Thread 16721115

Anyone read this book able to discern heads or tails? Quaternions have always been interesting, but the logic behind them is non-intuitive. Thoughts?

>>16721115 (OP)
Hamilton’s original question that led to the real quaternions was basically “you know i, the square root of -1? what if we put two i’s instead of just one”. He was obsessed with this question apparently. Yeah I think he was autistic. Anyway that’s one way of looking at the matter. The frobenius theorem is its natural endpoint.

The other common thing is just the quaternion group (as opposed to the real quaternions) as representing these rotations of a cube. You can verify with an actual cube if you want that the defining identities hold ij=k^-1 etc.

>>16721115 (OP)
>the logic behind them is non-intuitive
What do you mean? There is no special logic needed for quaternion algebras; it's just classical logic.
Anyway, quaternion algebras are just a type of algebra. What is it you want to understand?

>can't into elementary hypercomplex analysis
we have a goner here lol
that shit is easy as hell

>>16721115 (OP)
Quaternions are basically just trigonometric identity property
1^2 = sin^2 + cos^2 ,but instead of 1^2 = 1/(sec^2) + 1/(csc^2) you can redefine the second as 1^2 = sec^2 + csc^2 because it's like a fidget spinner, where you can reverse the poles without having to associate.

>>16721115 (OP)
thanks for looking out for me, not sure why

>>16721115 (OP)
>NOOOOO WHAT DO YOU MEAN THESE THINGS DON’T COMMUTE
>LITERAL BLACK MAGIC
Quaternions are baby’s first non-commutative ring.

>>16722616
>he didnt learn linear algebra.

>>16722695
Every ring is automatically a module, anon.

>>16721115 (OP)
>This stuff is black magic and I refuse to believe otherwise
the proper term is african-american magic

>>16722704
fine tautology, anon.
your point being?

>>16722704

>[math]\textbf{Rng}\subset \textbf{Mod}[/math]

Proof?

>>16721115 (OP)
https://marctenbosch.com/quaternions/

>a special case of geometric algebra
Of course it doesnt makes sense
You can use geometric algebra to create a rotation from two reflections. This is why you multiply the vector by the quaternion twice. Quaternions only work for rotations in 3 dimensions where geometric algebra can do more than rotations and works in any dimensions and has a logical constructions instead of just making stuff up

>>16723085
Rings trivially satisfy all module axioms. I never implied any kind of subcategory inclusion.
>>16723084
>tautology

>>16721115 (OP)
>paper beats rock beats snake beats scissors beats paper
this filters the midwit

>>16723107
>Geometric algebra
Midwit detected.

>>16723090
>>16723107
back to >>>/g/ vibecoders
quaternion algebras are a big boy topic for mathematicians

>>16721115 (OP)
If the imaginary numbers can represent 2D space, how can you represent 3D space using a similar approach.

>>16723445
>this simpleton doesnt know calculus on manifold
lmao ahahahaha
>>16723482
quaternions algebra is like counting from 1 to 10 in contrast to the algebra of k vectors and p forms.

>>16725221
Differential geometry is like counting from 1 to 10 in contrast to generic projective modules.

>>16725221
>algebra is like counting from 1 to 10
he's got it

>>16721115 (OP)
It's black magic...but it works

i might be talking out of my since i haven't studied quaternions yet, but from what i can assume, the 3D rotation group, a nonabelian group, is non-commutative. quaternions encode this while preserving a Hamiltonian of some kind? something like conserving energy but with far greater generality. like i said I'm not sure if that was even the original intention for quaternions, but i think that's the key point. it can encapsulate the greater constraint of non-commutativity that normal complex numbers can't.