The point of an undergraduate degree isn't to teach you everything in mathematics. It's to give you the tools to figure it out.
Re:Random Points on a Sphere
by crasshopper on Wednesday July 11, 2018 @06:48AM
(#17505890)
read his post on 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)
> read his post on octonions http://math.ucr.edu/home/baez/octonions/
Not for a discussion on associative normed algebras ….
Aren't there other strange facts about dimensions 1, 3 and 7, particularly in linear algebra?
I'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.
Not 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
Re:Random Points on a Sphere
by mmmmmmmike on Wednesday July 11, 2018 @05:58AM
(#17505677)
Sort 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.
`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!