Snoflakes

During my Mont-Royal camping adventure with Nerdlord and 20hunert, the last two questionned whether no two snowflakes could be alike.

Here is an answer:

''It's a funny question, almost like a Zen koan -- if two identical snowflakes fell, my inquisitive friend, who would know? And can you ever be sure that no two are alike, since you cannot check them all to find out?

Although there is indeed a certain level of unknowability to the question of snowflake alikeness, as a physicist I find that I can address this issue with some confidence. As I will demonstrate, the answer depends to a large degree on what you mean by the question. (Yes, physics does occasionally have its Zen-like qualities.)

The short answer to the question is yes -- it is indeed extremely unlikely that two complex snowflakes will look exactly alike. It's so extremely unlikely, in fact, that even if you looked at every one ever made you would not find any exact duplicates.

The long answer is a bit more involved -- it depends on just what you mean by "alike," and on just what you mean by "snowflake." Let's look at the possibilities....''

Let's not. You can check out the complete answer here.




''about one molecule out of every 5000 naturally occurring water molecules will contain an atom of deuterium in place of one of the hydrogens, and about one in 500 will contain an atom of 18O instead of the more common 16O. These rogues are not exactly the same as their more common cousins.

Since a typical small snow crystal might contain 1018 water molecules, we see that about 1015 of these molecules will be different from the rest. These unusual molecules will be randomly scattered throughout the snow crystal, giving it a unique design. The probability that two snow crystals would have exactly the same layout of these molecules is very, very, very small. Even with 1024 crystals per year, the odds of it happening within the lifetime of the Universe is indistinguishable from zero.''


...





''The number of possible ways of making a complex snowflake is staggeringly large. To see just how much so, consider a simpler question -- how many ways can you arrange 15 books on your bookshelf? Well, there's 15 choices for the first book, 14 for the second, 13 for the third, etc. Multiply it out and there are over a trillion ways to arrange just 15 books. With a hundred books, the number of possible arrangements goes up to just under 10158 (that's a 1 followed by 158 zeros). That number is about 1070 times larger than the total number of atoms in the entire universe!

Now when you look at a complex snow crystal, you can often pick out a hundred separate features if you look closely. Since all those features could have grown differently, or ended up in slightly different places, the math is similar to that with the books. Thus the number of ways to make a complex snow crystal is absolutely huge.

And thus it's unlikely that any two complex snow crystals, out of all those made over the entire history of the planet, have ever looked completely alike.''

There.