The basis vectors can be represented by matrices, this algebra was worked out independently by Pauli for his work on quantum mechanics. Murray Gell-Mann defined an extention of Pauli matricies to 3x3 matricies:
The scalar would be the identity matrix.
The structure constant is antisymmetric in the three indicies and has values:
f123 = 2
f147 = f165 = f246 = f257 = f345 = f376 = 1
f458 = f678 =√ 3
The bivectors can be calculated by multiplying the matrices:
The tri-vectors are:
So the complete geometric multiplication table is:
a*b |
b.e |
b.e1 |
b.e2 |
b.e3 |
b.e12 |
b.e31 |
b.e23 |
b.e123 |
| a.e |
|
|
|
|
e12 |
e31 |
e23 |
e123 |
| a.e1 |
|
|
e12 |
-e31 |
|
|
e123 |
e23 |
| a.e2 |
|
-e12 |
|
e23 |
|
e123 |
|
e31 |
| a.e3 |
|
e31 |
-e23 |
|
e123 |
|
|
e12 |
| a.e12 |
e12 |
|
|
e123 |
|
e23 |
-e31 |
|
| a.e31 |
e31 |
|
e123 |
|
-e23 |
|
e12 |
|
| a.e23 |
e23 |
e123 |
|
|
e31 |
-e12 |
|
|
| a.e123 |
e123 |
e23 |
e31 |
e12 |
|
|
|
|
which is equivalent to the table derived here.
Further Reading
Other uses of Pauli Matrix:
Related Concepts:
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| see also: |
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| Correspondence about this page |
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Book Shop - Further reading.
Where I can, I have put links to Amazon for books that are relevant to
the subject, click on the appropriate country flag to get more details
of the book or to buy it from them. |
Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and
Physics (Fundamental Theories of Physics). This book is intended for mathematicians
and physicists rather than programmers, it is very theoretical. It covers the
algebra and calculus of multivectors of any dimension and is not specific to 3D modelling.
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Commercial Software Shop
Where I can, I have put links to Amazon for commercial software, not
directly related to the software project, but related to the subject being
discussed, click on the appropriate country flag to get more details of
the software or to buy it from them. |
Mathmatica
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Can you help?
Please send me any improvements to here. I would appreciate ideas to make the pages more useful including
error correction, ideas for new pages, improvements to wording. It helps
if you quote the full URL of the page. |
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Terminology and Notation
Specific to this page here:
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program
I am working on a project which uses these principles, if you would like
to help me with this you are welcome to join in, here: |
http://sourceforge.net/projects/mjbworld/ |
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