Message #2470

From: Melinda Green <melinda@superliminal.com>
Subject: Re: [MC4D] Message 2465 repeated
Date: Mon, 05 Nov 2012 18:22:13 -0800

I hope I am not becoming tiresome by coming back to IRPs again, but my
hunch as to why genus 3 tilings are so common has to do with their
relationship with the common cubic packing. I feel that constructing
your genus 3 polyhedra on this
<http://superliminal.com/geometry/infinite/4_6a.htm> surface may give
more illustrative and symmetrical figures than on one that is bent in
order to reconnect with itself. I.E. to create the torus handles. If I
am right I would expect this to be generally the case when dimension
equals genus.

-Melinda

On 11/5/2012 5:52 PM, Roice Nelson wrote:
>
>
> 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
> <http://www.cs.berkeley.edu/%7Esequin/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
> <mailto: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
>
>
>
>