Lets talk numbers
Posted 06 April 2006 - 10:38 AM
Any ideas on how many crystals might be in a cubic inch of space in a field after a snowfall? In a cubic foot?
Posted 28 December 2006 - 02:24 AM
But there is a nice way to make the estimate. The way is to consider the weight of the snow and compare this weight to the weight per snow crystal. A typical star-shaped crystal weights about one millionth of a gram.
So, you could just weight the snow, convert the weight to grams, and multiply by a million.
Or, if we are too lazy for that, consider how much volume the snow would occupy if you melted it. We know that one cubic centimeter of water is about one gram, so for each cubic centimeter of melt water, we'd have the remains of about a million crystals, assuming that they were once star-shaped.
A cubic inch of snow is about 16 cubic centimeters, but when snow melts, the volume decreases. Say it decreases to one-tenth the volume. Then, one cubic inch of snow melts into about one and a half grams of water, which is about one and a half million crystals.
A cubic foot is more than a thousand cubic inches. So, we are talking about 2 billion crystals (using the US meaning of billion). And that is a lot!
While we are talking numbers, consider the problem of checking to see if any two of the crystals look the same. If you have only two crystals, then you have just one pair to check. If you have three crystals, you have three pairs to check. This number goes up fast: 5 crystals, 10 pairs; 10 crystals, 45 pairs.... If you have n crystals, then you have to check n times (n-1) divided by 2 pairs. So, the million and a half crystals in a cubic inch would require checking about 1 followed by 12 zeros (a million million) pairs, which, if you could check one pair per second, day and night, 7 days a week, then you would have to keep checking for nearly 100 thousand years.
So, if you really like big numbers, consider the estimated number of snow crystals that have ever fallen on Earth. Charles Knight estimated this as the number formed by writing 1 followed by 35 zeros. Checking one pair per second, it would take about ... well, let't just say that the estimated age of the universe would seem completely insignificant in comparison.
Posted 28 December 2006 - 01:17 PM
And with those astronomical numbers of flakes that have fallen, (when was that esitmation and how much is added each year?) What is the probobility of finding a duplicate, if time was not a factor???
Posted 28 December 2006 - 07:28 PM
Tsuneya Takahashi and co-workers found out the weights by carefully growing a crystal, catching the crystal in an oily liquid, and then melting them. The crystals melted into small spherical droplets from which they could easily calculate the weight. But, unfortunately, I don't have their report here with me, so the one-tenth that I quoted above is based on my (often poor) memory.
The other important thing is the packing density. As we know from daily experience, smaller things often pack more densely than larger things. If the snow packs into solid ice, as it does in glaciers, then the density is close to that of liquid water. That is, nearly 10 times heavier than that in my previous post.
So, that cubic inch of snow, now a cubic inch of solid ice, originally made from smaller crystals, may be the remains of nearly 100 times as many crystals. We're talking about 150 million crystals.
As a result, instead of taking nearly 100 thousand years, we now have 100 x 100 = 10,000 times longer to sit their comparing crystal pairs. That is, um, about a billion years, which is getting close to the age of the Earth.
About the likelihood that any two snow crystals that have ever fallen on Earth look the same, well, as you can see from my estimate above, it would be a very safe bet to make! That is, it is possible to prove true (assuming we can agree on what "look the same" means) but essentially impossible to disprove because the time to actually check all of them is beyond us.
This answer is not very satisfactory though, as I am evading the question. OK, so we assume that the time to check is no factor. Well, I've looked into this question in a little more detail and found some scientific justification for saying 'yes', particularly if we include those smaller crystals. However, it turns out that a lot of basic knowledge about snow and clouds (from which they form) is just plain lacking. So, we probably won't be confident about a 'yes' or 'no' answer for some time to come.
Posted 18 July 2009 - 11:03 PM
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