Re: This Week's Finds in Mathematical Physics (Week 215)

1. Oct 11, 2006

Greg Egan

In article <4288E2CE.9050502@univie.ac.at>, Arnold Neumaier
<Arnold.Neumaier@univie.ac.at> wrote:

> Greg Egan wrote:
> > http://gregegan.customer.netspace.net.au/KleinQuartic/KleinDualOrder7.gif [Broken]
> >
> > shows a rotation around a heptavalent vertex in the triangular tiling
> > of Klein's quartic curve, mapped on to a 3d figure with tetrahedral
> > symmetry.
> >
> > This is a wireframe image that only shows the edges, because it allows
> > you to see that there are 3 vertices kept fixed by this rotation, and
> > that the rotation around the other 2 vertices is faster than around the
> > first!

>
> Could you give 7 different colors to the edges emanating from the
> fixed vertex on the top? This would help interpret the movie!
>
> Arnold Neumaier

I could do that, but it would make the movie 7 times bigger as a file,
because then I'd have to show a complete rotation instead of just (1/7)
of one. For various technical reasons that would be quite arduous (due
to Mathematica's inadequacies I have to cut and paste each frame manually
into another application, and while that's tolerable for 27 frames, for
189 it's too much), as well as make the whole thing slower to download.

It *is* quite hard to see what's happening here, because a number of the
edges are moving largely towards/away from the viewpoint. Probably it
would be better if I can try to find an embedding of the surface with
order-7 symmetry at some of the vertices, just as the tetrahedral
embedding has order 3 symmetry at 8 of the faces.

I don't want to give anyone the false impression, though, that it's even
possible to enact this symmetry by a continuous motion of the tiling
along Klein's quartic curve itself. Klein's quartic curve has no
continuous symmetries, only discrete ones, so there is no truly correct
movie to be made of what happens for anything less than (1/7) of a
rotation. And in contrast to the order-2 symmetry shown in:

http://gregegan.customer.netspace.net.au/KleinQuartic/KleinDualInsideOut.gi [Broken]
f

where the embedded surface can at least retain its topology in a smooth
sequence of embeddings that ends by returning to the initial range in R^3
with the correct permutation of faces/edges/vertices, I don't think
that's possible for the order 7 rotation. (If I put faces over the
wireframe of the current model, the surface undergoes self-intersections
that seem to be unavoidable.)

Anyway, I'll see if I can come up with something clearer.

Greg Egan

gregegan netspace zebra net au

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