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Pedantic nitpick, but I might as well bring it up first so we can have it out of the way: “There’s also an uncountable infinity of other patterns that all give
by evincarofautumn 15y ago
Pedantic nitpick, but I might as well bring it up first so we can have it out of the way:
“There’s also an uncountable infinity of other patterns that all give you equally dense packings.”
According to the author’s examples using only {a, b, c}, the set of possible packings would be countably infinite.
Unfortunately, the article has kind of a breathless tone because it tries to cover quite a bit of ground all at once. Perhaps it’d’ve been better broken up, but it was still an interesting read. I’d never really considered that there must be different crystal structures to ice, though it seems perfectly obvious once explained.
- robinhouston 15y agoI think the idea is roughly this: each packing pattern is represented by an infinite string over the alphabet {a,b,c} that has no doubled letters. And there is an uncountable infinity of such strings – continuum many – because these strings are in one-one correspondence with infinite binary strings: there are two possibilities for each letter, because it can’t be the same as its predecessor but it can be either of the other two.
- evincarofautumn 15y agoI considered that, then I figured it was reasonable to discount infinitely large pieces of ice.
- leif 15y agoIn that case the number of ice configurations would be finite, not countably infinite. EDIT: lest someone believe me, this is NOT TRUE, the parent is completely right