Manifolds

Higher-dimensional geometry (previously "Polyshapes").

Manifolds

Postby papernuke » Sun Jul 09, 2006 11:07 pm

what is a klein bottle? and how can it only not be seen in 3d? how does it work?

Edit by Rob: Changed the topic title to "Manifolds", it seems we are discussing more than the klein bottle here ;)
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Postby Marek14 » Mon Jul 10, 2006 6:14 am

http://en.wikipedia.org/wiki/Klein_Bottle is a good place to start :)

Klein bottle can only be realized in 3D by letting it intersect itself. In four dimensions, you can get rid of the intersection.
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Postby quickfur » Tue Jul 11, 2006 9:07 pm

Marek14 wrote:http://en.wikipedia.org/wiki/Klein_Bottle is a good place to start :)

Klein bottle can only be realized in 3D by letting it intersect itself. In four dimensions, you can get rid of the intersection.

Keeping in mind, of course, that in 4D, a Klein bottle can't hold any liquid because in 4D you need a 3D manifold to hold fluid in place.
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Postby bo198214 » Tue Jul 11, 2006 10:13 pm

hey, quickfur is back! :)
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Postby houserichichi » Wed Jul 12, 2006 2:48 am

My brother bought me a Klein bottle for my birthday after years of pestering and angry requests. Get yours here. I believe mine to be the "big Klein bottle".
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Postby wendy » Wed Jul 12, 2006 8:14 am

A klein bottle is given to a way of connecting the edges of a square together. It is a hedrix, or 2d-cloth, but because ( a ) 2d-cloths divide in 3d, and ( b ) the klein bottle is one-sided, it must cross.

It works well in four-dimensions, rather like knots work in three dimensions, but not in 2d. A knotted loop can not be used as a boundary for a polygon in 3d, nor can a klein-bottle be used to bound a planid (ie 3d shape, as "bounding solid" in 4d.

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Postby quickfur » Wed Jul 12, 2006 12:48 pm

wendy wrote:A klein bottle is given to a way of connecting the edges of a square together.

So how many possible combinations are there? IIRC, you get a klein bottle when you twist one pair of edges, and a real projective plane when you twist both pairs. Are there any other possible topological combinations? Is it possible to use hexagons instead of squares, and have a triple twist?
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Postby bo198214 » Wed Jul 12, 2006 4:39 pm

quickfur wrote: a real projective plane when you twist both pairs. Are there any other possible topological combinations? Is it possible to use hexagons instead of squares, and have a triple twist?


The real projective plane is originally even simpler. Simply take a diangle, with both edges directed clockwise and identify both edges. (A sphere is a diangle with one edge clock- and the other counterclockwise directed and both identified.)

In general every closed 2-surface can be given by an n-angle with the edges identified in some way.
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Postby pat » Wed Jul 12, 2006 8:02 pm

Another answer to the hexagon question....

Take any 2n-gon. Pair up the sides. For each pair, decide whether they'll be identified with or without a twist.

Now, you've got a 2-manifold. If it's got a twist, you cannot embed it in 3-dimensional space without self-intersection.

If it has no twists, it's a sphere with handles.
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Postby Keiji » Wed Jul 12, 2006 11:10 pm

What is the manifold without any twists from the hexagon, anyway? I can't visualize that, let alone anything more complex...

To me, it seems like such an object must be self-intersecting...
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Postby bo198214 » Wed Jul 12, 2006 11:54 pm

A hexagon with opposite edges identified without twist is also the torus.
Geometrically we first glue one pair of edges. This gives a hose with equally directed both ends which we can then glue edge by edge with a twist of 90 degrees.

Algebraicly it can be seen as follows. The identifications are given by the word
abca<sup>-1</sup>b<sup>-1</sup>c<sup>-1</sup>
Now let d=ab especially a<sup>-1</sup>=bd<sup>-1</sup>, then our word is equivalent to
dcbd<sup>-1</sup>b<sup>-1</sup>c<sup>-1</sup>=d(cb)d<sup>-1</sup>(cb)<sup>-1</sup>
which is the torus (given by the word xyx<sup>-1</sup>y<sup>-1</sup>).

We can reinterpret this geometrically as it does not matter whether we cut the hose at "a" or at "ab" (consider this as vector addition of "a" and "b" on the hose) but if we cut it at "ab" then we can consider "cb" one edge to be glued.
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Postby Keiji » Thu Jul 13, 2006 10:26 am

Okay that made no sense..

Suppose we folded the hexagon up into a square prism shape, like this:

Image

Sides 1 and 4 join at the cyan dot. The red sides are 3 at the top and 5 at the bottom. The blue sides are 2 at the top and 6 at the bottom. Now, we have to attach side 2 to side 5, which is easily done as we simply take it done the middle and onto side 5. But if we try to do the same with 3 and 6, it becomes self-intersecting, definately not a torus.
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Postby bo198214 » Thu Jul 13, 2006 12:38 pm

Sorry, but you make no sense. I provide a picture:
Image
We fold the right "a" over to the left and get a hose. Everything else as in my previous description (with the correction that it is not a 90 degree twist but a 180 degree twist).
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Postby Keiji » Thu Jul 13, 2006 6:32 pm

Oh right, I get it now! So it's a sort of twisted torus.
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Postby quickfur » Fri Jul 14, 2006 1:34 am

Now, what happens if one pair of edges in the hexagon are flipped? What exactly would that object be? Would it correspond with a Klein bottle, or something else?
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Postby Keiji » Fri Jul 14, 2006 1:52 am

Well, if we form the Klein bottle the other way (make the Möbius strip first, instead of making the uncapped cylinder first) we get the figure-8 Klein bottle...

As for the hexagon with one twist, hmm. That would probably be a figure-8 Klein bottle too, as we make the Möbius strip first, then we attach the sides of the strip to themselves while also putting a 180 degree twist in that, though I'm not sure if this twist makes any difference.

Edit: I just noticed that with the hexagon-torus thing, there are two ways to make it. Either you extend the ends along a circle until they meet, in which case the twist goes along the torus, or you bend them inwards until they meet, in which case the twist goes through the middle of the torus.
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Postby bo198214 » Sat Jul 15, 2006 11:33 am

hey quickfur, that would be a nice algebraic topology home work.
How about you doing it ;)
It seems quite sure that it is a sphere with attached crosscaps.
But how many crosscaps does it have?
We already ascertained that the Klein bottle is a sphere with 2 attached crosscaps.
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Postby Keiji » Sat Jul 15, 2006 11:48 am

I think I have found all of the cubic manifolds, there are nine of them:

Image

Hope you can understand my diagrams ;)

Note that all of these have a 3D net space, because although I have drawn a net of the cube, the cube you start with is solid, not hollow.
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