Quaternions & Octonions: Definition & Uses

[marlon]

Can anyone give me a definition of quaternions and octonions? What are these things and what are they used for.

regards
marlon

 

[Nenad]

chech this site out. It gives a good mathematical interpritation:
http://mathworld.wolfram.com/Quaternion.html

 

[marlon]

chech this site out. It gives a good mathematical interpritation:
http://mathworld.wolfram.com/Quaternion.html

So they are basically elements of some algebra with specific multiplication-properties. The Pauli-matrices who obey these rules are quaternions...

ok, can anyone give some more physical uses of these things


thanks, nenad for your quick repley

regards
marlon

 

[Dominguez Scaramanga]

looks like it can be implimented in quantum physics... have a look here, it's all way over my head and I didnt read much of it, but they're certainly using quaternions!

 

[marlon]

looks like it can be implimented in quantum physics... have a look here, it's all way over my head and I didnt read much of it, but they're certainly using quaternions!

This is what i was looking for...

grazie, grazie, grazie

marlonissimo

 

[Dominguez Scaramanga]

This is what i was looking for...

grazie, grazie, grazie

marlonissimo

no problem

 

[Gonzolo]

The short answer is that they are an extension to complex numbers. It is apparently a quite "natural" extension. I didn't know they were useful for something else than pure math, especially not QM!

 

[Dominguez Scaramanga]

I didn't know they were useful for something else than pure math, especially not QM!

ahh the wonders of google

... there's even a quaternions.com!

 

[Dr Transport]

They have been used in Reletivity Theory successfully. See the books by Mendel Sachs.

 

[Goalie_Ca]

Quaternions are also used by 3d programmers a lot for "camera" classes. Many people now argue that these things are useless because vector algebra and calculus can do the exact same thing. It is my understanding that this is a precurser to that branch of mathematics.

 

[laserblue]

Maxwell's original formulation of Electromagnetics was in terms of quaternions but there were several individuals such as Gibbs and Heaviside who questioned the necessity and pragmatics of using quaternions and promoted a version employing only VECTORS. The dispute got quite heated with insults being hurled back and forth. In NEW FOUNDATIONS FOR CLASSICAL MECHANICS, Dr. David Hestenes mentions this controversy and how quaternions reappeared in physics in the form of MATRICES and CLIFFORD ALGEBRA in Quantum Mechanics. (http://modelingnts.la.asu.edu/GC_R&D.html )
Quaternions have been used very extensively by engineers and computer programmers for modelling rotations as mentioned in some of the other postings.
Expressing GTR in a quaternionic formulation is at the root of Mendel Sach's approach to a unified theory of electromagnetism and gravity but quaternions are not a comprehensive mathematical language for expressing geometrical concepts and relations in physics. Some relationships such as between Yang Mills theory and GTR show up better when expressed in Geometric Algebra.

 

[thecolor11]

Here's a link to one of my professor's description. You can click the link to some papers. http://www.physics.orst.edu/~tevian/octonions/