
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December 13th, 2011 at 4:11:19 PM permalink
Quote: Face

"No two snowflakes are ever the same". Me being me, I had to challenge the idea and as a result now believe this cliche to be false.


Were some two, or more, things exactly the same, wouldn't those also occupy the same, or a common, space and time? Ie, as one entity, or one which has many such vantage points? As opposed to that old lament of the ancients, "You can never swim in the same river (again)."

Another thing that comes up in such discussions is the enantiomorphism paradox.
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odiousgambit
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January 16th, 2012 at 1:33:47 PM permalink
So you want to be a Scientist, eh. Well, speaking for myself anyway, prepared to be humbled if you really study it.

Something unexplained to my satisfaction anyway is the way sunrise and sunset do not change evenly this time of year. We just came away from the shortest day late in December, so the days are indeed getting longer. But not in a balanced way: the sunrise actually gets later in the morning, then is the very same time of the morning way into January. The effect of longer days comes purely from the evening sunset being later, and there was no earlier sunrise until just yesterday or so [around here]. [edited]

I have puzzled over this for a long time and have come across this, which clearly leads to some pretty damn complicated stuff. It still baffles me that the facts 'Earth 's tilt is about 23.5 degrees' and "Earth's orbit isnot a perfect circle, but is instead an ellipse' causes mornings to be shorter than evenings, yet we do not have to keep resetting our clocks. Instead sunrise and sunset change in an unbalanced way.

Perhaps someone smarter has some insight?
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Doc
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January 16th, 2012 at 6:05:56 PM permalink
I am not an astronomer at all, though I think the Wizard has some moderately-serious interests in this area, so perhaps he should be speaking up here. I have never looked into this topic enough to be able to explain the answer, but I think I can give a hint as to why such a unbalanced effect should not be too surprising.

Suppose you are looking down perpendicularly upon the earth's orbit from very high above and visualize it as a definite ellipse (exaggerated from what it really is) with the sun at one of the foci. The earth's rotational axis is tilted, so from your perspective it forms a line rather than a single point. Think about the direction of that rotational axis line and about the major axis of the orbital ellipse, then consider the point in time when the earth passes one end of the major axis.

At that point in time, the rotational axis line, even if extended, does not pass through the sun; i.e., the earth's rotational axis is not aligned with it's orbital axis. Although the earth is nearest to (or farthest from) the sun when it crosses the orbital major axis, that date is not the winter or summer solstice, which is the date that the rotational axis (extended) would appear, from your "over-orbit" perspective, to pass through the sun. I believe this lack of axis/orbit symmetry is what gives rise to the lack of symmetry in the sunrise/sunset pattern.

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