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+--- Thread: Octonion Hyper-Operations (/showthread.php?tid=1517)
Octonion Hyper-Operations - Catullus - 06/11/2022
How does tetration work for quaternions? What about for octonions? How do pentation and higher hyper-operations work for those?
RE: Octonion Hyper-Operations - tommy1729 - 06/12/2022
We try to walk before we can run
We do not even understand tetration well for octonions ( or even complex numbers lol )
And there are many tetrations and many hyper-operators.
So I assume it might be case specific.
I like the question , but it seems very very general.
Wish I could say more ... sorry.
regards
tommy1729
RE: Octonion Hyper-Operations - tommy1729 - 06/12/2022
I think most (known) methods rely on properties such as commutative associative etc.
So that might imply they all fail for most non-complex numbers.
Note that it is also problematic to define derivatives for non-complex functions. ( zero-divisors , depending on the type of infinitesimal , ... )
so that makes formulas that use calculus concepts * uncomfortable *.
Functional composition is associative ... but if we plug in non-associative numbers as parameters ... feels * delicate *.
Also we might get extra fixpoints in say the quaternions and such that harrass our convergeance domains and such.
Or as we learned before ; functions do no agree on the fixpoint. ( see the bummer thread )
There are also issues with combinatorial interpretations/methods but those are complicated.
Best I hope for is that the taylor series expansions of the solutions we have can be used ;
plug in a power-associative number x instead of the complex z and just compute the values.
... It should agree with the complex solution and by extension also the field/ring/whatever of x.
hopefully.
regards
tommy1729
RE: Octonion Hyper-Operations - Catullus - 07/05/2022
How do they work for matrices?