# Questions tagged [klein-bottle]

The Klein bottle is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. It was first described in 1882 by the German mathematician Felix Klein.

90 questions
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### How can I plot an animated Klein Bottle in Graphing Calculator 3D? [on hold]

How can I plot an animated Klein Bottle in Graphing Calculator 3D? I mean, is it possible to change the shape to cut it and see Moebious bands?
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### How can I make a Klein's bottle 2x wider?

I'm using this formula to draw a parameterized Klein's bottle, with $a = 6$, and $b = 16$. It looks pretty. Now I want to "wrap" this shape into another, but a $c$-times bigger one. This task is easy ...
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### Klein bottle as two Möbius strips with fundamental polygon

My question is the following. If we operate with the fundamental polygon of a Klein bottle in order to obtain two Möbius bands, really you don´t obtain two new Klein bottles? I think that the ...
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### Is there a non-trivial covering of the Klein bottle by the Klein bottle?

Let K be the Klein bottle obtained by the quotient of $[0, 1] × [0; 1]$ by the equivalence relation $(x, 0) ∼ (1 − x, 1)$ and $(0, y) ∼ (1, y)$. Is there a non trivial covering of $K$ by $K$? The ...
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### Homology of Klein bottle with Mayer-Vietoris

I'm practicing with using the Mayer-Vietoris sequence, and found this computation. I thought it would be a good exercise to try cutting the Klein bottle into two cylinders, instead of into two mobius ...
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### How Klein Bottle in R⁴ does not have self intersection?

This is the screenshot from Feko's book on Differential Geometry. In 1.5.10 in the given hint how he deduced that in R^4 the Klein Bottle has no self intersection? Also,in 1.5.11, what does square ...
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### The fundamental group of Klein bottle/{3pt}

The fundamental group of Klein bottle has been well discussed in many materials, I'm think if it is requested to calculate the fundamental group of Klein bottle/{3pt} or even more generally Klein ...
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### Twisted Strip to boundary of surface A\disc\disc is equal A#K\disc

I have struggles to understand the outline of the proof given in "Donaldson-Riemann surfaces". I work with this source: http://wwwf.imperial.ac.uk/~skdona/RSPREF.PDF. On page 21 bottom he mentions the ...
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### Topological embedding of Klein bottle into $\mathbb{R}^4$ that projects to usual “beer-bottle” surface in $\mathbb{R}^3$?

What is an explicit topological embedding of the Klein bottle into $\mathbb{R}^4$ whose projection, of some sort, down to $\mathbb{R}^3$ gives the usual "beer-bottle" immersed surface (https://upload....
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### Using polygonal representations to show that a Klein bottle can be constructed by connecting two Mobius strips

I am reading introductory material w.r.t. Differential Geometry and I came across a problem that asks to show that a Klein Bottle can be seen as the connected sum of two Mobius strips using polygonal ...
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### Does Day and Night on a Klein bottle have a steady state?

Place a $m \times n$ ($m,n \ge 3$) square grid on a Klein bottle. On each square, we select an arbitrary non-mirror symmetric marker, and arrange them on the Klein bottle in some way. This arrangement ...
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### From $[0,1]\times [0,1]$ construct the Klein bottle

The argument by Wikipedia is enough or complete for the solution of the exercise? Exercise: From $[0,1]\times [0,1]$ construct the topological space known as Klein bottle. Wikipedia says ...More ...
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### Why an ordered pair gives a torus?

I saw a video (Who cares about topology) that explained the inscribed square problem. The problem say's that you have a simple closed loop in the plane, prove that there's at least one square such ...
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### Simplicial complex underlying space homeomorphic to torus/klein bottle

I have read a lot of books and a lot of posts in the Internet and I still can't understand few problems. Professor, who presents us a lecture "Algebraic topology", is doing his job like he must to do ...
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### Whether there exists some compact manifold whose continuous map must have a fixed point?

Whether there is some compact manifold $M$ such that $f:M\rightarrow M$ is continuous $\Rightarrow f$ has a fix point. I mean that any continuous map from $M$ to $M$ must have a fixed point. I ...
1answer
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### Klein Bottle as the connected sum of two projective planes

Can someone explain intuitively why the Klein bottle is diffeomorphic to the connected sum of two projective planes? I can do this using origamis/fundamental graphs )w/e they are called. is it ...
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I know that homeomorhic spaces are homotopic, but am not sure if this applies, since I think they are not homeomorphic due to orientability. I know $f$ and $g$ are homotopic if they represent: X$\... 1answer 208 views ### Connected sum of two “same” Klein bottles If I take sphere and remove two open disks from it and on the boundary of that space I make identification like on the picture, what do I get? Are both of those objects Klein's bottles? This is what ... 1answer 280 views ### Making a Klein bottle from 2 Möbius bands I thimk this can be done by idemtifying points on the boundary but I am not sure how to show this Any ideas? E.g. By drawing nets.. 1answer 58 views ### Klein Bottles in the Levine traffic model The Biham–Middleton–Levine traffic model has recently fascinated me. I started learning about it on the Wikipedia Page found here. One way to run this simulation is on a Klein bottle surface. When ... 0answers 63 views ### Why is the immersed image of the Klein bottle$S^2$with three closed discs identified? Someone said that there are three discs glued together on the Klein bottle self-intersecting itself in a circle. Here is a picture: They said that one from the "near" side of the tiny connecting tube,... 1answer 228 views ### Properties of a closed solid klein bottle? Okay, so I'll denote the topological space of the klein bottle as$Kb$(because I don't know what the proper notation is). I'm curious about a closed version of the solid klein bottle, which I believe ... 0answers 122 views ### Why is$\mathbb{R}^2/G$homeomorphic to the Klein bottle? Let$G$be the group of transformation generated by$a,b:\mathbb{R}^2\to \mathbb{R}^2$where$a(x,y)=(x+1,y-1)$and$b(x,y)=(x,y+1)$. We note than$bab=a$and that$G$acts properly discontinuously ... 0answers 473 views ### The Klein bottle can be immersed in$\mathbb{R}^3$and embedded in$\mathbb{R}^4$How can one show that the Klein bottle can be immersed in$\mathbb{R}^3$and embedded in$\mathbb{R}^4$? We define the Klein bottle as the quotient space of$I^2=[0,1]\times [0,1]$under the relation ... 1answer 151 views ### How can I write Klein bottle as an adjunction space? I want to find the homology groups of the Klein bottle by Mayer-Vietoris. For this I want to describe the klein bottle as an adjunction space. I think it can written as a pushout$S^1\cup_f D^2$but I ... 1answer 921 views ### Computing Klein bottle's cohomology ring in$\mathbb{Z}\$

Well I've been struggling with this one. This is the picture of the Klein Bottle. It has two triangles (U upper, V lower), three edges (the middle one is "c") and only one vertex repeated 4x. So my ...