# How many snowflake patterns are there?

Someone has commented on my previous post asking whether there is a limit to the number of patterns that can be formed in a snowflake. They have named themselves Jack Frost, although I question the veracity of this.

I always struggled with the idea that was promoted at school, suggesting that no two snowflakes are alike. No one ever offered a proof, so I questioned its truth.

If indeed the proposition is true, then the answer to Jack’s question is infinite. Assuming snow keeps falling, and that its rate of descent doesn’t slow over time, then the number of flakes that have ever existed will continue to rise, and will not tend to any limit. Under the assumption that each one is different, then the number of different snowflakes will continue unabated towards infinity (and beyond).

I prefer a somewhat simpler scenario. Let’s assume that snowflakes are a certain size, and within this volume, each space is filled either with ice or air, and this is what makes snowflakes so different from one another.

I’m imagining a three-dimensional version of a black-and-white favicon, a cube. For the sake of argument, let’s assume that it’s 16 "pixels" wide, high and deep. So in total, there are 4,096 positions, each of which can take a value of "ice" or "air". That would make about 10^1232 possible snowflakes (10 followed by 1,231 zeros).

Now this would be kerbed somewhat by the fact that I assume all ice particles need to be connected to one another, but even at such a small scale (16×16), the numbers are astronomical. Add in the fact that snowflakes can come in a variety of sizes, and that the 16 could be increased hugely, then there’s lots of possibilities.

There’s my attempt at solving the problem. Any advance? Anyone? Bueller? Anyone?

2 Responses to “How many snowflake patterns are there?”

1. elise on January 6th, 2007 11:07

If you like snowflakes, there’s a physicist who does amazing snowflake photographs…
http://www.its.caltech.edu/~atomic/snowcrystals/photos/photos.htm

His photographs are also showcased in a number of books (see website) and in a special series of holiday stamps released by the US Postal Service.

I really dug ’em.

2. uyt on October 22nd, 2007 11:08

how are snow flake patterns are formed?