A Straight Dope Classic from Cecil's Storehouse of Human Knowledge

How can they be sure no two snowflakes are alike?

February 5, 1988

Dear Cecil:

How do they know with any degree of certainty that no two snowflakes are alike? When I took statistics I was taught that to draw a valid conclusion one had to take a representative sample of the entire population. But considering the impossibly large number of flakes in a single snowfall, let alone that have ever fallen, how could snowologists have possibly taken a sample large enough to conclude that no two are alike?

Cecil replies:

They didn't, of course. Chances are, in fact, that there are lots of duplicates. What the snowologists really mean is that your chance of finding duplicates is virtually zero. It's been calculated that in a volume of snow two feet square by ten inches deep there are roughly one million flakes. Multiply that by the millions of square miles that are covered by snow each year (nearly one fourth of the earth's land surface), and then multiply that by the billions of winters that have occurred since the dawn of time, and it's obvious we're talking unimaginable googols of flakes. Some of these are surely repeats.

On the other hand, a single snow crystal contains perhaps 100 million molecules, which can be arranged in a gigajillion different ways. By contrast, the number of flakes that have ever been photographed in the history of snow research amounts to a few tens of thousands. So it seems pretty safe to say nobody's ever going to get documentary evidence of duplication. Still, it could happen, and what's more, Leslie, it could happen to you. The way I figure, anybody who could dream up a question like this has got to have a lot of time on his hands. Get out and start looking.

We encounter a little problem

Dear Cecil:

Considering some of the dope you dish out, I'd expect your mistakes to be equally spectacular, and you've certainly outdone yourself this time. The mind (mine, anyway) boggles at the magnitude of error in your recent dissertation on snowflakes in which you said that over the history of the earth there have been "unimaginable googols of flakes." A googol is one followed by 100 zeroes (10^100). My calculations show that since the earth was formed four billion years ago, the estimated number of flakes (not counting you and me and your other readers) is only about 10^28. That leaves a difference of 10^72.

Let's try to get a handle on the size of that error. The difference between the diameter of a carbon atom's nucleus and the diameter of the known universe is about 40 orders of magnitude. That still leaves about 32 orders of magnitude to sweep under the rug, or about the difference between a carbon atom and the Milky Way. To put it another way, the number of protons, neutrons and electrons in the known universe is much less than one googol. You've exceeded that by a margin of unimaginable to the unimaginable power. I knew you could do it, Cecil. Congratulations.

Cecil replies:

I knew some smartass was going to call me on this. I am well aware that the number of snowflakes falls short of a googol by a considerable margin. However, swept up in a fit of literary grandiosity —  I mean, come on, how often do you get to use a word like "googol" in a sentence? — I decided to fudge it. I'm so embarrassed. Incidentally, by my calculations, the number of flakes is actually about 1030, a difference of 102 from your figure. (You goofed up the multiplication for the number of square feet in a square mile, judging from your work sheet.)

Another little problem

Dear Cecil:

I am a senior electrical engineering student at Northwestern University. Regarding the number of snowflakes that have fallen since the dawn of time, I have no problem with Josef Prall's point that there have been 10^28 to 10^30, as opposed to your estimate of a googol (10^100). However, I feel compelled to point out that the difference between the two amounts is not 10^72. Obviously neither Prall nor you learned manipulation of exponents correctly in high school. 10^3 (or 1,000) minus 10^2 (100) doesn't equal 10^1 (10), it equals 9 times 10^2, or 900. Likewise, 10^100 minus 10^28 isn't 10^72, it's 10^28(10^72 - 1) or 10^28(10^71 x 9.999 ...) or 9.999 ... x 10^99. Get it straight.

Cecil replies:

I hate senior electrical engineering students. Whatever his many other moral failings, I think it is reasonably clear from his letter that Josef Prall knows 10^100 minus 10^28 doesn't equal 10^72. He was using  — certainly I was using  — the expression "a difference of 10so-and-so" as a shorthand way of saying "a difference of so-and-so orders of magnitude." This may seem a bit careless, but in today's fast-paced world, every microsecond counts.

Vindication!

Some months ago, Straight Dope fiends will recall, this column struck a mighty blow for truth and freedom by attacking the belief that no two snowflakes are alike, a superstition that has blighted the lives of millions. Not having time to inspect all the world's snowflakes (besides, I lost the tweezers), Cecil relied instead on the crushing logic of mathematics, arguing that so many flakes had fallen since the dawn of time that there were bound to be a few duplicates.

Naturally, many scoffed. One peanut-brain called to say he knew for sure no two snowflakes were alike because he had heard it on Nova. There was also the unfortunate business with the googols, which we won't go into here. My defense in all cases was couched strictly in theoretical terms, since I did not expect any actual cases of twin flakes to turn up (although I must say the cast of characters in those Doublemint commercials certainly came close).

I was therefore pleasantly surprised to read in the bulletin of the American Meteorological Society that matching snow crystals were recently discovered by Nancy Knight of the National Center for Atmospheric Research. The crystals in question admittedly aren't flakes in the usual sense but rather hollow hexagonal prisms. They are also not absolutely identical, but come on, if you insist on getting down to the molecular level, nothing's identical. They're close enough for me. Just shows you, not only is this column at the cutting edge of science, sometimes we have to wait for the cutting edge to catch up.

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