What has a shape but no volume?

[MevsEinstein]

TL;DR Summary

what the title says.

I was reading a Chemistry book when I read about the three states of matter. Everyone knows what they are, but I didn't know the simplest way to describe each of the three until I read this book. It said that a solid has a shape and a volume, a liquid has no shape but has a volume, and a gas has no shape or volume. How about something that has a shape but has no volume? It couldn't be plasma, since it doesn't have a shape.

 

[ergospherical]

I guess it means "not fixed" instead of "no", etc. (But even that's not really true - have you ever made pizza?).

 

[Frabjous]

A non-rigorous definition
in a container
A solid has a constant shape and volume
A liquid has the shape of the bottom of the container and a constant volume
A gas has the shape of the container and the volume of the container
If you look at the ordering, you need to have a constant volume before having a constant shape.

 

[phinds]

What has a shape but no volume?

A Klein bottle

 

[Frabjous]

A Klein bottle

mine does

 

[phinds]

mine does

Does what? It can't have a volume because it has no inside.

 

[Frabjous]

Does what? It can't have a volume because it has no inside.

Really? I can fill it water.

 

[Frabjous]

In case you want to get your own. It’s the best physics thing my parents ever got me.
https://www.kleinbottle.com/
It is also a very amusing website.

”At last, Acme Klein Bottle has conquered topological and engineering frontiers to manufacture genuine glass Klein bottles. These are the finest closed, non-orientable, boundary-free manifolds sold anywhere in our three spatial dimensions.”

 

[phinds]

Really? I can fill it water.

Yes, you can put some water on part of the outside. That doesn't mean it has a volume.

 

[Frabjous]

Yes, you can put some water on part of the outside. That doesn't mean it has a volume.

If I place it in a bowl of water, the level rises; however, Acme does advertise it as volume and magnetic monopole free.

 

[DaveE]

Really? I can fill it water.

Volume is a sloppy idea in everyday speech for open surfaces. We would all agree that a measuring cup has some volume, but what about a watch glass? What about an upside down watch glass?

How flat does a surface have to be to have no volume? Wouldn't you also have to specify other things like how it's filled, how it's oriented, what the boundaries are?

 

[Frabjous]

Volume is a sloppy idea in everyday speech for open surfaces. We would all agree that a measuring cup has some volume, but what about a watch glass? What about an upside down watch glass?

View attachment 298040

How flat does a surface have to be to have no volume? Wouldn't you also have to specify other things like how it's filled, how it's oriented, what the boundaries are?

I was being flippant. You did raise some interesting questions about open surfaces. Here’s a picture from their website with volume calibration marks. They clearly anticipated your and @phinds concerns.

Check out the numerical values.

 

[phinds]

Volume is a sloppy idea in everyday speech for open surfaces.

Indeed.

In everyday English I suppose you can say it has a volume but mathematically (i.e. topologically) it is identical to a Mobius strip --- it's a one-sided construct. You can get from any point on the surface to any other point on the surface without ever crossing an edge. Even a flat sheet of paper can't do that.

 

[Tom.G]

What has a shape but no volume?​

A shadow!

 

[sci0927]

Menger sponge.

 

[Mayhem]

A shadow!

Anything 2D really.

 

[bob012345]

I agree. Specifically I would say any 2D surface in 3D or higher space has shape but no volume. It may enclose a volume of space, such as an infinitely thin spherical shell, or not such as the Möbius strip or Klein bottle but it has no volume itself.

 

[Hornbein]

I agree. Specifically I would say any 2D surface in 3D or higher space has shape but no volume. It may enclose a volume of space, such as an infinitely thin spherical shell, or not such as the Möbius strip or Klein bottle but it has no volume itself.

In 3D a 1D object can't hold water. It will leak out into the second horizontal dimension.

A 2D object embedded in 4D can't hold water. It would leak out into the third horizontal dimension.

So in 4D a Klein "bottle" couldn't hold water. Models of the surface in 3D can hold water because the third horizontal dimension doesn't exist in 3D.

"It's only a model." -- Monty Python and the Holy Grail.

 

[valenumr]

Anything of 2 dimensions?

 

[Hornbein]

Anything of 2 dimensions?

Right. It will fail to hold water everywhere no matter what you do. You could gin up some highly convoluted object to simulate 3D but that's cheating.

 

[valenumr]

Right. It will fail to hold water everywhere no matter what you do. You could gin up some highly convoluted object to simulate 3D but that's cheating.

It's still an interesting question in a sense. The closest thing I can think is something like a sheet of graphene, but it still necissarily has volume (think mass density as a concept). But physically, I guess one would have to talk about point particles like an electron to ponder the question.

 

[Hornbein]

It's still an interesting question in a sense. The closest thing I can think is something like a sheet of graphene, but it still necissarily has volume (think mass density as a concept). But physically, I guess one would have to talk about point particles like an electron to ponder the question.

In our world, in order for an object to exist it has to have 3 dimensions greater than zero. When I say 2D this means that one of the dimensions is insignificant.

In the 4D world then a 2D object has two dimensions are insignificant. There can be only one vertical dimension so that means a 2D object has a horizontal dimension that is insignificant. A liquid would then flow out in this dimension/direction. Either that or the object can hold only an insignificant amount of liquid.

But it is neater to imagine an ideal 2D object that magically is impermeable to fluids than to continually have to hedge with these messy qualifying statements.

---

In 4D tortillas have to be 3D. Who wants a tortilla that can support only insignificant toppings? The natural shape would be a 3D ball (solid sphere). It's flat in the 4th dimension.

 

[pbuk]

I was reading a Chemistry book when I read about the three states of matter. Everyone knows what they are, but I didn't know the simplest way to describe each of the three until I read this book. It said that a solid has a shape and a volume, a liquid has no shape but has a volume, and a gas has no shape or volume. How about something that has a shape but has no volume? It couldn't be plasma, since it doesn't have a shape.

I think that that is a terrible way to describe the three states, in the UK we like to stick to the facts: https://www.bbc.co.uk/bitesize/guides/z93jfcw/revision/1

But I think you are misquoting: I expect that the book talks about fixed or specific shapes and volumes and in particular says a gas has no specific shape or fixed volume.

 

[bob012345]

There are no solids, liquids or gasses, only fields. :)

 

[Filip Larsen]

What has a shape but no volume?​

Taken as an isolated general question, then if shape is considered to be surface area one could argue that an object with a fractal dimension between 2 and 3 would have shape but no volume.

 

[dextercioby]

What has shape and no volume? This is an interesting question in pure mathematics (not the place for it), but in chemistry we can say that a gas has no fixed volume, but has the volume of the enclosure fully surrounding it, while a liquid of the enclosure supporting it against gravity.

 

[bob012345]

What has shape and no volume? This is an interesting question in pure mathematics (not the place for it), but in chemistry we can say that a gas has no fixed volume, but has the volume of the enclosure fully surrounding it, while a liquid of the enclosure supporting it against gravity.

This appears to be true on some scales but not all. The atmosphere around the Earth acts more like the liquid example you give. It is not fully contained yet has a finite volume while a liquid drop may be contained by surface tension and not fill a small enough container. These behaviours are scale dependent.

 

[valenumr]

This appears to be true on some scales but not all. The atmosphere around the Earth acts more like the liquid example you give. It is not fully contained yet has a finite volume while a liquid drop may be contained by surface tension and not fill a small enough container. These behaviours are scale dependent.

I was thinking about the same (atmosphere). It is I'll defined. My favorite part is how it drives flat-earthers nuts though.

 

[dextercioby]

Why would you say the atmosphere has a finite volume? What is it?

 

[TeethWhitener]

I think that that is a terrible way to describe the three states, in the UK we like to stick to the facts: https://www.bbc.co.uk/bitesize/guides/z93jfcw/revision/1

But I think you are misquoting: I expect that the book talks about fixed or specific shapes and volumes and in particular says a gas has no specific shape or fixed volume.

Apparently glasses and plastics aren't solid in the UK...

 

[jbriggs444]

Hoberman sphere.

 

[Filip Larsen]

Hoberman sphere.

I suggested a fractal volume in #25, but that suggestion must have fallen through the cracks ...

 

[sysprog]

Really? I can fill it water.

a bit more specifically (from kleinbottle.com):

A Klein Bottle, although it is a closed surface with no edge, does not enclose any volume. Ignoring the thickness of the walls, my glass Klein Bottles have zero volume because they do not divide the universe into an inside and an outside. They have no boundary.​

 

[Averagesupernova]

...because they do not divide the universe into an inside and an outside. They have no boundary.

I find this all a bit silly. There are other ways to assemble objects together and say this. A Klein jar intersects itself. Big deal. The boundary may be blurred, but I cannot buy that there is no boundary. Maybe the definition of 'edge' needs to be revisited.

 

[sysprog]

I find this all a bit silly. There are other ways to assemble objects together and say this. A Klein jar intersects itself. Big deal. The boundary may be blurred, but I cannot buy that there is no boundary. Maybe the definition of 'edge' needs to be revisited.

You seem to me to be perhaps hastily overly dismissive of the special characterics of the Klein bottle $-$ do you think that we can reconstruct all of topology to make it no longer special? $-$ wouldn't that be a big job?

From nsf.gov:

​

​

A two-dimensional representation of a Klein bottle--a shape with no inside or outside, just one continuous surface. A true Klein bottle needs at least four dimensions; in other words, it can't be blown from glass. Two- and three-dimensional representations like this one exist to help us visualize the topology, but they are not completely faithful to the original shape. The surface cannot be built in two- or three-dimensional space without self-intersection, as shown here with the "handle" passing through the side of the surface.​

 

[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
-
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
-
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
-
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.
-
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.
-
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.
-

-
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.