In article <428092DE.1040306@univie.ac.at>, Arnold Neumaier(adsbygoogle = window.adsbygoogle || []).push({});

<Arnold.Neumaier@univie.ac.at> wrote:

[snip]

> I'd like to see a movie of the 7-fold symmetry around a vertex!

> This should be telling as well!

[snip]

> > If you look carefully, you'll see each corner of his tetrahedral

> > gadget is made of a little triangular prism with one triangle facing

> > out and one facing in. Since 4 x 2 = 8, there are 8 of these triangles.

> > Abstractly, we can think of these as the 8 corners of a cube!

>

> This could probably also be shown in a movie deforming the surface

> into a cube and back.

I've been contemplating doing both of these movies! So far I haven't had

time, but I think they'd be interesting, so hopefully I'll get around to

it soon.

In the second case, though, I can show you an image of the final, cubic

form -- an object with cubic symmetry where the 8 "nice" triangles sit at

the corners of a truncated cube, while the other 48 triangles form a

topologically inscrutable, if clearly symmetrical, self-intersecting mess.

http://gregegan.customer.netspace.net.au/KleinQuartic/KleinDualCube.gif [Broken]

Mathematica's dodgy rendering of intersecting polygons gives this a weird

scratchy look, as if it was photographed on celluloid and roughly handled.

In article <4281D6A1.7020802@jupiter.t30.physik.tu-muenchen.de>, Florian

Dufey <dufey@jupiter.t30.physik.tu-muenchen.de> wrote:

> Well, the rotational symmetry group of a cube is also the rotational

> symmetry group of an octahedron. You can also inscribe a cube in an

> octahedron in that way that the corners of the cube are the centers of

> the faces of the octahedron. Pretty like you inscribed the cube in the

> Klein quartic.

> Could it be that the triangles of the Klein quartic form directly the

> octahedron?

Not quite an octahedron, not quite a cube, but the 8 triangular faces of

a truncated cube! This figure has the same rotational symmetry as either

of those two dual polyhedra.

Greg Egan

Email address (remove name of animal and add standard punctuation):

gregegan netspace zebra net au

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

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