i remember reading at some point about how the cross product can only exist in three or seven dimensions and i want to understand why that is
anyone here know much about esoteric vector math?
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@Felthry *blinks* weird, we're curious too now!
@IceWolf (of course you can define an exterior product on a vector space of any dimension but the cross product is special)
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@IceWolf (special in that it results in a pseudovector of the same dimension as the original vectors, that is)
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@IceWolf (math gets really weird in the weirdest ways. you don't even have to go to like banach-tarski or anything, just look into vector math in higher dimensions, or affine geometry, or anything at all to do with fractional calculus)
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@Felthry Not sure if this will explain it clearly enough but https://arxiv.org/abs/math/0204357 goes into what you need for a cross product, and how 3 and 7 dimensions wrap things up. And yeah, it does tie in to octonions being as far as you can really go for analogues of complex numbers in more dimensions.
(There's sedenions, 16-dimensional, but they're not division algebras and are only partly multiplication-associative.)
i feel like this has something to do with the fact that there aren't any analogues of complex numbers in any dimensions other than four, eight, and (maybe) sixteen, maybe?
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