Thread: "Message 2465 repeated"

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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 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
>

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Yeah, the 3-torus seems to be more popular. =A0Even pretzels seem to always=
prefer 3 holes :)

el" target=3D"_blank">http://www.google.com/images?q=3Dpretzel

G=
enus-3 tilings feel like they show up more, but part of the reason for focu=
s on them may also be the "tetrus", a symmetrical representation =
of a 3-torus which takes the form of a thickened tetrahedron. =A0It's i=
n some of the pictures you posted, but check out this paper by Carlos Sequi=
n as well for discussion:



OnTetrus.pdf" target=3D"_blank">http://www.cs.berkeley.edu/~sequin/PAPERS/B=
ridges06_PatternsOnTetrus.pdf

When you look at a tetrus, it appe=
ars to have 4 holes rather than 3, one for each face of the tetrahedron. =
=A0What's going on? =A0Consider a tetrahedron stereographically project=
ed on the plane. =A0One 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

Lik=
ewise, the 3-torus, in some sense, has 4 holes. =A0The tetrus is bent such =
that the "outside" or "inverted" hole looks like all th=
e others. =A0But it's still the same as a sphere with three handles. =
=A0All the tetrus shapes in the pictures you posted are genus-3.



For the 4-torus, can we find a corresponding symmetric shape like the t=
etrus? =A0We need to base it on the thickened skeleton of something with >5 faces. =A0But alas, no platonic solid has 5 faces. =A0You could use =
a triangular prism, or a rectangular pyramid, even though they are not regu=
lar. =A0



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 te=
trus significantly warps the heptagons, like in the scuplture Nan posted ab=
out last year:



target=3D"_blank">http://games.groups.yahoo.com/group/4D_Cubing/message/191=
7

You'd have to embed the faces in a higher dimensional spac=
e to get them connected up in a geometrically regular, angular way. =A0That=
or stick with the IRP, like Melinda said :)



seeya,
Roice

P.S. Sorry for what appeared to be wacky formatt=
ing in my last post. =A0The only cause I could figure was the new gmail com=
pose feature. =A0Hopefully this one is better.

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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
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
>
>
> 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 > > 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
>
>
>
>


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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 href="http://superliminal.com/geometry/infinite/4_6a.htm">this
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