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Not trueFrom: Steven
Subject: General
Date/Time 2008-07-08 17:14:12
Remote IP: 66.21.151.155
MessageFor one, octonions are NOT even associative, so if you want an
associative algebra, the best you can do is quaternions.
Octonions are a generalization, but you LOSE associativity.
You CAN generalize further, but you get nastier
power associative algebras like sedenions.
All of this is ultimately cumbersome though.
However, if you are not using this machinery, and all you are doing is defining a "multiplication of arcs", then in reality how is that any different from
standard vector addition? In reality, you are really describing a path product,
which is OK to define, but you won't necessarily obtain a path you want . . .
The composition of two distance-minimizing geodesics is
not necessarily a distance-minimizing geodesic!
Geodesics are only LOCALLY distance-minimizing.
Example: consider two sequential geodesics on the same great circle--
if they are long enough, the composition creates a geodesic that is
not the length-minimizer you want.
Steven
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