[HN Gopher] Quaternion Differentiation (2012)
       ___________________________________________________________________
        
       Quaternion Differentiation (2012)
        
       Author : niborgen
       Score  : 58 points
       Date   : 2024-06-28 12:22 UTC (10 hours ago)
        
 (HTM) web link (fgiesen.wordpress.com)
 (TXT) w3m dump (fgiesen.wordpress.com)
        
       | ykonstant wrote:
       | The general theory of this differentiation process is given here:
       | https://en.wikipedia.org/wiki/Exponential_map_(Lie_theory)
        
       | bbor wrote:
       | A) This is one of the best pieces of math pedagogy I've seen in
       | ages. A true "explanation" instead of a "reference" -- thanks for
       | sharing.
       | 
       | B) I think we should pass a hacker news law via referendum
       | (that's a thing, right? It should be!) that any article
       | mentioning quaternions must also generalize to Octonions, or at
       | least gesture in that vague direction to pay respect. As the
       | first paper I found on Octonion Differentiation says best:
       | Each Cayley-Dickson algebra Ar+1 is obtained from the preceding
       | Ar with the help of the so called doubling procedure [1, 14, 17].
       | This gives the family of embedded algebras: Ar -- Ar+1 -- ....
       | For a unification of notation it is convenient to put: A0 = R for
       | the real field, A1 = C for the complex field, A2 = H denotes the
       | quaternion skew field, A3 = O is the octonion algebra, A4 denotes
       | the sedenion algebra.             The quaternion skew field is
       | associative, but non-commutative. The octonion algebra is the
       | alternative division algebra with the multiplicative norm. The
       | sedenion algebra and Cayley-Dickson algebras of higher order r >=
       | 4 are not division algebras and have not any non-trivial
       | multiplicative norm. Each equation of the form ax = b with non-
       | zero octonion a and any octonion b can be resolved in the
       | octonion algebra: x = a-1b, but it may be non-resolvable in
       | Cayley-Dickson algebras of higher order r >= 4 because of
       | divisors of zero.            Therefore, in this article
       | differential equations are considered [only] with octonion or
       | quaternion variables for octonion or quaternion valued functions.
       | 
       | https://arxiv.org/pdf/1003.2620 FWIW the octonion answer seems to
       | be much more dependent on what kind of analysis you're doing. AKA
       | "it's complicated"
        
       ___________________________________________________________________
       (page generated 2024-06-28 23:00 UTC)