Message #2469

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Message 2465 repeated
Date: Mon, 05 Nov 2012 19:52:55 -0600

Yeah, the 3-torus seems to be more popular. Even pretzels seem to always
prefer 3 holes :)

http://www.google.com/images?q=pretzel

Genus-3 tilings feel like they show up more, but part of the reason for
focus on them may also be the "tetrus", a symmetrical representation of a
3-torus which takes the form of a thickened tetrahedron. It’s in some of
the pictures you posted, but check out this paper by Carlos Sequin as well
for discussion:

http://www.cs.berkeley.edu/~sequin/PAPERS/Bridges06_PatternsOnTetrus.pdf

When you look at a tetrus, it appears to have 4 holes rather than 3, one
for each face of the tetrahedron. What’s going on? Consider a tetrahedron
stereographically projected on the plane. One might think it only has 3
faces with a quick glance, but of course it has 4, the last taking up the
entire background.

http://www.gravitation3d.com/magictile/pics/tetrahedron.png

Likewise, the 3-torus, in some sense, has 4 holes. The tetrus is bent such
that the "outside" or "inverted" hole looks like all the others. But it’s
still the same as a sphere with three handles. All the tetrus shapes in
the pictures you posted are genus-3.

For the 4-torus, can we find a corresponding symmetric shape like the
tetrus? We need to base it on the thickened skeleton of something with *5
faces*. But alas, no platonic solid has 5 faces. You could use a
triangular prism, or a rectangular pyramid, even though they are not
regular.

Whatever shape is chosen, painting these 30C puzzles on the resulting
4-torus will need to warp the 30 faces, just as painting the {7,3} onto a
tetrus significantly warps the heptagons, like in the scuplture Nan posted
about last year:

http://games.groups.yahoo.com/group/4D_Cubing/message/1917

You’d have to embed the faces in a higher dimensional space to get them
connected up in a geometrically regular, angular way. That or stick with
the IRP, like Melinda said :)

seeya,
Roice

P.S. Sorry for what appeared to be wacky formatting in my last post. The
only cause I could figure was the new gmail compose feature. Hopefully
this one is better.

On Mon, Nov 5, 2012 at 9:27 AM, Eduard <baumann@mcnet.ch> wrote:

> You wrote:
> "So both of your face adjacency graphs will naturally live on the surface
> of a 4-torus (four holed donut)."
>
> What a dream :
> A 4-torus (four holed donut) having the coloring of a30 and b30
>
> It is not easy to find beautyfull pictures of genus 4 manifolds (many are
> only genus 3) :
> http://wiki.superliminal.com/wiki/File:Genus_4_1.PNG
> http://wiki.superliminal.com/wiki/File:Genus_4_2.PNG
> http://wiki.superliminal.com/wiki/File:Genus_4_3.PNG
> http://wiki.superliminal.com/wiki/File:Genus_4_4.PNG
> http://wiki.superliminal.com/wiki/File:Genus_4_5.PNG
>
> Have you better ones ?
>
> Make angular wrl samples ?
>
> Ed
>