Message #2471
From: Eduard Baumann <baumann@mcnet.ch>
Subject: Re: [MC4D] Message 2465 repeated
Date: Tue, 06 Nov 2012 17:49:54 +0100
Thanks for these awsome "Klein quartic celebrations" of Sequin !
—– Original Message —–
From: Roice Nelson
To: 4D_Cubing@yahoogroups.com
Sent: Tuesday, November 06, 2012 2:52 AM
Subject: Re: [MC4D] Message 2465 repeated
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:<br>
"So both of your face adjacency graphs will naturally live on the surface of a 4-torus (four holed donut)."
What a dream :<br>
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) :<br>
http://wiki.superliminal.com/wiki/File:Genus_4_1.PNG<br>
http://wiki.superliminal.com/wiki/File:Genus_4_2.PNG<br>
http://wiki.superliminal.com/wiki/File:Genus_4_3.PNG<br>
http://wiki.superliminal.com/wiki/File:Genus_4_4.PNG<br>
http://wiki.superliminal.com/wiki/File:Genus_4_5.PNG
Have you better ones ?
Make angular wrl samples ?
Ed