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Sure, I guess that's a consequence of general rotational symmetry. The main thing I was pointing out were the different geometric behaviors of multiplication b
by westoncb 8y ago
Sure, I guess that's a consequence of general rotational symmetry.
The main thing I was pointing out were the different geometric behaviors of multiplication by real vs imaginary units—not so much the difference between negative and positive units on either axis.
- Jtsummers 8y agoSeparate out the components, rotation and reflection and scaling. -1 is reflection. i is rotation by 90 degrees CCW. -i is both rotation by 90 degrees and reflection. Positive reals are scaling. So -ai is really rotation by 90 degrees, reflection, and then scaling by a.
- westoncb 8y agoBut none of that is related to my key point: multiplication by unitary values is significantly different depending on which axis the unit comes from (real vs imaginary), hence my usage of 'asymmetric'. In the last part of my first comment I point out the benefit (that I perceive) arising from the asymmetry. I also still wonder if a symmetric version (like I describe in the first comment) would be possible. Splitting the geometric behavior into rotation, reflection, and scaling is interesting—so thanks for the description—I just don't see how it relates to my comment.