1. ## Quaterions vs. geometry

This is a continuation from the thread drift in the mathematics section. integers-semiring-quotient. I think it's important to talk about computer processor speed when selecting the best mathematical tools.

2. ## Re: Quaterions vs. geometry

Originally Posted by mathnerd
Speaking of that "quaternion vs. vector war", it seems both sides held bizarre points of view. Hamilton thought that quaternions could do everything and if you weren't using quaternions, then there was something wrong with you. On the other hand, Heaviside and Gibbs just figured that you could dissect the quaternion, pull out the cross product and dot product, and kick away the ladder, pretending you never needed the quaternions in the first place. Unfortunately, when you do that, the dot product and cross product become mysterious and unmotivated.
I think it's better to put the cross product and dot product where they belong, on the unit position vector instead of every nonsensical node. I also think that frees up computer processing speed because it is fewer steps.
Originally Posted by mathnerd
the quaternions are interesting for their own sake as an extension of even if there are more concise alternatives currently available in physics.
I'm not aware of a completed model.

3. ## Re: Quaterions vs. geometry

Originally Posted by KickLaBuka
This is a continuation from the thread drift in the mathematics section. integers-semiring-quotient. I think it's important to talk about computer processor speed when selecting the best mathematical tools.
That's only true if you try to apply it in a computer.

4. ## Re: Quaterions vs. geometry

Originally Posted by KickLaBuka
I think it's better to put the cross product and dot product where they belong, on the unit position vector instead of every nonsensical node. I also think that frees up computer processing speed because it is fewer steps.

I'm not aware of a completed model.
It should be noted historically that vector algebra came FROM the quaternions.

5. ## Re: Quaterions vs. geometry

Originally Posted by emperorzelos
It should be noted historically that vector algebra came FROM the quaternions.
Quaterions have no reasoning in Cartesian vector space. Quaterions introduce four vectors into a geometric one, three of which are imaginary. There is no x,y,z reasoning for the existence of the imaginary Quaterions. They just pop into form. If something turns imaginary, it aught to be because the vector crossed an axis. We aren't seeing that with Quaterions, so I don't comprehend the defense. I admire your passion.

6. ## Re: Quaterions vs. geometry

Originally Posted by emperorzelos
if you try to apply it in a computer.
Even the brains computer writing out an equation takes time to accomplish. Quaterions are extra writing, half of which gets thrown out anyway.

7. ## Re: Quaterions vs. geometry

Originally Posted by KickLaBuka
Even the brains computer writing out an equation takes time to accomplish. Quaterions are extra writing, half of which gets thrown out anyway.
But, to be clear, what sorts of calculations are we interested in doing? Vector addition and scaling are no more difficult when comparing ordinary 3-vectors and quaternions, so are we talking about cross and dot products?

In that case, you could just define dot and cross products in terms of quaternion multiplication, couldn't you? That is, if you have quaternions which have a 0 in the scalar part, the dot product is just (AB+BA)/-2, where A and B are the two quaternions. And then the cross product is (AB-BA)/2.

8. ## Re: Quaterions vs. geometry

Originally Posted by mathnerd
what sorts of calculations are we interested in doing?
Everything I know about the things; I learned in an hour sitting at a booth drinking mojitos. Could you rather come up with a geometry where you think Quaterions are required and I can set out to show why they are invalid.

9. ## Re: Quaterions vs. geometry

Originally Posted by KickLaBuka
Everything I know about the things; I learned in an hour sitting at a booth drinking mojitos. Could you rather come up with a geometry where you think Quaterions are required and I can set out to show why they are invalid.
I'm not sure I can think of an example of where they are absolutely required. I have no particular bone to pick with quaternions, nor do I have any evangelical position to advance like Hamilton did. I'm mainly interested in exploring the algebraic properties.

10. ## Re: Quaterions vs. geometry

Originally Posted by KickLaBuka
Quaterions have no reasoning in Cartesian vector space. Quaterions introduce four vectors into a geometric one, three of which are imaginary. There is no x,y,z reasoning for the existence of the imaginary Quaterions. They just pop into form. If something turns imaginary, it aught to be because the vector crossed an axis. We aren't seeing that with Quaterions, so I don't comprehend the defense. I admire your passion.
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.

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