Understanding Kleins Bottle: Explained in Layman's Terms

[srijithju]

Can anyone explain in laymans terms what a Kleins bottle is . I have encountered the definition that it is a non orientable surface , that it is a 2 dimensional manifold in 4 dimensions.

I don't have a clue what a non orientable surface is , or what a manifold is . I also don't have any idea how to think in 4 dimensions.

Could anyone help here . Maybe if its not possible to expain in simple terms , you could provide links from where I can learn the math required.

 

[Bill Simpson]

I'll give it a shot.

First, imagine you life in a 2d world on a flat sheet that goes as far as you want, but you cannot imagine ever being able to get into the 3rd dimension. Now create a ribbon of clear plastic say 1cm by 20 cm. Put an arrow across the ribbon at each end so both arrows point in the same direction. Now gently bend the ribbon into a circle, but you must stay completely in the 2d world, so you have to stretch it. You should see when the opposite ends touch the arrows are next to each other and still pointing in the same direction. You have made a ring. If you take a tiny scrap of clear plastic, draw a tiny picture of a left hand on it, lay it on the ring and slide slide or rotate it any way you want you always still see it is a left hand. So it remains "oriented" and doesn't become a right hand.

Suppose come crazed math person comes along and asks you to make a ring, BUT have the arrows point in the opposite direction when they come together. You try this lots of different ways, but you cannot find any way to do it. Then the math person asks what happens if you lift one end of the ribbon off into this thing called "the third dimension", turn it 180 degrees, and bring the end back into the plane. Now when the ends touch the arrows are pointing in the opposite direction. But you live in a 2d world and cannot imagine where the third dimension is. However, if you cheat for a minute and do this in 3d by twisting the ribbon you see you can do this. That thing is called a "Mobius strip." And if you take the tiny picture of the left hand and slide it along the ribbon after you have made a complete circle, you are looking through the back of the ribbon and picture, BUT if you do this carefully and look suddenly you realize it is a right hand now. It is no longer "oriented."

Now we bump everything up a dimension, from 2 or 3 to 3 or 4. Take a hose. Put an arrow around each end so both arrows point the same direction. Bend the hose into a donut, sometimes called a torus. The arrows touch and are going the same direction. And the math person asks if you can do this with the arrows ending up going the opposite direction. You try, you can't. You make a tiny clay model of your left hand, put it inside the tube, let it go round and round the ring and it remains a left hand. It is oriented.

Suppose you take one end of the tube, lift it up into the 4th dimension, "flip it over" and bring it back into the 3rd. When the ends meet the arrows are going in opposite directions and you have a klein bottle. Put your little clay model inside, let it slide around and after one trip it becomes a right hand. It is not oriented.

Get some supplies, actually try this, really try it, well, everything up to the "lift it up into the 4th dimension" step, that is optional and difficult to do. And carefully compare the steps to see if you can imagine how this works.

Charles Howard Hinton, mathematician about a century ago, incredible memory. He created a "victorian toy" that was a box of colored cubes with instructions. Those cubes were "3d cross sections" of a 4d cube. The colors showed which faces mated with which faces. Old university libraries may still have a few of Hinton's books. I read those 40 years ago. You might still be able to find and read those, but treat them very gently. But I never did find a table in the books telling me how to construct the box of cubes and color them to match Hinton's. Rudy Rucker, recent mathematician, wrote a few books on the fourth dimension and supposedly constructed his own set of matching cubes. But I never did find a table in his books telling me how to construct the cubes to match. Hinton claimed that with the directions and lots of practice you could learn to think in 4d. Either he actually could or he just got very very good at solving problems in 4d in his head.

If anyone is aware of an antique shop somewhere with Hinton's cubes and instructions I'd like to know. I'd pay a serious price for that.

 

[Hurkyl]

A Klein bottle is a two-dimensional shape (like the surface of a sphere), but one that cannot exist in the usual three-dimensional Euclidean space you're familiar with.

If you google "Klein bottle" you'll find pictures of Klein bottles -- you just have to imagine that the bottle doesn't actually intersect itself.



People these days know about old video games, right?

The world of pac-man is played on a cylinder -- if you go off the right-hand edge of the screen, you appear on the left.

If ghosts^* get flipped upside-down every time that happened, then the world would be a Mobius strip.

*: Pac-man is symmetric, so you couldn't tell if he got flipped upside-down



The world of asteroids is played on a torus (the surface of a donut) -- if you go off the right-hand edge, you reappear on the left. If you go off the top, you reappear on the bottom.

If you were flipped up-side down every time you went off the right side and reappeared on the left... but you were not flipped every time you went off the top and reappeared on the bottom, then that world would be a Klein bottle.

 

[srijithju]

Thank You Hurky and Simpson very much for giving nice examples and helping understand the things .

But I disagree that when you go round the mobius strip once , the shape disorients.
I feel that by going round the strip once would not leave you in the same place . You have 2 go round the strip twice to reach the same place , after which there is no disorientation.

When looking at it from 3 dimensions you may feel like you reached same place after one round, but is this true in 2 dimensions ?

To illustrate what I am saying consider a sphere balloon . I don't have any mathematical knowledge of curved geometries , but I am aware of the fact that you can assume the surface of balloon to be a curved 2 D geometry unlike the eulcidean geometry , in the sense that if you start from a point then after a finite distance traveled you end up at same point.
Now let's consider 2 diametrically opposite points of sphere A and B . If you squeeze balloon by apllyin pressure on A and B , your shape will change from a sphere , but the rule is still valid ( that if you start from a point then after a finite distance traveled you end up at same point). Eventually you can squeeze it such that A and B touch .

Now on the 2 D surface of sphere if I travel from A to B do I reach back at where I started. Well it looks like I reach same place from 3 dimensions but in 2 D i still have to complete one revolution of sphere ( well its not a sphere now) to reach my original point . I don't think A and B are same point in 2 D .

They are same point on different sides in 3D , but in 2D there is no sides .

If you assume that A and B are same points in 2 D you will have to bring in logically paradoxical concepts like disorientation into picture .

Consider 2 dogs jack and jill which are at point A . let us say jill travels to B .
Now according to your perspective , you are saying jack and jill can see each other but they look disoriented to each other . If one was clockwise , other would look anticlockwise.

What I say is that jack and jill cannot see each other as they are at different points in 2D space.

Please explain what i think is wrong or flawed

 

[Hurkyl]

You have 2 go round the strip twice to reach the same place , after which there is no disorientation.

That's because you're visualizing it wrong.

A plane doesn't have a front and a back -- it is just a plane.


A sheet of paper isn't a plane -- it is a (really thin) solid rectangular prism. We can draw a point on one side, and draw a point at "the same place" on the other side, and see them as different points.

But if we are using the sheet of paper to help us visualize a plane, we are supposed to think of those two points as really being the same.


The same is true of a Möbius strip. If we cut a strip of paper to make one, we don't really have a Möbius strip -- instead, we have a (rather squished) solid torus, the same sort of shape as a donut. But if we want to use this strip of paper to help us visualize a Möbius strip, we have to think of the opposite sides as being the same -- so once around takes you back where you started, but everything that stayed put is now reflected in orientation as compared to you.

 

[srijithju]

That's because you're visualizing it wrong.

A plane doesn't have a front and a back -- it is just a plane.


A sheet of paper isn't a plane -- it is a (really thin) solid rectangular prism. We can draw a point on one side, and draw a point at "the same place" on the other side, and see them as different points.

But if we are using the sheet of paper to help us visualize a plane, we are supposed to think of those two points as really being the same.


The same is true of a Möbius strip. If we cut a strip of paper to make one, we don't really have a Möbius strip -- instead, we have a (rather squished) solid torus, the same sort of shape as a donut. But if we want to use this strip of paper to help us visualize a Möbius strip, we have to think of the opposite sides as being the same -- so once around takes you back where you started, but everything that stayed put is now reflected in orientation as compared to you.

Yes , I can see that a sheet of paper is not a perfect analogy and the 2 points on 2 different sides are distinct if we look at them as being part of a paper .

I also agree with you that if the strip of paper were of 0 thickness and it was transparent also let's say , the 2 points on opposite sides would exactly coincide. But alas , this is true only as long as we view it from a 3 dimensional space. When I move to the 2 dimensional space these points are distinct.

What I am claiming is that 2 points that coincide in 3 dimensions , need not coincide in 2 dimensions.

Lets get back to my sphere example where i squished the sphere such that 2 of its opposite points now coincide ( when viewed from 3 dimensions ) . Let say you have a point Q , 1 cm away from A on same side as A ( left side let's say) Also there is a point R , 1cm away from B on same side as B ( right side ) . Let us say the circumference of the sphere is very large compared to these distances.
.

Now if I move into 2 dimensions , you are saying that the distance between point Q and point R is only 1+1 = 2cm ?

But the problem is that there is no way I can go from Q to R covering only 2 cm . The path from Q to A , then A to B ( 0 distance ) then B to R is invalid. There is no way an object of finite size could pass from A to B . There is not even a way of a point object of 0 size could go from A to B in 0 distance . In 2 dimensions points A and B are completely distinct .

Let me conduct an experiment in this 2d world and try to measure the distance between A and B . I would be using light rays to measure distance . Now light emitted from A would travel in a straight line around the circumference of the sphere to reach B ( this is a straight line in my 2d world although it may not look straight from 3dimensions ) . There is no way light travels 0 distance from A to B . Hence any distance measurement I do based on this light would lead me to conclude that A and B are very far apart.

As you can see , it is impossible to prove from within 2 dimensions that A and B are the same point . Hence A and B are distinct in 2 dimensions.

On a similar line of reasoning as above , I also have lots of doubt regarding the curvature of space time and expansion of space time.

For example we say space is expanding because we see galaxies moving apart .I Just don't get this idea. If space were expanding , there is no way we would know for we too are a part of this space.

As regards to bending of light rays due to curvature of space. - If light rays bend due to curved space , we cannot see the light to be bending because we too are a part of this curved space , for us the light rays always moves in a straight line.


What I want to say here is that from a particular N dimensional space / geometry, if it is impossible to prove certain facts regarding the nature of this space , from within this space ; and you have to resort to looking at this space from higher dimensions to prove things about it , then what you may be seeing ( proving ) in higher dimensions are not necessarilly true in the lower N dimensional space from which you perceive things, because their truth is immaterial in our N dimensional space.

So though you think that the 2 sides of mobius strip coincide from a 3 dimensional space , this coincidence has no observable counterpart from within the 2 dimensional realm of the mobius strip.


There are of course things that you could prove from within this mobius strip / my sphere example that also are readily seen from higher dimensions . For example the fact that the geometry is a closed one is evident because I can reach the same point from where i start by moving in a straight line.

 

[Hurkyl]

What I am claiming is that 2 points that coincide in 3 dimensions , need not coincide in 2 dimensions.

Two points either do coincide or they do not. There is no in-between.

Your visualization is obviously not helping you understand the idea of identifying points. It either needs to be discarded entirely, tweaked to eliminate the features confusing you, or you have to adjust what information you are extracting from it.


Incidentally, you might find it interesting to note that you can construct the same surface you're considering by taking the surface of a donut, and deforming it by decreasing the size of the donut hole until it vanishes.

But the problem is that there is no way I can go from Q to R covering only 2 cm . The path from Q to A , then A to B ( 0 distance ) then B to R is invalid.

Invalid in what sense? It is a continuous path, and measurable by the length measure (with length 2 cm).

Let me conduct an experiment in this 2d world and try to measure the distance between A and B.

One thing to understand is that this point is a singularity. Physics, as you know it, explicitly forbids space-time from having such points, so it's unlikely you can get much out of your physical intuition.

For example we say space is expanding because we see galaxies moving apart .I Just don't get this idea. If space were expanding , there is no way we would know for we too are a part of this space.

Except we can measure it, through redshift.

Anyways, our rulers don't expand because they are being held together by the electromagnetic force. We don't expand for the same reason. The Earth-Moon system doesn't expand with the rest of the universe, because gravity keeps them together. The same for the solar system, I believe for our galaxy, and I imagine even the local galaxy cluster is gravitationally bound. But really you should ask someone better versed in cosmology (read: in the appropriate forum) about what scales expansion finally becomes significant enough to overcome the various forces holding objects together.

As regards to bending of light rays due to curvature of space. - If light rays bend due to curved space , we cannot see the light to be bending because we too are a part of this curved space , for us the light rays always moves in a straight line.

Wrong and right. Light always moves in straight lines. But when we draw maps, straight lines are usually curved.

What I want to say here is ...

Agreed. This is the reason differential geometry exists -- to study the intrinsic geometry of various shapes without making any reference to any sort of external geometry. And because of the way general relativity uses differential geometry, it cannot possibly make use of "higher dimensions". The curvature that GR describes is purely intrinsic, and can actually be measured, in principle. A two-dimensional purely spatial analog is Gaussian curvature.


If you're interested in general relativity, you might do better specifically asking about it, and possibly in the relativity forum. (though straight differential geometry would be appropriate in the math forums)


Incidentally, here is a badly drawn picture of what one might see in a universe that has the shape of a Möbius strip prism, with the left and right edges of the universe painted black (they're the same side, of course), and the floor of the universe painted in colors. You are a brown stick figure with your right hand raised and looking slightly downward. The universe is small, so you can see the back of your head (several times).

 

[srijithju]

One thing to understand is that this point is a singularity. Physics, as you know it, explicitly forbids space-time from having such points, so it's unlikely you can get much out of your physical intuition.

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Ok .. so going by this you would like to say that the mobius strip has all points as 'singularity' ??

Hence the laws of physics as we know them are no longer valid in this mobius strip.

Well I just think we should be open to two different perspectives, or strings of thought .

First : As you mention , the 2 faces of mobius strip are the same even in the 2D world of the mobius strip . This perspective means that physics laws cannot be formulated when we have so called singularities .

Second : As i am suggesting , consider that physics holds good , but geometry is interpreted differently which is opposite of your notion ( and I think your notion is the one that is commonly accepted by mathematicians ) . So with these so called new rules of geometry , you would not have to worry about physics breaking down. Because there is no singularity . The so called points of singularity are distinct points after all.

By the way , thank you very much for your mobius strip universe picture . Assuming what you are saying is in fact correct , your picture clearly demonstrates how I would see reflections of myself and that all the odd reflections are disoriented ( normally a mirror will show your left hand as right and so on , but in the mobius strip nearest / odd reflection , I see right hand as right and left as left ) .

I am also starting to see some sort of consistencies within the 2d geometry from your picture , but still I seem to be having lot of difficulties .

if I had only looked at this from your pictures perspective I may have readliy accepted the fact that one revolution of the circle leaves me at the same place , but somehow my preconceived pictures still seem to point that your picture is an illusion - I would only be able to see the even numbered correctly oriented reflections.

Moreover the laws of physics are some fundamental laws that should not break down at any special points like 'singularity' . The only reason I can see why these are breaking down are due to limitations in geometry to explain these points. As long as the maths is consistent , natures laws which are formulated in maths should also be consistent shouldn't they ?

 

[Hurkyl]

Ok .. so going by this you would like to say that the mobius strip has all points as 'singularity' ??

Er, no. Well, I suppose there can be Möbius strips with singularities (e.g. if you fold it to have a sharp corner), but none of them have the kind of singularity your sphere had.

If you pick any point on the Möbius strip and a sufficiently small distance, the set of all points within that distance of your chosen point is a disk.

For the pinched point of your sphere, it's not.


As embedded in 3-space, a (small) ball around a point is cut into two hemispheres. These hemispheres define different "sides" locally! But that doesn't mean that one point is actually two points!



I think what you're doing is you're looking at this two-dimensional surface in Euclidean space (e.g. the Möbius strip, or the pinched sphere), but what you are actually thinking of is a covering space.

You have in your head a sphere which is immersed in three dimensional space in an odd way such that two of the points of the sphere have been drawn at the same place. You have in your head an annulus which is immersed in three dimensional space in such a way that once around the annulus goes twice around the Möbius strip.

When we talk about the Möbius strip, and we embed it in Euclidean space to visualize it, we are thinking about a two-dimensional shape that looks exactly like the image drawn in Euclidean space. However, you are thinking about the covering space -- you have the annulus in your head.

That's why you see things differently -- you are thinking of something truly different!




Now, covering spaces are an important tool. Many shapes are more easily studied by lifting to a covering space. For example, the line covers the circle, e.g. by sending the point x on the line to the point on the circle with angular position x. If you are interested in functions whose domain is the circle, they are conveniently described as functions on the line that are periodic with period 2pi.

So don't forget the idea you have, but save it for the right time and place.

 

[srijithju]

I think what you're doing is you're looking at this two-dimensional surface in Euclidean space (e.g. the Möbius strip, or the pinched sphere), but what you are actually thinking of is a covering space.

I could not understand what you are saying completely . I tried looking up what a covering space was but I didnt understand much in the definitions as they were framed in mathematical language.

Anyway from what you have said I assume that a covering space is something like a mapping from one object /shape to the other . Or its rather a function whose domain and range are spaces / shapes ( or more specifically the domain contains a set of points in a certain space , and the range another set of points in another space).

If you think I am trying to map the sphere to a mobius strip, that is not at all what I am doing.

The only reason I brought up the sphere example was because if an object moves from A to B , then it gets disoriented just as in the case of mobius strip . But on the sphere there is only one point / points - A and B where this disorientation can happen .

But on the mobius strip disorientation is possible on each and every point . This is why i assumed all the points of mobius strip as singularity.


Maybe to give another example consider a cube in 3 dimensions . The surface of the cube could be considered a 2 dimensional space. It is a closed and continuous space.

Yes this is square when viewed from 3d does have edges , but I if i were completely living in the 2d world of the surface of cube , I would not be able to directly observe edges.. ( I would be able to detect their presence by measurements though )

Now let's squish this sphere by constantly reducing its height . We keep doing this till the height is 0 . then every point on this surface becomes a singularity.

It might look like a finite plane from 3d , but its not a plane in 2d .. it actually wraps around itself at the 'boundaries' that we see from 3d euclidean space.

Now if a person were to start at one side of square and the go to the other end of square , he would disorient in shape... of course he would have to go around these 'edges/boundary' and make an infinite turn there ... but as far as he is concerned , he is still moving continously forward.


this shape is pretty similar to a mobius strip in that each point ( except the boundaries ) are capable of disorientation , but of course there are vast differences between the two ( for eg mobius strip has no edges when viewed from 3d , the symmetries of the 2 shapes are different etc. )


I am curious to know from your perspective , would you consider all the points on this square as a singularity , or would you only consider the edges as singularity ??

 

[Hurkyl]

Now let's squish this sphere by constantly reducing its height . We keep doing this till the height is 0 . then every point on this surface becomes a singularity.

It might look like a finite plane from 3d , but its not a plane in 2d .. it actually wraps around itself at the 'boundaries' that we see from 3d euclidean space.

If the height is zero, then it really is a plane, and nonsingular. In order for the two faces to be distinct, there actually has to be a positive distance between them.

(Of course, a three-dimensional someone standing on the plane could talk about there being two different sides. A two-dimensional someone living in the plane could not)


The result is non-singular, because if you pick any point, the set of points near it is a disk. Actually that's not quite true -- if you pick a point on the edge, the set of points near it is a half-disk. So in the stronger sense, the edges of the square are singular points.

It is useful to weaken the notion of singularity so that edges are non-singular (to allow for half-disks). The stricter notion of a non-singular space is a "manifold", the weaker notion is a "manifold with boundary".

(Topologically, the corner of filled solid square is the same as any other point. I won't talk geometrically, because I'm not sure I can get the terms exactly right)



It is, however, a singularity of something else -- this is obviously a point where the transformation breaks down. So we might say the plane is a singularity of the "squishing the cube" transformation.

 

[srijithju]

Thanks a lot Hurky for your explanations .. I guess I am discussing a topic which I don't really understand about .
I still am curious why a sphere squished at 2 opposite points would eventually give a singularity , but if squished on all points ( which would result in the formation of a hemisphere eventually ) has no singularities . To me in the first case 2 points become one ...in the second case .. infinite pairs of points become one separately.

I guess I should try and understand topology from some book .. maybe things would become clearer then .