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This article is from the sci.fractals FAQ, by Michael C. Taylor and Jean-Pierre Louvet with numerous contributions by others.

26 How does quaternion arithmetic work?

quaternions have 4 components (a + "i_b + "j_c + "k_d) compared
to the two of complex numbers. Operations such as addition and
multiplication can be performed on quaternions, but multiplication is
not commutative.

Quaternions satisfy the rules

* i^2 = j^2 = k^2 = -1
* ij = -ji = k
* jk = -kj = i,
* ki = -ik = j

See:

Frode Gill's quaternions page
http://www.krs.hia.no/~fgill/quatern.html

 

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