On 11/27/07, metaweta@gmail.com <metaweta@gmail.com> wrote:

A relevant link:

http://math.ucr.edu/home/baez/square.root.html -- Mike Stay

I've read this webpage three times, and I still don't understand it. He seems to be attempting to reconstruct the Cayley-Hamilton derivation of the quaternions, which extends the complex numbers via a new element j such that j^{-1} i j = -i; but since he never states exactly how his new element k is supposed to act on i, it's hard to be sure. Taking into account his cavalier attitude towards what is now non-commutative multiplication, it's scarcely surprising that confusion reigns at the conclusion. Has anybody else managed to make more sense of it than yours truly? WFL

On Nov 28, 2007 2:04 PM, Mike Stay <metaweta@gmail.com> wrote:

Well, he says explicitly that it's not the quaternions. I won't ruin the puzzle by saying what it is, though.

I will do my best to ruin the puzzle, if I can ... attempted spoilers below! Another way of thinking about all this "i" stuff is with a matrix. We can map 1 to the identity matrix, [[1 0] [0 1]]. We can map i to the 90 degree rotation matrix, [[0 1] [-1 0]]. Complex conjugation, the way he (confusingly!) describes it, is then the reflection in the x-axis, [[1 0] [0 -1]]. The confusion comes from the fact that we're talking about a point, a + bi, but then we're also describing i as a transformation, 90 degree rotation. So I think we need to do a better job of keeping track of which one we mean! Anyway, if we take this matrix approach, it's pretty clear that what he calls j corresponds to the reflection in the y = -x line, and what he calls -j is the reflection in the y = x line. Since they're all reflections, it's also clear that j^2 = k^2 = 1. Now, h^2 = k turns out to be quite tricky. There's no real 2x2 matrix that squares to a reflection -- determinants give an easy enough proof of that! So h must be something like [[1 0] [0 i]], treating i now as a number instead of as a matrix or transformation. Sounds like it's time to head into a 4D world of complex cross complex to give some interpretation of that ... So in answer to his last question, what's "really going on", I would say that we take something (the real line) and a transformation (reflection through the origin) and interpret the square root of the reflection as a rotation by adding extra dimensions. Then complex conjugation is a reflection in 2D, so it in turn needs an extra dimension so that its square root can turn into a reflection (Is a third dimension enough, or do I need to look at complex 2x2 matrices and thus need 4D?). Now, wasn't someone just saying exactly this (here on this list? Or was I reading it elsewhere?) about Flatlanders interpreting a mirror as a reflection tool while we higher-dimensional people could see that it's a 3D rotation? --Joshua Zucker