How do they know that no two snowflakes are alike? They can't really check, can they?
The anecdotal evidence offered below shows there's a lot of types at least, prompting a researcher to state there's infinite types and (even more boldly) no two are the same. Knight only proved that there may be subclasses of simple flakes that are likely to have more members (like the plain hexagonal). The poster may be right though, apart from picking at differences at the molecular level, there may even be identical snowflakes at the crystalline level.
It's like saying no two people have the same fingerprints. You can't prove it, but you can show how many variations there can theoretically be. Then you can make an educated guess about how unlikely it is for there to be two or more identical ones.
It's theoretically impossible to prove no two snowflakes are the same. Even if you did have access to all the snowflakes on the planet, there would still be H2O-snowflakes on other planets, in space, in the past, in the future, etc. And besides, it's pointless to prove such a thing.
In fact, it's not what people mean when they say "no two snowflakes are the same". What they're saying is: "the number of possible variations of snowflakes is nearly limitless and most (not all, cue the hexagon) variations seem to have an equal likelihood to occur". But that just doesn't sound like something you'd be telling your kids in the snow :).
