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.