- #1
marlon
- 3,792
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Can anyone give me a definition of quaternions and octonions? What are these things and what are they used for.
regards
marlon
regards
marlon
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Quaternions & Octonions: Definition & Uses
- Thread starter marlon
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- Quaternions
In summary, quaternions are used in mathematics, physics, and programming, and they are a more natural extension to complex numbers. They are also used in relation to Yang Mills theory and GTR.
- #2
Nenad
- 698
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chech this site out. It gives a good mathematical interpritation:
http://mathworld.wolfram.com/Quaternion.html
http://mathworld.wolfram.com/Quaternion.html
- #3
marlon
- 3,792
- 11
Nenad said:
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
- #4
Dominguez Scaramanga
- 15
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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! 
- #5
marlon
- 3,792
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Dominguez Scaramanga said: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
- #6
Dominguez Scaramanga
- 15
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marlon said:
no problem
- #7
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!
- #8
Dominguez Scaramanga
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- #9
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They have been used in Reletivity Theory successfully. See the books by Mendel Sachs.
- #10
Goalie_Ca
- 102
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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.
- #11
laserblue
- 64
- 1
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.
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.
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- #12
thecolor11
- 12
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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/
1. What are quaternions and octonions?
Quaternions and octonions are mathematical concepts that extend the idea of complex numbers. They involve a combination of real and imaginary numbers and are used to represent rotations in three and four-dimensional space, respectively.
2. What are some practical uses of quaternions and octonions?
Quaternions and octonions have many practical applications in fields such as mechanics, physics, computer graphics, and robotics. They are used to represent rotations and orientations in 3D and 4D spaces, making them useful in computer animation and game development. They are also used in quantum mechanics to describe the properties of particles.
3. How are quaternions and octonions different from complex numbers?
Quaternions and octonions differ from complex numbers in several ways. Unlike complex numbers, which have two imaginary units (i and -i), quaternions have three imaginary units (i, j, and k), and octonions have seven imaginary units. Also, while complex numbers can be commutative, quaternions and octonions are non-commutative, meaning the order in which they are multiplied matters.
4. Can quaternions and octonions be visualized?
Yes, quaternions and octonions can be visualized using a geometric representation called the "Cayley-Dickson construction." This involves plotting the real and imaginary components of the numbers on different axes, similar to how complex numbers can be plotted on a 2D plane.
5. Are quaternions and octonions difficult to understand?
While quaternions and octonions may seem complex at first, they can be understood with a basic understanding of algebra and geometry. With practice and familiarity, they can become useful tools in various fields of mathematics and science.
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