Mathematics

I think the claim that "no two snowflakes are alike" is fairly common. The idea is that there are so many possible configurations of snow crystals that it is improbable that any two flakes sitting next to each other would have the same configuration. I wanted to know: Is there any truth to this claim?

Kenneth Libbrecht, a physics professor at Caltech, thinks so, and makes the following argument:

Now when you look at a complex snow crystal, you can often pick out a hundred separate features if you look closely.

He goes on to explain that there are $10^{158}$ different configurations of those features. That's a 1 followed by 158 zeros, which is about $10^{70}$ times larger than the total number of atoms in the universe. Dr. Libbrecht concludes

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.

Being the skeptic that I am, I decided to rigorously investigate the true probability of two snowflakes having possessed the same configuration over the entire history of the Earth. Read on to find out.

Is it true that everyone on earth is separated by at most six degrees? There's plenty of empirical evidence to support this claim already, so I am going to take a different, more theoretical approach.

Read on to see my results.

I am subscribed to David Horvitz's new project entitled IDEA SUBSCRIPTION in which he posts almost-daily simple instructions. Yesterday's instructions read as follows:

CALCULATE HOW FAR THE HORIZON IS FROM YOU. THIS CAN BE DONE BY THE FOLLOWING: DETERMINE THE HEIGHT OF YOUR EYES FROM THE GROUND (IN FEET). MULTIPLY THIS BY 1.5. FIND THE SQUARE ROOT OF THIS NUMBER. THE FINAL NUMBER IS THE DISTANCE IN MILES TO THE HORIZON SPECIFICALLY FROM YOU. NOW WALK THIS EXACT DISTANCE. AS YOU DO THIS THINK ABOUT HOW YOU ARE WALKING THE ENTIRE RANGE OF WHAT IS VISIBLE FROM WHERE YOU STARTED. AT THE END TAKE ONE LAST STEP AND ENTER OVER THE HORIZON, INTO DISAPPEARANCE. NOW WALK BACK.

I do not profess to have spent much time researching this in the past, but I had never heard of this approximation before. The approximation is so concise that I was curious as to its error. The approximation is obviously incorrect for very tall heights since it is unbounded:

I therefore spent the last 5 minutes formalizing a bound on the error of this approximation. The results, which follow, were quite surprising.

This evening I finally got around to doing some forensic data recovery from a broken (i.e., horribly clicking) hard drive. Most of the data I had backed up, but there are a couple non-vital files for which it would be nice to recover. That and I've never done something like this before and it's quite fun. It's especially fun that the partition I'd like to recover was formatted in ReiserFS, for which no free and few commercial recovery tools exist.