Re: Quaternions, Penrose Road to Reality ch. 11.3



Just d' FAQs wrote:
> On 25 Jun 2005 08:53:41 -0700, jg.campbell.ng@xxxxxxxxx wrote:
> >I'm attempting to gain a physical feeling for quaternions and have got
> >stuck in chapter 11.3 of Penrose's Road to Reality; actually, reading
> >Road to Reality is a second goal that I want to pursue in parallel.
> >
> >i, j, and k, taken as operators (terminology correct?), rotate Pi about
> >the appropriate axis, in the right-handed sense.
> >
> >Penrose argues that one can show that ij = k, provided we take ij and j
> >followed by i.
> >
> >I was attempting to follow the argument /without/ using Penrose's
> >spinor book-belt construction (*) and got quite confused. It looks like

[...]

>
> Can't comment on your mental state, nor on Conway's book, having no
> access to either. :)
>

Penrose cited J.H. Conway (International Congress of Mathematicians,
Helsinki, 1978) as his introduction to the device.

> As for quaternions and the belt trick: The belt is a fun insight by
> physicist Paul Dirac, and an SGI graphic implementation of it by John
> Hart is mentioned in the FAQ (along with a paper describing it).

Thanks, I'll see about porting that to my system. Perhaps it can become
a student exercise :-)

>
>
> <http://graphics.stanford.edu/courses/cs348c-95-fall/software/quatdemo/>

I looks like the link to the paper is dead; however, there is a PDF
version at:

http://graphics.cs.uiuc.edu/~jch/papers/vqr.pdf


>
> The authors of "Gravitation" (ISBN 0716703440) discuss it in terms of
> entanglement, the object being linked to the environment.
>
> It is an attempt to make the 2-to-1 cover and the RP^3 topology of 3D
> rotations more tangible. The problem is as follows. We all "know" from
> common experience that if we rotate an object about a fixed axis by an
> amount of 360 degrees, that everything is exactly as before. This is
> true if we look at the object in isolation, but *not* if we tie it up.
>
> The space of 3D rotations, the group we call SO(3), has a topology all
> its own. In fact, the topology is the same as that of 3D projective
> space, which we call RP^3. And one of the intrinsic properties of any
> topological space is its fundamental group. To get this group, we pick
> a point, and draw loops starting and ending there. Next we define an
> addition, which is merely making big loops out of little ones, glueing
> the end of one to the start of the next. Finally, we merge loops into
> equivalence classes, where two loops are equivalent if we can deform
> one into the other within the topological space. Arithmetic is still
> defined on classes, and satisfies all the properties of a group. We
> have closure. We have associativity. We have an identity, the null
> loop. And we have inverses, reversing the direction of traversal.
>
> Unit quaternions, the 3-sphere S^3, have the simplest possible such
> group, because every loop can be contracted to the null loop. That is,
> the fundamental group of S^3 is the trivial group with one element.
>
> But the 3D rotations, and RP^3, have a *different* fundamental group,
> the "even-odd" group of two elements. The physical implication of this
> is that a 360 degree rotation path is *not* equivalent to a null path!
> However, the sum of two, a 720 degree rotation, *is* equivalent to a
> null path.
>
> There is a subtle but vital difference between what we "know" and what
> the fundamental group tells us. What we "know" is about the object;
> but the group is talking about the path. The belt *is* the path. When
> we use the belt trick, we leave the object alone (it really does end
> up as it began); we only manipulate the belt, deforming the path.
>
> The fixed end of the belt is the point representing the identity. The
> free end is the point representing our final pose. The belt between is
> a representation of the path between them. A null path is equivalent
> to an untwisted belt.
>
> Our difficult here is that we rarely have prior experience, and thus
> no intuition, with this whole idea of deforming a rotation path.
>
> Even so, our bodies know. Hold a book in the palm of your hand above
> your shoulder, palm up, fingers back, thumb in. Rotate away from you,
> and continue 360 degrees. The book will now be below your shoulder,
> translated, but with book and hand in the same orientation as before.
> Now keep turning in the *same direction*, another 360 degrees. This
> does not induce a spiral fracture, but restores the original pose!

Finally, I managed it -- I think; and the arm is still attached! But
the neighbours think I've taken up Tai Chi.

Any cricket fans here? It's not unlike bowling the googly in leg-spin
bowling, at least the way I attempted it; Murilitheran (Sri Lankan
demon bowler) would find it easy :-)

>
> Now let's relate this to quaternions. We claim that k should represent
> a 180 degree rotation around the z axis. Now we will further stipulate
> the *direction* of rotation. We also claim that kk should represent a
> composition, thus a 360 degree rotation all in the same direction. It
> is true we cannot distinguish k from -k looking only at the object; we
> also cannot distinguish 1 (no rotation) from -1 (360 degree rotation).
> But when we look at paths, these are different. Thus kk=-1, leaving a
> twisted arm; but -k k=1, turning one direction then back, leaving an
> untwisted arm. And, to our great satisfaction, kkkk=1, four 180 degree
> turns (two 360 degree turns) in *the same direction* restore the arm.
>
> It's lovely how everything comes together. The arm works, the rotation
> works, the quaternion multiplication works, the topology works, and
> the fundamental group works.
>
> We can even go one step further, and draw paths on the quaternion unit
> sphere. First draw a small path that starts and ends at the identity.
> This is the same for S^3 and SO(3). Now draw a path that starts at 1
> and ends at -1. This is *not* a loop for S^3; but for SO(3) we make no
> distinction between q and -q, so this *is* a loop! We have created a
> 360 degree rotation path we cannot contract. Now on S^3 continue with
> a path from -1 back to 1. This completes a full loop for S^3, and also
> one for SO(3); but this loop we *can* contract, in both spaces.

Plenty to work on! Including, I think, a more thorough visit to Mac
Lane and Birkoff.

Many thanks, Jd'F; as usual, an answer that the question hardly
merited.

Incidentally, as well as preparing a introductory course on Computer
Graphics (mentioned before, but now forgotten about for some time), I'm
setting myself the task of giving, before the end of his bi-centenary,
a little general talk about Hamilton and quaternions, to include also
mention of Grassmann, Clifford, Gibbs and vector analysis, and finally
Hestenes and geometric algebra. Can be done in an hour? Well, if I do
no more than show a few equations and diagrams and give an arm-wavy
(literally) introduction to quaternions, plus give a good bibliography,
it will be some use.

Best regards,

Jon C.

.

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