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

  1. Oct 11, 2006 #1
    In article <428092DE.1040306@univie.ac.at>, Arnold Neumaier
    <Arnold.Neumaier@univie.ac.at> wrote:


    > I'd like to see a movie of the 7-fold symmetry around a vertex!
    > This should be telling as well!


    > > 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
    Last edited by a moderator: May 2, 2017
  2. jcsd
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