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

1. Oct 11, 2006

### Greg Egan

In article <428092DE.1040306@univie.ac.at>, Arnold Neumaier
<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

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