On Mar 11, 2015, at 1:52 PM, Allan Wechsler <acwacw@gmail.com> wrote:
The thread started out talking about quaternions and Hurwitz integers, so I assume we are still multiplying quadruplets with the quaternion rule.
Right, all I meant was quaternion multiplication, which I guess I'm so used to on S^3 = unit quaternions that I neglected to mention it.
On Wed, Mar 11, 2015 at 4:41 PM, Andy Latto <andy.latto@pobox.com> wrote:
I'm not sure how you're multiplying 4-vectors. If you're multiplying them componentwise,
(root2/2, root2/2,0,0) ^2 = (1/2, 1/2, 0, 0).
If you're multiplying them as quaternions,
roott2/2 + root2/2 * i = i
So I can't tell in what sense the set of points you give is a multiplicative system.
In S^3, (√½ + √½ i)^2 = i indeed, I screwed up yet again. (I was going by the observations that the integer quaternions Z + iZ + jZ + kZ with an even sum of coefficients (generated by ±1±i, ±i±j, ±j±k) is clearly closed under multiplication, and same for arbitrary quaternions of norm = 1.) Oh, well. To make up for wasting everyone's time, here is a research announcement: (with Joseph Gerver): There are analogues in the quaternions and octonions of the map-of-France fractal and others. The map-of-France analogue in the quaternions is based on taking a cluster of one 24-cell and its 24 neighboring ones in the 24-cell honeycomb of R^4, rotating, rescaling, and iterating, much as one does for the hex tiling of R^2 to get the map-of-France. --Dan P.S. So far we have not been able to get this to work in some other dimensions, like R^3: taking a cluster of one truncated octahedron surrounded by its 14 neighbors in the truncated octahedron honeycomb and trying to proceed mutatis mutandis does not seem to work as one might hope.
On Wed, Mar 11, 2015 at 4:32 PM, Dan Asimov <asimov@msri.org> wrote:
It appears to me that the 24 points on the unit sphere S^3 in R^4, of form
(±√½, ±√½, 0, 0) (and permutations)
form a multiplicative system with inverses.
QUESTION:
How many such 24-element multiplicative systems with inverses are there in S^3 ?
--Dan
On Mar 10, 2015, at 9:11 AM, Adam P. Goucher <apgoucher@gmx.com> wrote:
. . . the fact that the Lipschitz quaternions of unit norm (abstractly Q_8) form an index-3 normal subgroup of the Hurwitz quaternions of unit norm (abstractly 2T):
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