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Name: benjamin p heroux
Status: N/A
Age: N/A
Location: N/A
Country: N/A
Date: Around 1995


Question:
I have been told many a time that imaginary and real numbers were othogonal. What third othogonal component would complete a description of three dimensions? A part of quaternions? If so, is there a simple algorithm for figuring the properties of hyperquaternions, hyperhyperquaternions, etc.?


Replies:
Yes, quaternion algebra does make a somewhat neat representation of cross and dot products in 3-dimensional space, just as complex algebra can represent some aspects of 2 dimensions very well. The way it works for quaternions is that a point x,y,z in space is represented by the quaternion

q = x i + y j + z k

and then q1 * q2 in quaternion algebra (where i^2 = -1, i*j = k etc)
gives

q1 * q2 = -(x1*x2 + y1*y2 + z1*z2) + (y1 z2 - z1 y2) i +

(z1 x2 - x1 z2) j + (x1 y2 - y1 x2) k

= - D + x3 i + y3 j + z3 k

where D is the 3-dimensional dot product of (x1,y1,z1) and (x2,y2,z2) and (x3,y3,z3) is the 3-dimensional cross product. However, I have never seen this used anywhere, or found to be particularly useful. One problem with quaternion algebra, unlike complex algebra is that it is non-commutative: q1 * q2 is not the same as q2 * q1. Any extension to hyper-quaternions etc. makes things even worse: the results are forced to be non-associative and therefore lose even the general "group" properties (I believe they are called "loops").

asmith


Nice explanation, asmith! Just for your information, quaternions are used in computational chemistry to enforce rigid-body constraints on classical dynamics simulations of certain large molecules . . . so there is an application you may not have seen.

However, it should be noted that there are more numerically efficient methods than quaternion methods (the SHAKE algorithm comes to mind) and so only a small fraction of compu-chemists use quaternions for this purpose.

rtopper



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