But I did reference Coxeter's paper, and for a good reason; I wasn't planning to copy his entire paper into my posting, but simply point out the relevant issues. You have to read these threads in context, and not be so quick to play "gotcha". At 11:10 AM 6/26/2014, Dan Asimov wrote:
I made no reference to Coxeter's paper.
--Dan
On Jun 26, 2014, at 11:01 AM, Henry Baker <hbaker1@pipeline.com> wrote:
I was responding to Smith's question. My point had to do with the _number_ of quaternion multiplications, not whether there were any other operations (in this case, conjugation). Since the question involved rotations, and since the conjugation^2=identity, I didn't bother getting into the whole detail of Coxeter's paper, since I had already given you the link.
At 10:04 AM 6/26/2014, Dan Asimov wrote:
As nice as Coxeter's paper is, I'm only comparing what you wrote that I quoted below with what I wrote (below).
--Dan
On Jun 26, 2014, at 9:45 AM, Henry Baker <hbaker1@pipeline.com> wrote:
Read Coxeter's paper.
Theorem 5.1. The reflection in the hyperplane sum(y_nu x_nu)=0 is represented by the transformation x -> -y conj(x) y. The product of two such reflections ... is a rotation.
Theorem 5.2. The general rotation through angle phi (about a plane [this is 4D, remember]) is x -> a x b, where N(a)=N(b)=1 and S(a)=S(b)=cos(phi/2). Conversely, [every] transformation x -> a x b (N(a)=N(b)=1) is a rotation whenever S(a)=S(b).
At 09:17 AM 6/26/2014, Dan Asimov wrote:
Perhaps I'm not understanding this, but left multiplication of points in R^4 (identified with H, the ring of quaternions) by any fixed unit quaternion q
L_q: R^4 -> R^4 via L_q(x) := qx
results in a rotation of R^4. (Same for right multiplication R_q(x) := xq.)
If instead we're talking about a pure unit quaternion u (Re(u) = 0), then identifying R^3 with the pure quaternions
H_0 := {x in H | Re(x) = 0}
results in left multiplication by q
f_q: R^3 -> R^3 via f_q(x) := qx
yielding the cross product of q with x, which of course is a projection of R^3 onto the 2-plane perpendicular to q.
So, I'm not sure in what sense a quaternion multiply computes a reflection.
--Dan
On Jun 26, 2014, at 6:07 AM, Henry Baker <hbaker1@pipeline.com> wrote:
. . . each quaternion multiply only computes a _reflection_, and you need 2 reflections to make a rotation.