1. ## Re: Quaterions vs. geometry

Originally Posted by emperorzelos
You might want to look into the history. The reason for quaternions existence was to solve the problems we attribute to vectors today. I have a book on it actually, it's facinating.
I haven't read about the history of quaternions vs. vectors in great detail (I'm more interested in the math, not the bickering) but from the little I've seen, it seems as though Hamilton overreached with his quaternions and turned into some sort of prophet for people favoring quaternions over vectors. It's one bizarre part of the history of math I never quite understood, how quaternions manage to create these cults. People are still bickering over them if you look online in the right places.

2. ## Re: Quaterions vs. geometry

Originally Posted by mathnerd
I haven't read about the history of quaternions vs. vectors in great detail (I'm more interested in the math, not the bickering) but from the little I've seen, it seems as though Hamilton overreached with his quaternions and turned into some sort of prophet for people favoring quaternions over vectors. It's one bizarre part of the history of math I never quite understood, how quaternions manage to create these cults. People are still bickering over them if you look online in the right places.
The history is that Hamilton wanted something that could rotate objects in 3 dimensional space, they had 2 dimensional in the form of complex numbers. However all attempts to extend complex numbers to 3D failed as it results in contradictions. He came up with the current relation which was consistent and worked and has a 3 dimensional subspace and a very natural way to rotate it.

and that is the big thing about quaternions over normal euclidean space with rotation matrices. Quaternions do not ever suffer from gimbal lock.

3. ## Re: Quaterions vs. geometry

Quaternions

My impression is that they add the "left hand" rule to the "right hand" rule of outer products.
(That is, a reflection in 3-space from the normal +/- polarity of the k vector formed by ) (opposite polarity in the direction of the thumb).... so the metric tensor never has a trace equal to zero.... for any set of scalars.....

Since the determinant ijk = -13 (a "negative definite" volume as opposed to 1*1*1 = 13 , a "positive definite" volume in 3 dimensions. (think of ijk as an inner product with orthogonal coordinates in the complex plane so that the trace of a matrix describing volumes can never go to zero, even though the dot product of any of the two components can be zero for (so no "gimbal lock")

gimbal lock

(This is off the top of my head; a conjecture, a proposal. I'll have to think about it some more before I can be sure...)

(And so characterize the "weak force" relativistically? I'll have to think a bit more on it.... )

For GTR, they may describe the guantum equivalent of "negative curvature" - "anti-polarization" of photons not on the geodesic because they are directed away from it, rather than toward it (i.e. "each other" for photons on the geodesic). So maybe correspond to "dark matter/energy" in three dimensions (the photons/inertial matter) you don't see...
(e.g., the photons inside an "Einstein Ring" that aren't on the geodesic that you actually observe, and can be going in any direction except the one that hits your sensor)..

That is, "anti-spin" polarization...

but I could be wrong....

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