What has a shape but no volume?

[Averagesupernova]

You seem to me to be perhaps hastily overly dismissive of the special characterics of the Klein bottle

Believe me, I've considered it. So what makes it so special? It has a continuous surface. So what? It intersects itself. So what? A sphere such as the Earth has one continuous surface and can be thought of to have 'sides' in the same way as a Klein bottle. China is on the opposite 'side' of the USA. It's quite common to refer to something 'on the other side of the world' no matter how correct or incorrect that may be. The Earth also has no defined amount of liquid it would hold. What happens if I decide to modify the jar in your post and make the smooth round top into a round edge that would look similar to the top of a glass? Still meet the definition? I just see this as getting really hung up on definitions. And believe me, I've tried to see it otherwise.

 

[sysprog]

Believe me, I've considered it. So what makes it so special? It has a continuous surface. So what? It intersects itself. So what? A sphere such as the Earth has one continuous surface and can be thought of to have 'sides' in the same way as a Klein bottle. China is on the opposite 'side' of the USA. It's quite common to refer to something 'on the other side of the world' no matter how correct or incorrect that may be. The Earth also has no defined amount of liquid it would hold. What happens if I decide to modify the jar in your post and make the smooth round top into a round edge that would look similar to the top of a glass? Still meet the definition? I just see this as getting really hung up on definitions. And believe me, I've tried to see it otherwise.

Although a sphere also has no boundary, it is not non-orientable, as a Klein bottle is $-$ if we grant your "hung up on definitions" critique as disposatory of the properterial specialness of certain manifolds, then how do we rescue topology? $-$ isn't every branch of mathematics founded upon definitions?

 

[Averagesupernova]

isn't every branch of mathematics founded upon definitions?

It certainly is and I would be silly to think that is the wrong approach. But I do have a few issues with the definitions concerning boundaries of the Klein bottle. My question wasn't answered concerning adding the edge to the bottle.

 

[sysprog]

It certainly is and I would be silly to think that is the wrong approach. But I do have a few issues with the definitions concerning boundaries of the Klein bottle. My question wasn't answered concerning adding the edge to the bottle.

The Klein bottle by part of its definition does not have an edge $-$ I think that adding an edge to it would make it a different kind of manifold, and I don't see how you propose hypothetically to add an edge $-$ if you split it vertically it becomes topologically 2 Moebius bands of opposite chirality, rejoinder of those bands at their edges exactly along those edges as they were created by severance makes the recombined objects a Klein bottle again, which does not have an edge, or a boundary, or orientability.

From http://www.ifp.illinois.edu/~sdickson/Klein/Klein.html#Klein_Halves_Real:

 

[Averagesupernova]

https://senseis.xmp.net/?KleinBottle
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The above link shows a gif that more accurately describes the Klein bottle in my opinion. It shows the little arrows like men marching around the surface. What has not been mentioned in this thread is that these little men go through the wall. Or, it has been poorly illustrated in other examples that there is a relief that allows this. This changes my opinion slightly, but to allow such a thing now implies that there is an edge.

 

[jbriggs444]

https://senseis.xmp.net/?KleinBottle
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The above link shows a gif that more accurately describes the Klein bottle in my opinion. It shows the little arrows like men marching around the surface. What has not been mentioned in this thread is that these little men go through the wall. Or, it has been poorly illustrated in other examples that there is a relief that allows this. This changes my opinion slightly, but to allow such a thing now implies that there is an edge.

The English language is not doing us any favors on this one. The men do not go through the wall exactly. Nor do they go around the wall exactly. Yet they end up on what, at least locally, seems like it should be the other side. From a global perspective, there is only the one side.

 

[sysprog]

https://senseis.xmp.net/?KleinBottle
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The above link shows a gif that more accurately describes the Klein bottle in my opinion. It shows the little arrows like men marching around the surface. What has not been mentioned in this thread is that these little men go through the wall. Or, it has been poorly illustrated in other examples that there is a relief that allows this. This changes my opinion slightly, but to allow such a thing now implies that there is an edge.

That site includes the remark:

A Klein bottle is a surface which has no edges, no outside or inside and cannot properly be constructed in three dimensions.​

It's a 4-dimensional surface. That's what requires 3D representations (and 2D pictures thereof) of it to include the 'self-penetrating' characteristic. It doesn't have an edge, because the edges of the 2 Moebius bands of which it is composed are eliminated by their joinder into its composition.

 

[Averagesupernova]

It seems a bit silly to argue about a device in a 3D world that only truly exists in 4D. I stand by my original points.
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That being said, I did a Google search for a double Klein jar. Came up with some examples but nothing that I had been envisioning in my head. I will try to hand draw it, but I'll warn everyone here that my artistic abilities in that area are quite limited.
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The basics of it would be:
Take the pic in post#35. Modify it by moving the vortex on the top to the right slightly. Modify the tube that come out the side by moving it off to the right on the bottom of the jar. Now add a vortex in the bottom on the left having it's tube come out the side opposite the already existing tube. This tube then goes into the top on the left.
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This allows trips round and round without breaking through a wall. Go in one vortex, come out another.

 

[MevsEinstein]

A sphere such as the earth

Not to be picky but Earth is actually not a sphere. It's actually more like a smudged ball, similar to a pear.

 

[phinds]

Not to be picky but Earth is actually not a sphere. It's actually more like a smudged ball, similar to a pear.

No, it is not. It is approximately an oblate spheroid, NOT pear-shaped.

 

[jbriggs444]

No, it is not. It is approximately an oblate spheroid, NOT pear-shaped.

I am not familiar with the etymology of the pear-shaped metaphor, but Isaac Asimov mentions it in an essay:

It seems that Tufts university took down their copy. I had to search to find a copy elsewhere. It is a fun read if you've not seen it before.

Even the oblate-spheroidal notion of the Earth is wrong, strictly speaking. In 1958, when the satellite Vanguard I was put into orbit about the earth, it was able to measure the local gravitational pull of the earth–and therefore its shape–with unprecedented precision. It turned out that the equatorial bulge south of the equator was slightly bulgier than the bulge north of the equator, and that the South Pole sea level was slightly nearer the center of the earth than the North Pole sea level was.

There seemed no other way of describing this than by saying the earth was pear-shaped, and at once many people decided that the Earth was nothing like a sphere but was shaped like a Bartlett pear dangling in space. Actually, the pear-like deviation from oblate-spheroid perfect was a matter of yards rather than miles, and the adjustment of curvature was in the millionths of an inch per mile.

 

[Averagesupernova]

I'm feeling a bit guilty about derailing this thread but picking over whether the Earth is a sphere or pear, or slightly squished sphere in the context of this thread relieves all my guilt.

 

[Doc Al]

I think it's time to close this thread as it has drifted far beyond chemistry or physics.