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Questions tagged [mobius-band]

The Möbius band or Möbius strip is a surface with only one side and only one boundary. The Möbius strip has the mathematical property of being non-orientable. It is named after the German mathematician August Ferdinand Möbius.

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The image $M=\gamma(S)$ is a Mobius Strip, what are the missing points of the parameterization of the interior and what is another parameterization?

Let $$S=\{(r,\theta) | -\frac12 <r<\frac12 , 0\leq\theta<2\pi \}$$ and $$S'=\{(r,\theta) | -\frac12 <r<\frac12 , 0<\theta<2\pi \}$$ so $S'$ is the interior of $S$. Define $\gamma ...
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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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Do parallel, angle, triangle, area etc still apply in Mobius band?

Normal geometry concepts, such as parallel, angle, area, triangle, do they still apply in Mobius band? If not, in which case will they fail to do so? For example, what would three lines on a Mobius ...
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possible number of sheets for a Moebius band covering

Let M be the Moebius band, identified by the quotient of $[0,1]\times [0,1]$ by the equivalence $(x,0) \sim (1-x,1)$. Let $p: M\to M$ be a covering and $n$ its number of sheets. Find the possible ...
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is every path-connected covering of the Moebius strip a Galois cover?

Let $p : E → M$ be a covering of $M$ the Moebius Sttrip such that $E$ is path connected. Is this a Galois covering? My intuition is there must be some non locally path connected coverings that are ...
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Mobius strip with constant negative curvature

Is there any simple model of the Mobius strip with a constant negative Gaussian curvature? There is an example on Wikipedia (https://en.wikipedia.org/wiki/M%C3%B6bius_strip#Open_M%C3%B6bius_band), but ...
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Twisted Ring homeomorphic to Möbius Band?

Is the following parameterized surface homeomorphic to a Möbius Band? ...
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Twisting the unit square n times before gluing( 2.1.6 in G&P).

The question is given below: I have made a Mobius band with a paper and twisted it 3-times but I could not describe what I see it may be a 3 knot shape, could anyone give me a hint for solving that ...
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Visualisation of an orientable surface bounded by the Möbius curve

I'm learning multivariable calculus on MIT OpenCourseWare. When the teacher explained Stokes theorem he mentioned the Möbius strip. He showed it was non-orientable. Then he showed a somehow twisted ...
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Embedding a special 10 vertices graph into mobius strip

In how many different ways we can embed the 10 vertices Graph below into a mobius strip which no two edges intersects and $A,B,C,D$ lies on the circle boundary of the mobius strip respectively?
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Group action of $\mathbb Z$ on infinite strip is homeomorphic to the Mobius Band

I am trying to prove that: Given $X=R×[-1, 1]$ and the action of $\mathbb Z$ as $m(x,y)=(x+m, (-1)^m y)$, prove that the space $X/Z$ is homeomorphic to the Moebius Band. Since there is no ...
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Exercise 10. Groups and Covering spaces. Lima

Let $X$ be the space obtained from the sphere $S^2$ by gluing the north pole to the south pole, let $Y=\mathbb{R}^3-S^1$, where $S^1=\left\{(x,y,0)\in\mathbb{R}^3:x^2+y^2=1\right\}$ and let $Z$ be ...
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Which surface is homeomorphism to mobius strip?

I'm a bit confused since I read many versions of this. First of all, I couldn't understand how to define homeomorphism. Intuitively, I think that if I can deform an object, without tearing it, to ...
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Paradromic rings and Mobius strip

I'm working on a project about the differences between the original Möbius strip, a strip with an additional even number of half-twists, and a strip with an additional odd number of half-twists. This ...
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What is the 'center circle' of a Mobius Band?

What is the 'center circle' of a Mobius band? The question I am working on asks me to cut (literally) a Mobius band in half 'along its center circle.' What exactly does this mean? I know the plane ...
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CW complex for Möbius strip

I was asked to find a CW complex for the Möbius strip with one 0-cell, two 1-cells, and one 2-cell. I can find a CW complex for a Möbius strip with more cells (two 0-cells, three 1-cells and a single ...
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Is there a way to prove algebraically that a Möbius strip is non-orientable?

I am doing my HL Maths coursework on non-orientability of surfaces and am trying to prove whether a möbius strip is orientable or not (of course it isn't) Is there a way to prove algebraically that a ...
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Is this a valid triangulation of Moebius strip?

This is a quick sanity check. I'd like to know if the diagram I've created is a valid triangulation of the Moebius strip or not.
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Contracting a solid torus to mobius band

A mobius band can be obtained from a solid torus by strong deformation retraction. Visually the mobius band is obtained by continuously rotating the diameter in the disk around the torus. If I ...
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Proving that $[0,1] \times X \cong [0,1] \times Y$, where $X$ is Möbius strip, $Y$ is curved surface of cylinder, and $\cong$ denotes homeomorphic

I want to prove that $[0,1] \times X \cong [0,1] \times Y$ where $[0,1] \subset \mathbb R$ has the usual Euclidean topology, $X$ is a Möbius strip and $Y$ is the curved surface of a cylinder. Here, $\...
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Moebius strip orientability

Let $\{M\}$ be the one-sided Moebius strip and $\{MM\}$ the corresponding two-sheeted Moebius strip. Let us assume $\{MM\}$ to be the doubling of $\{M\}$; hence it is orientable. Then, $\{M\}$ is ...
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Write two vector bundles of $S^1$, not diffeomorphic between them

I just want your opinion on this one: "1. Write two vector bundles with rank 1 on $S^1$, such that they are not isomorphic between them". "2. Prove that $S^1\times S^1$ is parallelizable For ...
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Tautological vector bundle over G1(R2) isomorphic to the Möbius bundle explicitily?

TShow that the tautological vector bundle over G1(R2) is smoothly isomorphic to the Möbius bundle. How can I write a isomorphism explicitily?
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Cutting a multiple twisted Möbius strip in half

At my workplace in the weekends we ususally play around with science and younger children. This months topic in mathemathics was a bit of soft topology. We of course made Möbius strips. What we called ...
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Möbius Strip is no orientable

This is an exercise from Do Carmo's Riemannian Geometry book. Let $G=\{Id,A\}$, $C= \{ (x, y, z) \in \mathbb{R}^3; x^2 + y^2 = 1, -1 < z < 1 \}$, where $A(p)=-p$. Define $\frac{C}{G}$ the ...
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Does this method determine if we are living in a Mobius strip or cylinder?

Inspired by this question. Suppose we are living on a two-dimensional walkway with railings. Gravity is always directed towards the surface. We know the walkway is either a cylinder or a Mobius strip....
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Is it possible to determine if you were on a Möbius strip?

I understand that if you were to walk on the surface of a Möbius strip you would have the same perspective as if you walked on the outer surface of a cylinder. However, would it be possible for ...
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graph on mobius strip is nonplanar? [duplicate]

Consider the closed curve on the mobius strip that covers both sides of the fundamental polygon: https://thumb9.shutterstock.com/display_pic_with_logo/1663882/462955276/stock-vector-blue-moebius-strip-...
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Is there a nice picture of the direct sum of two Möbius bundles?

This question is not really about mathematics, but I'm looking for a picture about a mathematical statement. The direct sum of two Möbius bundles is a trivial bundle. Do you know of a nice graphic ...
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No closed surface diffeomorphic to the Moebius trip without boundaries

Consider the Moebius strip $M$ as a quotient space of $\mathbb{R}\times [0,1]$ with opposite lines glued together with reverse orientation. I would like to prove that a closed submanifold $N$ in $\...
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An elementary but formal proof that the Moebius strip is not orientable

We define surfaces as images of $C^1$ functions from $K \subset\mathbb{R}^2 \rightarrow \mathbb{R}^3$, with $K$ compact, and we say a surface is orientable is we can pick continuously a normal vector ...
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Möbius squared, a better illustration or not?

Making a Möbius sling from a slip of paper makes very different effects on the direction that gets the the short ends (North - South) and direction that get the long sides (East – West). The North – ...
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Is'nt the paper Möbius strip a 3d object that is very different from the 2d Möbius band?

The reason why the paper Möbius strip is twice as long as the paper strip used, is, as far as I can underständ it, that paper has two more sides, apart from the two dimensional. A common paperslip cut ...
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What is this logarithm doing in Wikipedia's formula for the unbounded Möbius surface?

According to Wikipedia, the cartesian coordinates of a Möbius strip of width $1$, with its medial circle of radius $1$ in the $(x,y)$ plane, is given parametrically by $$x=(1+\tfrac12v\,\cos\tfrac12u)\...
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Homemorphism from projective plane and Moebius strip

I am looking for an explicit homeomorphism from $ \mathbb{R} P^2=\mathbb{S}^2/(x \sim -x) $ to $M/ \partial M$ with M the Moebius strip and $\partial M$ its boundary.
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Mobius band gluing construction

Can somebody help me understand how the mobius band can be viewed as $\mathbb{R}\times [0,1]/\sim$ where $(x,y)\sim (x+1,1-y)$? (also, if somebody can help me with the appropriate tags)
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How does a Möbius strip not have an area?

The self contained version of this question is: "Given a Möbius strip $M$ do there exist two curves $\gamma_1,\gamma_2$ in $M$ such that $M\setminus\gamma_1$, $M\setminus\gamma_2$ are orientable and ...
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How to define the “full-twist” Möbius strip

I would like to have a precise definition of the topological space called "full-twist" Möbius strip $M$, i.e. a Möbius band with a $360$ degree twist (the usual Möbius strip has just a half-twist). ...
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Is it possible to obtain a torus by gluing 2 Möbius bands?

Is this transform actually possible? Which is gluing edges of two Mobius bands to make a torus? I tried to do it physically with pieces of paper, but I couldn't complete it. The detailed explanation ...
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Fundamental group of two Moebius bands identified on boundary circles

Let $M_1, M_2$ be two copies of the M\"obius band, and let $\partial > M_i$ its boundary circle for $i=1,2$. Let $f:M_1\to M_2$ be an homeomorphism, and consider the topological space $$X=(M_1\...
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Mobius strip to the Klein bottle

From the Youtube, I noticed that the Klein bottle can be sliced into two Mobius strips. Or this experiment is also explained in this following link: http://www.kleinbottle.com/sliced_klein_bottles.htm ...
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Difference between Cross-Cap, Mobius Band, and Real Projective Plane

I understand how the mobius band and the real projective plane are different (the first is a manifold with boundary, for example, while the second is a compact, closed surface and a true manifold). 1....
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How to integrate surface area of the Mobius strip using 'density'?

https://www.quora.com/Can-you-do-a-surface-integral-on-a-mobius-strip According to this link, it is possible to integrate surface area of the non-orientable Mobius strip by using density. However, I'...
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Möbius bundle no global trivialization

I there! I am trying to write down why the Möbius bundle has no global trivialization. I just read that there is none but I want to see a written prove for this. I am not even sure which definition of ...
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Möbius Band Bundle $(Mo,\mathbb{S}^1,\text{proj}_1,\mathbb{R}) $ is not a Principal $\mathbb{R}$-bundle

This is claimed in various places. The problem seems to be with finding a free and transitive group action that has the fibers of $Mo$ as its orbits. I construct $Mo$ as $$ Mo = \mathbb{S}^1 \times \...
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Confusion of classification of closed surfaces

I read that we can distinguish closed topological spaces without boundary up to homeomorphism by orientability and euler characteristic - is this correct? But what confuses me is that the Klein ...
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Which is the actual surface of a Möbius strip?

Usually, the pop explanations of the properties of the Möbius strip say that if you take a pencil and start drawing a line in the middle of the strip, you'll have to make two complete loops to return ...
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Mobius strip/band

What happens when we cut Mobius band at "2/3" of its height? Will we get two pieces which are tangled and one shorter than other? Any help will be appreciated
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Is the Klein bottle homeomorphic to the union of two Mobius bands attached along boundary circle?

Question: Determine whether the Klein bottle is homeomorphic to the union of two Mobius bands attached along their boundary circles. The Klein bottle is the quotient space $$ K=I^2 /{\sim}, \quad (x,...
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In how many dimensions is the full-twisted “Mobius” band isotopic to the cylinder?

There is a question on this site about the distinctions between the full-twisted Mobius band and the cylinder, but I would like to ask something different, so I start a new question. Let us call $C$ ...