Wednesday, March 14, 2012

No Two Alike

Another meandering post.  You've been warned.

I'm re-reading Isaac Asimov's informal autobiography, I. Asimov (a play on his collection of robot stories, entitled I, Robot, and to be distinguished from his formal autobiographies published earlier in his life), and finding it quite entertaining.  Partly, this is because I'm an inveterate re-reader and re-watcher.  My enjoyment of a piece of writing or a movie or a TV program doesn't diminish because I know how it goes.  If I enjoyed it the first time, I'll enjoy it just as much the seventh time, or the fifty-seventh.  Even a sporting event isn't diminished because I know how the final score (although I do prefer to watch it live the first time, if I can).  All this just by the way.

Anyway, in this book, Asimov mentions his facility at giving impromptu talks, and mentions by way of illustration that he has given a couple of thousand talks, no two exactly alike.

And that phrase, "no two exactly alike," is so characteristic of snowflakes that I immediately thought of them.  In fact, I'd go so far as to wager that if you asked people what the first thing was that they thought about snowflakes, it would be that no two are alike.

But is that actually so?  Have there really never been two snowflakes alike?  If you're like most people, you'd probably just as soon leave well enough alone and assume it's true.  For the heck of it, though, take a trip with me down the rabbit hole.

The whole idea that no two snowflakes are exactly alike has been around for time immemorial, but things really got moving with a man named Wilson Bentley (1865-1931), who grew up in Vermont.  When he was fifteen, his mother gave him an old microscope to experiment with.  Well, Vermont winters being what they were, I suppose it's natural that Wilson should have been drawn to snowflakes.  And so he took to maneuvering snowflakes under his microscope and sketching them.

It turned out, however, that they melted quickly—far too quickly for him to sketch in time.  So Bentley assembled a contrivance, a camera attached to a microscope attached to a board covered with black velvet, which permitted him to take pictures of the snowflakes before they melted.  Over his lifetime, he took images of over five thousand snowflakes, and sure enough, no two of them were exactly alike.

Five thousand, though a lot to take pictures of, is still a minuscule fraction of all the snowflakes that ever were, or even of those that are currently in existence (a constantly changing population, to be sure).  Surely there is no way that we could possibly take pictures of all the ones that currently exist, let alone those that have ever existed.  Is there perhaps another way to answering the question?

Consider: Each year, a substantial portion of the Earth is hit by snowstorms sufficient to dump several meters of snow on the ground.  I'm not sure of my statistics, but we probably wouldn't be far off if we assumed that the total annual snowfall amounted to a depth of, let's say, two tenths of a meter over the entire surface of the Earth, if it was spread around evenly.  Since the surface area of the Earth is about 5×10^14 square meters, we're talking about 10^14 cubic meters of snow.  When packed tightly (tightly enough to crush them), snowflakes might occupy a cube about a tenth of a millimeter on a side.  So each year, we get something like 10^26 snowflakes.  Taking into account the fact that there has been snowfall for a few billion years, there have been perhaps 10^36 snowflakes, ever, in the Earth's history.  That's a lot of snowflakes.

However, there are also lots of different shapes that any particular snowflake might take on.  Snowflakes exhibit six-fold symmetry because they're constructed from ice crystals, which have six-fold symmetry.  (You can find a picture of one in this article.)  So let's represent a snowflake as a hexagonal lattice, a bit like a honeycomb of cells, each of which might be occupied by an ice crystal, or not.  An individual hexagonal ice crystal is a few tenths of a nanometer across, whereas an entire snowflake might be a few tenths of a millimeter across.  So the hexagonal lattice representing our snowflake would have a diameter of about a million cells, and would contain about 750 billion cells in all.

Does that mean that there are nearly a trillion possible snowflakes?  No, because each one of those cells could either have an ice crystal, or not.  We could represent the snowflake by filling each one of those cells with a 1 if it had an ice crystal, or a 0 if it did not.  In other words, each snowflake would be represented by a huge binary number with 750 billion digits.  Such a tremendous number is on the order of 10 raised to the 230 billionth power.

It's hard to overstate how big a number this is.  Even if you were, somehow, to write a 100 digits a second, every second of every hour of every day, without interruption for sleep or eating, you have perhaps only an even-money chance of just writing this number out during your entire lifetime.  It goes without saying that it's much, much, much larger than 10^36.  (It is, however, much smaller than a googleplex.  I just thought I'd point that out.)

However, we're not playing quite fair, because we've completely neglected the symmetry exhibited by most snowflakes.  If we take that into account, it turns out that the number of possible snowflakes drop to something more like 10 raised to the 40 billionth power.  Quite a bit smaller, but still much larger than 10^36.

There's another thing, too.  Bentley took his photographs with an optical microscope, which was of course incapable of resolving ice crystals down to the individual molecular level.  These days, we're now capable of doing that, but it would be unfair to insist that snow crystals, which in an ordinary atmospheric environment would be constantly changing anyway, be identical to that level of precision.  A typical photograph of a snowflake might be able to resolve crystals to a level of detail that would take a hundred thousand cells to fill the entire snowflake.  Remembering to take into account the symmetry of snowflakes, there could still be on the order of 10^5,000 different snowflakes, at this reduced level of resolution.  Still much larger than 10^36.

OK, how about this?  If one looks at an array of Bentley's photographs, one notices that the ice crystals are not arranged haphazardly around the snowflakes, even after one takes into account the six-fold symmetry.  Instead, there is order at all different scales.  In fact, people have likened snowflakes to fractals; there are even simulations of snowflake generation that build upon the fractal arrangement.

That reduces the level of variation accessible to the snowflake.  It's hard to say for sure, but in most of the Bentley images, I think one can make out about six levels of detail.  (That's consistent with a scale ratio of about two to three.)  What's more, each unit of detail has within it detail that only goes about three or four levels down, which means that each level can be represented using about fifty bits or so.  That means a total of three hundred bits might suffice to denote a snowflake to the level of precision needed to figure out whether they match or not.  That would still mean about 10^90 distinct snowflakes, though.

All right, one last thing, which at first will seem to be a significant digression.  There is, in probability, something called the birthday paradox, which goes something like this: Suppose you get fifty otherwise randomly selected people together in a room.  What are the odds that at least one pair of them will share the same birthday (possibly different year)?  One in four?  One in ten?  How many people do you think you need to make the odds even?  Would forty do it?  How about sixty?  A hundred?

The answer, surprising to most people who haven't heard this question before, is that the odds are about even that out of just 23 people, at least one pair will share a birthday in common.  It's a bit surprising because there are 365 days in a year (not counting leap day), but consider what happens if you choose the people one by one.  The first, of course, can have any birthday at all.  In order to avoid a pair sharing the birthday, the second must not share a birthday with the first.  The third must avoid sharing a birthday with both the first and the second.  The fourth must avoid sharing a birthday with the first, the second, and the third.  And so on.  By the time you get to 23 people, there are about 250 birthday sharings that must be independently avoided.  It's not surprising that such sharings are avoided only half the time.

It turns out that this "paradox" (not truly a paradox at all, naturally, but just a counter-intuitive result of probability theory) has very wide applicability.  The number of samples that can be randomly selected before you stand a good chance of getting a pair is much smaller than the total number of choices.  In fact, it's on the same order as the square root of the number of choices.  (There's that square root again!)  The square root of 365 is a bit over 19, and sure enough, 23 isn't very far over 19.  If one takes into account the year of birth over the course of a century, then there are about 36,500 different birthdates, but the square root of 36,500 is only about 191, so that only about 200 randomly selected people are needed before you have a good chance of matching the entire birthdate.  And the square root of 10^90 is 10^45, so the size of the collection of snowflakes you need to have a good chance of pairing two of them is about 10^45.

It's more than 10^36, but not much more.  (What's a factor of a billion between friends?)  And there are a lot of back-of-the-envelope manipulations in what I wrote, so perhaps there are other deeper symmetries to take advantage of.  But I think it's rather magical that the numbers work out nicely so that it's quite possible that somewhere, across the vast mists of time, there were at (probably very different) points, two identical snowflakes!