There are six sides to a snowflake. Snowflakes are complicated. Snowflakes have intricate designs that can be seen with microscopes. That much we know. But then there is the assertion: no two snowflakes are exactly alike. And how do we know that? Everyone seems to know it. It’s just common knowledge. It gets used as an analogy. “Just like no two snowflakes are alike, neither are any two people.”
But nobody has ever, ever proved that no two snowflakes are alike. How could you? Well, turns out you can, if you are willing to work in the realm of probability. Which, by the way, is the realm in which modern physics works anyway. As a sidenote, did you know that there’s a nonzero chance that your entire body will suddenly disappear, and reappear on the top of the Taj Mahal? Nonzero, but unbelievably small. Like so small that it would (almost) never happen even if you lived for a hundred billion billion billion billion billion billion years. But it still, technically, could happen. And on the scale of microscopic particles, those probabilities become far more important. The chance that one (or six?) of the particles in your body will suddenly “tunnel” to India is not nearly as small.
Back to snowflakes. It turns out that there are approximately 10^18 water molecules in a snowflake. (That’s 1,000,000,000,000,000,000, or one quintillion.) While each electron, proton, and neutron within those water molecules are identical, the molecules themselves are not. Water molecules are made of two hydrogen atoms and one oxygen atom. About one in every 3000 water molecules actually contains a variant of hydrogen called deuterium (which has an extra neutron). So, there are approximately 3.33 x 10^14 deuterium-water molecules roaming around in there. Working out the probability of where those three hundred trillion molecules would be within one quintillion makes my brain hurt. It’s somewhere on the order of HOLY CRAP THERE ARE NO NUMBERS THIS HIGH. And that’s just the beginning.
That deuterium calculation only needs to be employed if you have two snowflakes that already look identical under a microscope, which is to say, a condition where all the molecules have arranged themselves identically. And there are so many ways that those molecules can do that. A core of six water molecules sits at the center, but the seventh molecule can connect in several different ways. Let’s just say for the sake of argument that there are three ways this seventh one can connect to one of the original six. That means there are a total of 18 possible places for the seventh molecule. The eighth one now has three ways to connect to seven molecules. That means 21 possibilities. Add them all together, up to one quintillion, and you find there are 2.6 x 10^3438 possibilities for where all the molecules might show up. (Don’t take my word for it — I completely made that number up. These probabilities are way beyond my ability.)
Now, there are approximately 10^24 (1,000,000,000,000,000,000,000,000 or one septillion) snowflakes that fall to earth each year. The earth is about four billion years old, so that means somewhere around 4 x 10^33 (four nonillion) or so snowflakes that have fallen since the dawn of time. Taking that number, and plowing it into the other numbers above gives the odds of two snowflakes being identical as:
1 in 314,343,699,450,453,234,665,435,233,111,342,658,537,309,314,245,438,365,298,717,451,894,845,616,848,945,416,519,719,648,674,161,749,846,849,064,008,746,516,400,198,474,894,848,949,084,654,984,987,984,894,564,987,898,946,565,000,848,481,651,688,646,876,456,511,184,654,840,897,841,654
That’s a very small chance. In layman’s terms, it’s roughly the same chance as
- being struck by lightning every day at 3:07 pm for forty-six years straight,
- winning every lottery in every state in the USA on the same day, despite not buying a ticket for any of them,
- finding the exhumed remains of President William Henry Harrison in your backyard, intermingled with the fossilized remains of a Tyrannosaurus Rex,
- your entire town “quantum tunneling” on top of the Taj Mahal, twice,
- waking up one morning to find that you are not only a cockroach, but also the star of the next Star Wars movie, as well as the true heir to the throne of Atlantis, or
- Donald Trump saying, “I’m sorry, I was wrong about that.”
Oh, wait. I forgot that snowflakes are always six-sided, so they have rotational symmetry. Divide all the numbers here by 6. Then they’ll be accurate. As you can see, every snowflake is unique. Just like I learned at camp: “I’m unique, just like everybody else.”