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Wow, mind blown. Unfortunately, even with an undergraduate degree in mathematics, I was lost after this paragraph: `This made me eager to find a proof that all
by max_likelihood 8y ago
Wow, mind blown. Unfortunately, even with an undergraduate degree in mathematics, I was lost after this paragraph:
`This made me eager to find a proof that all the even moments of the probability distribution of distances between points on the unit sphere in \mathbb{R}^d are integers when \mathbb{R}^d is an associatve normed division algebra.`
Nonetheless, very interesting!
- ggggtez 8y agoThe point of an undergraduate degree isn't to teach you everything in mathematics. It's to give you the tools to figure it out.
- jhanschoo 8y agoI'm pretty sure that by > \mathbb{R}^d is an associatve normed division algebra the author is referring simply to a generalization of the euclidean structure.
- jvkersch 8y agoNot quite -- the comment refers to the fact that R^d has the structure of a normed division algebra in dimensions 1, 2, 4, and 8. This means that you can multiply things together in a nice way when you're in one of those spaces. For R^1, this is just multiplying real numbers, for R^2 it's multiplying complex numbers, R^4 is quaternions, and R^8 is octonions. As you go up in dimensionality, you lose more and more nice properties: the quaternions are not commutative and the octonions are also not associative (which is why there's no mention of them in the blog post). The point is that in dimension 1, 2, and 4 all sorts of interesting things happen. John Baez has a paper about this: http://math.ucr.edu/home/baez/octonions/node1.html http://math.ucr.edu/home/baez/octonions/node1.html
- mmmmmmmike 8y agoSort of. A Euclidean structure has “addition” but not “multiplication”. Having an additional operation that’s required to be associative, have inverses, distribute over addition, and play nicely with the norm turns out to be such a constraint that, as he mentions in the previous paragraph, the real numbers, complex numbers, and quaternions are the only such structures. Literally speaking then, “... when R^d is an associative normed division algebra” just means “when d = 1, 2, or 4”, except of course that the idea is to use the multiplicative structure in the proof.
- crasshopper 8y agoread his post on octonions http://math.ucr.edu/home/baez/octonions/ http://math.ucr.edu/home/baez/octonions/ The 1, 2, 4, 8 phenomenon surprised 20th-century mathematicians, and derives from weird facts about how S7, S3, and S1 fiber. (fibration is lining one shape with other shapes)