What is the 4D Equivalent of a Möbius Strip or Klein Bottle?

[blackholesarecool]

TL;DR

if 2d = mobius strip
3d = klien bottle
what could 4d be??

if 2d = mobius strip
3d = klien bottle
what could 4d be??

 

[jedishrfu]

maybe this:

https://en.wikipedia.org/wiki/Real_projective_space

In mathematics, real projective space, denoted ⁠

⁠ or ⁠

⁠ is the topological space of lines passing through the origin 0 in the real space ⁠

⁠ It is a compact, smooth manifold of dimension n, and is a special case ⁠

⁠ of a Grassmannian space.

 

[PeroK]

TL;DR Summary: if 2d = mobius strip
3d = klien bottle
what could 4d be??

if 2d = mobius strip
3d = klien bottle
what could 4d be??

It's a higher dimensional Klein bottle:

https://www.lehigh.edu/~dmd1/kleinn4.pdf

 

[fresh_42]

TL;DR Summary: if 2d = mobius strip
3d = klien bottle
what could 4d be??

if 2d = mobius strip
3d = klien bottle
what could 4d be??

a) It is Klein bottle, not klien. It is constructed as a quotient space $\{[0,1]\times [0,1]\}/\sim$ where $(0,y)\sim (1,y)$ and $(x,0)\sim (1-x,1).$

b) The Möbius strip is constructed as quotient space $\{[0,1]\times [0,1]\}/\sim$ where $(0,y) \sim (1,1-y).$

Both are quotient spaces of a two-dimensional space. The question about "4D" doesn't make much sense. What do you want to generalize?

 

[blackholesarecool]

connect edges of a cube in some way, wouldnt that work

 

[fresh_42]

connect edges of a cube in some way, wouldnt that work

Sure, you will get something. As you get something by the generalizations of the Klein bottle that @PeroK referenced. Whether this can be called a Möbius strip is questionable. You might get different answers depending on whether you pose this question in topology or in differential geometry, and certainly different solutions depending on how you define the quotient space of the cube.

 

[blackholesarecool]

well i would say Klein bottle is 3d mobius strip, also mobius strip is 2d mobius strip fyi

 

[fresh_42]

well i would say Klein bottle is 3d mobius strip, also mobius strip is 2d mobius strip fyi

The Klein bottle is a surface, not three-dimensional. Have you ever cut a Möbius strip in half along the long side?

 

[blackholesarecool]

ok then klein bottle is 4d and mobius strip is 3d, thats the dimension they are like needed to actually show them, i think thats the dimension they embed in, so what about 5d

 

[Klystron]

TL;DR Summary: if 2d = mobius strip
3d = klien bottle
what could 4d be??

if 2d = mobius strip
3d = klien bottle
what could 4d be??

While I think I understand the OP's question, a Klein bottle is a two dimensional manifold. It only has one side, at least in Euclidean geometry.

This excerpt from Wikipedia may be relevant:

Like the Möbius strip, the Klein bottle is a two-dimensional manifold which is not orientable. Unlike the Möbius strip, it is a closed manifold, meaning it is a compact manifold without boundary. While the Möbius strip can be embedded in three-dimensional Euclidean space R3, the Klein bottle cannot. It can be embedded in R4, however.[4]

Continuing this sequence, for example creating a 3-manifold which cannot be embedded in R4 but can be in R5, is possible; in this case, connecting two ends of a spherinder to each other in the same manner as the two ends of a cylinder for a Klein bottle, creates a figure, referred to as a "spherinder Klein bottle", that cannot fully be embedded in R4.[5]

The aforementioned spherinder Klein bottle likely meets the OP's criteria.

 

[lavinia]

connect edges of a cube in some way, wouldnt that work

yes. This can be done in more than one way. If one generalizes the Klein bottle to a three dimensional flat manifold then one can show that every such manifold can be made from a cube by identifying pairs of faces.

I think that there are ten closed flat three manifolds.

Since the Klein bottle is a non-orientable flat two dimensional manifold, all of the non-orientable flat three manifolds could be considered to be generalizations of it. Or any flat 3 manifold where the identifications of faces of the cube have some half twists could be generalizations as well. Such a manifold could be orientable.

There are a few examples of these manifolds in post #15


- By flat is meant that the manifold can be given a metric whose Riemann curvature tensor is identically zero. Intuitively one can think of flatness as making the identifications of pairs of faces by bending the cube rather than distorting it by stretching or warping. This generalizes the bending of a strip of paper to make a Mobius band. The same goes for the Klein bottle although I am not sure if a flat Klein bottle can be realized in four dimensions.

- Not all closed the 3 manifolds can be embedded in R^4 but I think all can be embedded in R^5.

 

[lavinia]

The diagram in post #4 describes the Klein bottle as a square with opposite edges pasted together, one pair of edges without a half twist and the other pair with a half twist. Otherwise said, the Klein bottle is made from a rectangle by making a cylinder in one direction and a Mobius band in the other.

If one makes a cylinder in both directions one gets a torus. If one makes a Mobius band in both directions one gets another two dimensional surface called the real projective plane. This also in some sense generalizes the Mobius band.

The projective plane IMO is a truly bizarre surface.

 

[lavinia]

A general theorem says that a closed n dimensional manifold that can be embedded in n+1 space must be orientable. For instance the Klein bottle, which is a 2 manifold, can not be embedded in R^3 because it is not orientable. There are proofs of this using homology theory e.g. Alexander Duality.

For two manifolds the reverse is also true. Every orientable 2 manifold can be embedded in R^3. But for higher dimensional manifolds this isn't true. SO(3),for example ,is an orientable 3 manifold that can not be embedded in R^4.

 

[WWGD]

TL;DR Summary: if 2d = mobius strip
3d = klien bottle
what could 4d be??

if 2d = mobius strip
3d = klien bottle
what could 4d be??

Maybe you'd be interested in Tesseracts
https://en.m.wikipedia.org/wiki/Tesseract ?

 

[lavinia]

Here are some reasonable generalizations of the Klein Bottle to a 3 manifold using the suggestion in post #5 to generalize it by identifying faces of a cube. Those which are non-orientable can not be embedded in R^4 but can in R^5.

Starting with Euclidean 3 space, one can consider its quotient by identifying all points whose (x,y,z) coordinates differ by an integer. Then every point is identified with a point in the unit cube(excluding opposite faces) and the quotient space is a three dimensional torus made by pasting opposite edges of the unit cube to each other without twists.

Suppose one additionally identifies points that differ by a translation by 1/2 in the x direction together with a reflection in either the y or z direction or even by a reflection in both directions at the same time. The quotient space is then made from idetifying pairs of faces of the unit cube split in half along the x-axis and gives natural generalizations of the Klein bottle since the reflections are just half twists. If one reflects simultaneously in both the y and z directions the manifold is orientable.

One can also instead add a translation by 1/2 in the x direction together with a reflection in the y direction and then a translation by 1/2 in the y direction together with a reflection in the z direction and this identifies the faces of the unit cube split in half along both the x and y directions. Thus is a sort of double generalized Klein Bottle. This manifold is not orientable.

 

[WWGD]

yes. This can be done in more than one way. If one generalizes the Klein bottle to a three dimensional flat manifold then one can show that every such manifold can be made from a cube by identifying pairs of faces.

I think that there are ten closed flat three manifolds.

Since the Klein bottle is a non-orientable flat two dimensional manifold, all of the non-orientable flat three manifolds could be considered to be generalizations of it. Or any flat 3 manifold where the identifications of faces of the cube has some twists could be generalizations as well. Such a manifold could be orientable.

There are a few examples of these manifolds in post #14


- By flat is meant that the manifold can be given a metric whose Riemann curvature tensor is identically zero. Intuitively one can think of flatness as making the identifications of pairs of faces by bending the cube rather than distorting it in any way. This generalizes the bending of a strip of paper to make a Mobius band. The same goes for the Klein bottle although I am not sure if a flat Klein bottle can be realized in four dimensions.

- Not all closed the 3 manifolds can be embedded in R^4 but I think all can be embedded in R^5.

Whitney Theorem guarantees embedding of $n$-manifolds in $\mathbb R^{2n}$ for smooth manifolds.

 

[lavinia]

Whitney Theorem guarantees embedding of $n$-manifolds in $\mathbb R^{2n}$ for smooth manifolds.

Right but for 3 manifolds one can do it in R^5 rather than R^6

 

[WWGD]

ok then klein bottle is 4d and mobius strip is 3d, thats the dimension they are like needed to actually show them, i think thats the dimension they embed in, so what about 5d

To show them embedded. You can otherwise display them with self-intersections.

 

[WWGD]

Right but for 3 manifolds one can do it in R^5 rather than R^6

Even non-orientable ones, or you're assuming orientability?

 

[lavinia]

well i would say Klein bottle is 3d mobius strip, also mobius strip is 2d mobius strip fyi

The Klein bottle is a two dimensional surface. However it can be filled to make a three dimensional manifold of which the Klein bottle is the boundary.

The Mobius strip is also a 2 dimensional manifold but unlike the Klein bottle it has a boundary. Its boundary is a circle.

 

[lavinia]

Even non-orientable ones, or you're assuming orientability?

yes even non-orientable ones

 

[lavinia]

ok then klein bottle is 4d and mobius strip is 3d, thats the dimension they are like needed to actually show them, i think thats the dimension they embed in, so what about 5d

Any closed orientable surface such as a sphere or a torus can also be embedded in three dimensions.

No non-orientable closed surface can be embedded in three dimensions but all can be in four. The Klein bottle is not the only example.

If one wants a manifold that cannot be embedded in four dimensions but can in five, one would need a three or four dimensional manifold rather than a surface. Among three dimensional manifolds there are many examples not all of which need to be non-orientable.

 

[lavinia]

Perhaps this way of describing how flat 3 manifolds naturally generalize the Mobius band and the Klein bottle will be more clear.

A cylinder may by thought of as a circle with lines pointing perpendicular to it. If one starts with the cylinder then identifies all pairs of points on it that are a 180 degree rotation together with a reflection of the line, one gets a Mobius band.

For a Klein bottle, start with a torus viewed as a circle of circles and again rotate by 180 degrees then reflect the perpendicular circle along its vertical axis.

Now start with a three dimensional torus viewed as a torus of circles.

1)Rotate the torus by 180 degrees then reflect the circle along its perpendicular axis. This turns out to be just a Klein bottle of circles.

2)Now view the 3d torus as a circle of tori. Rotate the circle 180 degrees then interchange the two axes of each perpendicular torus.

3)Or reflect each of the axes of the perpendicular torus.

4) A more complicated example is to rotate the circle by 90 degrees then map the first axis to the second without a reflection but the second axis to the the first with a reflection.

This same idea generalizes to four dimensional tori and all tori of higher dimensions.

A more mathematical viewpoint:

If one views the circle algebraically as R/Z, and the torus as R^2/Z^2 then the Klein bottle is the quotient of the torus by the the involution (x+1/2,-y).
For three manifolds one starts with the 3 dimensional torus R^3/Z^3
For the three manifold 1) the involution is (x+1/2,y,-z) which looks just like a Klein bottle cross a circle.
For manifold 2) the involution is (x+1/2,z,y).
For manifold 3) the involution is (x+1/2,-z,-y).
For manifold 4) it is (x+1/2,-z,y) and this is not an involution but is a rotation of order 4.

 

[lavinia]

well i would say Klein bottle is 3d mobius strip, also mobius strip is 2d mobius strip fyi

A Klein bottle is not a 3d Mobius strip. It is a 2d surface just like a Mobius strip. The difference is that a Mobius strip has a boundary circle and the Klein bottle has no boundary. If one glues two Mobius strips together by pasting their boundary circles to each other one gets a Klein bottle.

While the abstract Klein bottle can not be incarnated in three dimensions there are plenty of 2d surfaces without boundaries that can, for instance the sphere.

If what you are getting at is that the Mobius band is a 2d surface that cannot be embedded in 2 dimensions and the Klein bottle is a 2d surface that cannot be embedded in three dimensions what are analogous manifolds that can not be embedded in four dimensions? It turns out that such manifolds cannot be surfaces. All surfaces can be embedded in four dimensions.

One might say that a 3d analogue of the Klein bottle is the solid Klein bottle which is a 3d manifold whose boundary is the Klein bottle. It is easy to construct an example of such a manifold. Then in the same way as one glues two Mobius bands together to make a Klein bottle one might glue two solid Klein bottles together along their Klein bottle boundaries to make a generalized Klein bottle as a three dimensional manifold.

It would be interesting to think about what such a manifold might look like and whether there is more than one way to make one.

More simply one might start with a torus cross a line segment which could be thought of as a 3d analogue of a cylinder then paste it to itself after a 180 degree rotation and a reflection of the line segment. This could be thought of as a 3d Mobius strip.If instead of a line one crosses the torus with a circle then one gets an analogue of the Klein bottle. This is the manifold 1) in post #23. It can not be embedded in 4 dimensions but can in 5.

I would like to say that this 3d analogue of the Mobius strip cannot be embedded in 3 dimensions but can in four but haven't thought it through. What do you think?

 

[mathwonk]

reference for post 21:
https://projecteuclid.org/journals/...nifolds-imbed-in-5-space/bams/1183526924.full

 

[Hornbein]

The Klein bottle is a 2D object embedded in a 4D space. I would quibble that in 4D surfaces are 3D so there are no 2D surfaces. However the English language has no word for such a thing, not surprising as such does not exist in our Universe. I can be explicit and say "2D surface." Maybe "semisurface" would be better. But in a four dimensional Euclidean space I'd say that 2D things have properties similar to 1D things in our Universe. Knife edges have to be 2D. In 4D it is better to get away from thinking about 2D things as surfaces. Hmmm, "2D curve" might be the ticket. If it doesn't curve then it's a 2D line. 2D lines would be used to do things like demarcate the boundary of a tennis court.

In 4D the simplest mathematical knot is our 2-sphere, analogous to the circle here in mundane 3D.

 

[WWGD]

The Klein bottle is a 2D object embedded in a 4D space. I would quibble that in 4D surfaces are 3D so there are no 2D surfaces. However the English language has no word for such a thing, not surprising as such does not exist in our Universe. I can be explicit and say "2D surface." Maybe "semisurface" would be better. But in a four dimensional Euclidean space I'd say that 2D things have properties similar to 1D things in our Universe. Knife edges have to be 2D. In 4D it is better to get away from thinking about 2D things as surfaces. Hmmm, "2D curve" might be the ticket. If it doesn't curve then it's a 2D line. 2D lines would be used to do things like demarcate the boundary of a tennis court.

In 4D the simplest mathematical knot is our 2-sphere, analogous to the circle here in mundane 3D.

In the usage I'm familiar with, a surface is a 2-dimensiomal manifold and we talk about n-manifolds embedded in k-space, with n<k.

 

[fresh_42]

In the usage I'm familiar with, a surface is a 2-dimensiomal manifold and we talk about n-manifolds embedded in k-space, with n<k.

And 3D in 4D, i.e. n-1 in n is a hypersurface. Hence we have: string, surface, hypersurface in 4D.

 

[mathwonk]

Hornbein, for what I believe you are thinking of, I use the term "codimension". So curves have codimension one in the plane, as do surfaces in 3 space, and 3 folds in 4 space; all are codimension one sub-varieties, also called hypersurfaces, as fresh_42 notes.

 

[martinbn]

And 3D in 4D, i.e. n-1 in n is a hypersurface. Hence we have: string, surface, hypersurface in 4D.

You would use "string" instead of "curve"?