#octonions — Public Fediverse posts
Live and recent posts from across the Fediverse tagged #octonions, aggregated by home.social.
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I think what the Cayley-Dickson construction is doing at least up to the #octonions is "teaching you" how to move stereographically (SGly) in a space you already know. this starts from R to C as you SGly move along the line R. but this can also be unprojected onto a circle, so alternatively you "learn" how to rotate in 2D *instead*, C. Next step up is rotation and SG motion in 2D, which can be unprojected to rotation in 3D, H. then rotation and SG motion in 3D and that's where it stops: O
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It's kinda lonely if one is interested in topics that few other people seem to care about. Need more friends who are into #octonions :)
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The #octonions are so damn fun. Still need to understand the Spin(8) triality better...
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Turns out that what the (aVa) ψ = a(V(a ψ)) moufang identity of the #octonions is really about is the translation between division-algebraic vector-reflection and clifford-algebraic vector-reflection (generated by chains of left-multiplication).
Similarly (aψ)(χa) = a(ψχ)a makes sure the decomposition of a vector into two spinors is compatible with their respective transformation behaviour (two transformed spinors on one side, the corresponding transformed vector on the other).
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TIL about an apparently notorious #OpenQuestion in #topology: is there a complex structure on the 6-sphere?
The Chern character rules out complex structures on 𝑆ⁿ for n>6. Apparently 𝑆⁴ doesn't even have an almost complex structure, although 𝑆⁶ does (coming as the unit sphere in the pure imaginary #octonions IIUC).
See https://doi.org/10.4153/CMB-1966-003-9 for some overview and refs. Learned from https://twitter.com/CihanPostsThms/status/1632549791036059648
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I have been engaging my fellows at Math 4 #Wisdom about working together. This has led to the idea of atomic learning exercises and how they fit within a map of #learning paths. I realized that for the sake of clarity I should distinguish what for me is not part of Math 4 Wisdom because it does not model cognitive frameworks for which I currently have evidence. So I made a map of Math for Fascination or Speculation which includes #octonions, #primes, exceptional Lie groups, Platonic solids.
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Octonions.jl v0.2.2 extends all complex analytic functions in the standard library to the #octonions!
Octonions are a type of hypercomplex number whose product is neither associative nor commutative. Besides their normal uses, they are also useful for testing numerical algorithms that are intended to generically work for even weird numbers.
Here we use them to check that the fallback QR and unpivoted LU decompositions in #JuliaLang do the right thing.
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In this talk I show how to multiply quaternions AND ALSO OCTONIONS using the dot product and cross product of vectors in 3 dimensions:
https://www.youtube.com/watch?v=JI5xPGN_sWo
For quaternions this is well-known, but for octonions it's less so.
I sketched the proof that both these algebras have a norm obeying
|ab| = |a| |b|
To see the details, go to the proof of Theorem 2 here:
https://golem.ph.utexas.edu/category/2020/07/octonions_and_the_standard_mod_1.html
I do the proof for octonions, but the same argument also works for quaternions!
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so it could turns out that the magic of the universe could be linked to an "Octarine" property of mathematics : Octonions looks like some fundamental key!
Cohl Furey : she is probably writing some history page here, so brilliant and "simple" (and not needing billions to explore) : https://www.quantamagazine.org/the-octonion-math-that-could-underpin-physics-20180720/
#physics #universe #42 #octonions
