Message #2472

From: Eduard Baumann <baumann@mcnet.ch>
Subject: Re: [MC4D] Message 2465 repeated
Date: Tue, 06 Nov 2012 17:59:19 +0100

Lets put a30 and b30 on
http://www.cs.berkeley.edu/~sequin/MVS/mvs.g4.D4.gif

Ed

—– Original Message —–
From: Eduard Baumann
To: 4D_Cubing@yahoogroups.com
Sent: Tuesday, November 06, 2012 5:49 PM
Subject: Re: [MC4D] Message 2465 repeated


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&#58;//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 &quot;tetrus&quot;, 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&#58;

http&#58;//www.cs.berkeley.edu/&#126;sequin/PAPERS/Bridges06&#95;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&#58;//www.gravitation3d.com/magictile/pics/tetrahedron.png

Likewise, the 3-torus, in some sense, has 4 holes.  The tetrus is bent such that the &quot;outside&quot; or &quot;inverted&quot; 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 &#123;7,3&#125; onto a tetrus significantly warps the heptagons, like in the scuplture Nan posted about last year&#58;

http&#58;//games.groups.yahoo.com/group/4D&#95;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 &#58;)

seeya,<br>
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&#58;27 AM, Eduard &lt;baumann@mcnet.ch&gt; wrote&#58;

  You wrote&#58;<br>
  &quot;So both of your face adjacency graphs will naturally live on the surface of a 4-torus (four holed donut).&quot;

  What a dream &#58;<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) &#58;<br>
  http&#58;//wiki.superliminal.com/wiki/File&#58;Genus&#95;4&#95;1.PNG<br>
  http&#58;//wiki.superliminal.com/wiki/File&#58;Genus&#95;4&#95;2.PNG<br>
  http&#58;//wiki.superliminal.com/wiki/File&#58;Genus&#95;4&#95;3.PNG<br>
  http&#58;//wiki.superliminal.com/wiki/File&#58;Genus&#95;4&#95;4.PNG<br>
  http&#58;//wiki.superliminal.com/wiki/File&#58;Genus&#95;4&#95;5.PNG

  Have you better ones ?

  Make angular wrl samples ?

  Ed