"No two snowflakes are alike." You've undoubtedly heard that a time or two. Seemingly, not an important math application. I got to thinking about it and that statement does bring up some important points.
First it brings up some lessons in basic logic. If we are trying to prove there are not two snowflakes alike, how would be prove or disprove a statement like this? Disproving it could be easy. Find two that are the same. If we could do this, we could put this issue to rest.
How about proving it to be true. We've all seen lots of pictures of snowflakes. None alike so far. The pioneer in this seems to be a Wilson Bentley from Vermont. He was born in 1865, a time when there weren't a lot of pictures being taken of anything. He had a collection of over 5,000 photographs of snowflakes. He was single (not a surprise) and had plenty of time to devote to his work. None of his matched. The fact that none of them match would not constitute a proof. This would be a good example of inductive reasoning. Here is an opportunity to discuss inductive verses deductive reasoning, and the benefits and drawbacks of each.
Can deductive reasoning be used here? Let me state that I'm well out of my area here. My lowest grade in high school was a C and that was in Chemistry. I was fine with that since I probably deserved lower. I really did try. Chemistry and I just do not click. Regardless, here we go.
I did some reading to try to figure this out. A water droplet might freeze onto a dust particle. They freeze in a hexagon shape. Most of what I saw kind of glossed over why that is. One statement explaining the snowflake pattern went like this. And I quote:
Hexagonal ice ([1969], ice Ih i see Phase Diagram), is in Space group P63/mmc, 194; symmetry D6h, Laue class symmetry 6/mmm; analogous to β-tridymite silica or lonsdaleite, having a a six fold screw axis (rotation around an axis in addition to a translation along the axis).
Curse you Chemistry. Anyway a hexagon is formed. Other water molecules latch onto the vertices of the hexagon, growing the snowflake as it falls through the air. Different shapes come about based on the temperature and humidity of the surrounding air. These flakes all take different paths to the ground, thus slightly changing its weather conditions, thus slightly changing the shapes as they grow. These different paths cause different shapes. At any one moment in time, the forming shape has the same weather conditions, giving the snowflake its symmetry.
One article states that there are 10,000,000,000,000,000,000 molecules of water in a snow flake and they can be rearranged in many different ways. A couple of articles likened this to factorials. The ways to arranged 6 books is 6! (= 720), 7 books is 7! (5,040), and 8 books is 8! is (40,320). I'm not sure finding the number of snowflake designs is as simple taking the factorial of the number of molecules, but I guess their point is that the number of patterns must be huge.
However, I'm still not convinced. Someone estimated there have been approximately 1,000,000,000,000,000,000,000,000,000,000,000,000 snowflakes. Really? No two alike in that bunch. We haven't looked at them all. Even if we did, even with global warming, I'm sure there will be a bunch more.
In fact, some of the scientists think there might have been duplicates. If the snowflake doesn't have far to fall, and thus doesn't have the chance do grow very much, that greatly increases the chance that two of them could be similar.
Actually, one scientist claimed she has found a pair. In 1998, Nancy Knight claims she found two alike. I saw a picture and they look pretty convincing. Nerdy scientists, though. have balked at this. Some of the hydrogen atoms (approximately 1 in 3,000) could be deuterium. (Hydrogen usually has just one proton in the nucleus. Deuterium has a proton and a neutron.) This would likely make snowflakes that looked the same, still not be identical.
Some people just can't admit defeat.
I hope this was helpful. I did my best to be as accurate and thorough as I could. (I'm sorry Mr. Gustafson. I really did try in Chemistry.)
