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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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Knowing that a Mobius strip is described by parametric equations how can I introduce a thickness element to show a constructable model? [closed]

I have tried various graphics packages to model a single twist Mobius strip, however none allowed me to identify a way of making the strip anything other than 2D. As an Electrical engineer my math is ...
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Proof construction for non-orientability of Mobius strip

I'm following Spivak's comprehensive guide to differential geometry and I'm getting a bit stuck on a calculation about orientability. The example is showing that the Mobius strip considered as a line ...
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Disjoint curves connecting specified pairs of points on the edge of a Möbius strip

Suppose $2n$ points along the edge of a Möbius strip are labeled by some given permutation of $[1,1,2,2,\dots,n,n]$. Is there a criterion or algorithm to determine whether we can draw $n$ non-...
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There is an embedding of the Mobius strip in $\mathbb{R}^3$, what is it?

Let $M= S^1 \times ]0,1[$. Define the Mobius strip to be $N:= M/ \mathbb{Z}_2$ where the non trivial element of $\mathbb{Z}_2$ acts on $M$ by sending $(x,r) \in M$ to $(-x,-r) \in M$. I red that the ...
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Surface area of Möbius strip

Since Möbius strips cannot be oriented, the calculation of surface area is meaningless; perhaps that is the reason why I have never seen such problem. However, the following φ is a parameter ...
Blue Various's user avatar
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Relative cohomology of open Mobius strip

Let $M=[0,1]\times (0,1)/\sim$ be the open Mobius strip, and consider the compact subspace $A=[0,1]\times[\frac{1}{3}, \frac{2}{3}]/\sim$. I am trying to compute the relative singular cohomology $H^{\...
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Topology of Mobius Strip

The mobius strip $M$ is topologically distinct from the cylinder $S^1\times I$ where $I$ is a finite segment of $\mathbb{R}$ (namely, one cannot be deformed into the other without cutting and pasting)....
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Spivak' proof that möbius band has not trivial tangent bundle.

I'm reading Spivak's Comprehensive introduction to differential geometry and i came across the proof that the tangent bundle of the Möbius band (as he defines it at an early stage i presume) is not ...
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Proof Concerning homeomorphisms of $\mathbb{P}^2$

Is the following proof valid? CLAIM: The space obtained by attaching a disc to a Mobius Strip along the boundary is homeomorphic to the projective plane. PROOF: We begin by showing that the boundary ...
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Mobius strip as fiber bundle

I am working through a couple of proofs concerning the Mobius Strip, and I am wondering if the following proof is valid. The claim is: CLAIM The Mobius Strip is defined by $M=${$(e^{i\phi},re^{i\frac{\...
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If Flatlanders walk around the Mobius string and we put a new one behind them every few steps, at which point they will be "reversing"?

Let's say that we put a Flatlander on the Mobius strip and make them move forward. Every X feet they travel, we put another one X feet behind them so that they can see each other. Then, after both of ...
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Is there a maximum width for a Möbius strip? [closed]

The Möbius strip is made by rotating one of the ends of a 2D strip 180° and then glueing these together. I was wondering, how wide can the strip be? For example, can we still make a Möbius strip from ...
Deschele Schilder's user avatar
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CW complex for Möbius strip and its homeomorfisams

I have to find CW complex for Mobius Strip. I can find a CW complex for a Möbius strip with more cells (one 0-cells, two 1-cells and a single 2-cell). My cells are points $A=(0,0) \approx (1,1)$, part ...
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Show that the Moebius band deformation retracts to a homeomorphic copy of $S^1$.

I can show that the unit square deformation retracts to a line and that line under a particular equivalence relation is homeomorphic to $S^1$. Let $X = [0,1] \times [0,1]$ and define an equivalence ...
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How do you calculate average circulation across a möbius band?

The surface is given by the parameterization shown in the picture. We are to find the average circulation across the mobius band with the circulation vector <-y,x,0> using stokes theorem if ...
Kevin Lorinc's user avatar
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CW-complex of Möbius band without 0-cells

So proofwiki, and some other sites, claim the Euler-characteristic of the Möbius strip is 0-1+1=0. Relying on the fact that a Möbius strip has no vertices, i.e. 0-cells. However I can nowhere find a ...
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The homeomorphism between $[0,1]\times[0,1]/\sim_{1}$ and $S^{1}\times[0,1]/\sim_{2}$

Let the quotient space be $[0,1]\times[0,1]/\sim_{1}$, where $\sim_{1}$ is defined as $(0,t)\sim_{1}(1,1-t)$. We say the given quotient space is the Mobius band. Now we need to prove this quotient ...
End points's user avatar
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Using Möbius band to define orientability of surfaces

Given a surface $S$, the usual definition of orientability is that there is an atlas of charts with all transition maps having non-negative Jacobian. Donaldson, in his book on Riemann surfaces, says a ...
Mohith Nagaraju's user avatar
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Deformation retraction of Möbius strip minus a point

Let the Möbius strip be $\mathcal M=[0,1]\times [0,1]/\sim$ where $(0,t)\sim (1-t,1)$. Let $$A=\{(x,y)|(x-1/2)^2+(y-1/2)^2=1/9\}\cup (\{1/2\}\times [0,1/6])\cup (\{1/2\}\times [5/6,1])$$ Show that $A/ ...
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Can I create a Möbius strip from two equilateral triangles?

Is it possible to take two equilateral triangles in the configuration depicted and create a moebius strip by connecting up the points of the traingles in a certain order. I'm working in a 3d app(...
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Deform the boundary of the Möbius band in the proper way

The boundary of a Möbius band is an unknot in $\mathbb{R}^3$, so we can deform it via an ambient isotopy to the standard circle in a plane. In this way, how does the Möbius band look like (i.e. how ...
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Day #2 of re-trying topology: Is the inner radius of an annulus used for a Mobius strip required exist to self intersect one?

In this figure I circled half of the inner radius of a Mobius band that I used to construct a holed cross cap. I don't know if it is required to exist to do the self intersection because I read ...
sheafpants4's user avatar
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Straight line on a mobius strip

I have a mobius strip that is twisted by 540° degrees (not only 180° as the usual mobius strip). Also it is one that can't be constructed with a paper-strip. I created it with this OpenSCAD code: <...
erichstuder's user avatar
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Fixed point property of Mobius strip

Does the mobius strip has fixed point property? If it isn't what is the map from mobius strip to itself witch lacks fixed point?
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Connection examples and embedding dimension theory

I've read about the "Utility Problem" (i.e. three utilities and three customers) requiring three dimensions to accomplish/attaching/embedding; and the Klein bottle requiring four (space-like)...
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How do I show that the Möbius band and $S^1$ are homotopy equivalent? [duplicate]

First of all I know that this post isn't really well since I don't have done much work but the problem is that I really have no idea how to deal with this problem, therefore I hope that nobody close ...
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Removing zero sections of n-twisted strips

First of all, a note: This question is motivated by my friend, who is a physics PhD student (I'm a math PhD student) and arose when I was trying to explain to him what line bundles are. (I wanted to ...
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Central circle of mobius band generates first homology

I was wondering, may someone please elaborate on why the central core in the mobius band generates the first homology of the mobius band? My thoughts are: $H_1(M)\cong H_1(\mathbb{S}^1)$ and $H_1(\...
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Are the following embeddings of the graph $K_{5}$ in the Möbius strip correct?

Are the following embeddings of $K_{5}$ in the Möbius strip correct? If not could you help me understand why?
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Exercise 5-15 from Spivak's Calculus on Manifolds

I came across the following question (Exercise 5-15) in Spivak's Calculus on Manifolds and am not sure how to solve it. Let $M$ be an $(n-1)$-dimensional manifold in $\mathbb{R}^n$. Let $M(\epsilon)$ ...
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Resulting surfaces of cutting Mobius strip at the boundary

I'm having a real hard time conceptualising the resulting surfaces of cutting a mobius strip along the boundary. I cut a 4cm Mobius strip 1cm along the boundary which resulted in interlinked 2cm ...
Major Redux's user avatar
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Number of half-twist for higher order hexa-flexagons

So, a hexa-flexagon is topologically equivalent to a Mobius Strip with 3 half-twist. Does this hold for a hexa-hexa-flexagon and higher order hexa-flexagons? How the heck do you unravel these and ...
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Which space do we obtain if we take a Möbius strip and identify its boundary circle to a point?

I know that the boundary circle of a Möbius strip is actually formed by the horizontal sides of $[0 ,1] \times [0,1]$.If we identify all the points of the 1st horizontal side to a single point and do ...
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How can the Möbius band be an image of a parametrization if it is not orientable?

I'm using a book of analysis on $\mathbb{R}^n$ with the following definitions: m-dimensional parametrization of class $C^k$ of $V\subseteq \mathbb{R}^n$: a homeomorphism $\phi:V_0\rightarrow V$ of ...
Fernando Nazario's user avatar
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Mobius strip and homeomorphism of a circle [closed]

I'm assigned a homework problem as follows: Let $X$ be the Möbius strip, obtained as a quotient space of $[0,1] \times [0,1]$ (with subspace topology of $\mathbb{R}^2$) by identifying the pairs of ...
Ryan Sinclair's user avatar
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Intuition behind equivalence of two identifications obtained from Möbius band.

Consider the real projective plane $\Bbb RP^2$ which can be realized as a quotient space obtained from a square by identifying points on each pair of it's opposite edges in reverse order. It has been ...
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Proving a result concerning quotient spaces.

Let $M$ be the Möbius strip and $C$ it's boundary circle. Prove that $M/C$ is homeomorphic to $\Bbb RP^2.$ I know that $\Bbb RP^2 \approx D^2/x \sim -x, x \in \partial D^2.$ So if we can able to get ...
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What is the tangent bundle of Mobius band?(It is trivial or not?)

I know that Mobius band is a quotient space of unit square with the equivalence relation $(0,t)\sim (1,1-t)$. I want to find the tangent bundle of Mobius band upto diffeomorphism. I am not getting any ...
Epsilon Delta's user avatar
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Is there a fixed-point free involution of the Möbius strip?

Is there a fixed-point free continuous involution of the Möbius strip? (Meaning a function $f:M\to M$ such that $f\circ f={\rm id}$ and $f(x)\ne x$ for all $x$.) The Lefschetz fixed-point theorem is ...
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Möbius strip is non-orientable from Gallot's "Riemannian Geometry"

I was working on the proof that Möbius strip is non-orientable using the following definition: A manifold $M$ is called orientable, if there exist a smooth atlas $(U_i, \phi_i)_{i \in I}$ such that $$\...
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immersion from mobius band to $\mathbb{R^3}$

find an immersion from mobius strip to $\mathbb{R^3}$ . One way to represent the Möbius strip embedded in $\mathbb{R^3}$ is by the parametrization. $$ \begin{array}{l} x(u, v)=\left(1+\frac{v}{2} \...
amir bahadory's user avatar
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1 answer
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Is a thick Möbius strip genus 2?

A cylinder has genus 0, but a thick cylinder has genus 1. This gets me into wondering, since a Möbius strip has genus 1, would a thick Möbius strip have genus 2? I can't quite comprehend if the ...
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The boundary of a closed Mobius band

The Mobius band is defined by the quotient: $r:I\times I\rightarrow M$, with the equivalence relation $(0,x)\sim (1,1-x)$ for all $x\in I$. The boundary of Mobius band $M$ is defined as the set of ...
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Attaching disks and contracting is equivalent to taking quotients.

Take a Mobius band $M$ and attached a disk $D^2$ along its boundary, which is a copy of $S^1$ embedded in $M$. It is known that what we obtain is something homeomorphic to the projective plane. Now ...
HP the Dancer's user avatar
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1 answer
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Topology of Mobius Ring

I'm having some difficulty with the finer points of the construction of the topology of the Mobius ring from Tu's manifolds. Given the square $ R = \{(x,y) \in \mathbb{R}^2 | \, 0 \leq x \leq 1, -1 &...
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2-dimensional manifold is orientable if and only if it does not contain the Möbius band

Criterion for non-orientability: a manifold $M$ is non-orientable if and only if it contains a "bad" sequence of charts, i.e. $\{U_i,\varphi_i\},i=1,\ldots,n$ such that the transition maps $\...
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how to chart a moebius strip

This is a question from a tutorial for topology manifold. It is stated as: How many charts do you need to cover the Moebius strip that has a river printed on it? The chart was defined as"Chart: -...
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Proving a function between quotient spaces is continuous

This is a question related to a problem previously asked here and also here. It states that the infinite strip $\mathbb{R}\times\left[\frac{-1}{2},\frac{1}{2}\right]$ under the group action $m\cdot(x,...
mate89's user avatar
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Möbius Strip + Möbius Strip = Klein Bottle, What about Klein Bottle + Klein Bottle =?

We know that 2 Möbius-Strips can be joined edge-wise to eliminate that edge producing 0 Edge topological structure. ML (Möbius Left) + MR (Möbius Right) = KOJ (Simple "inverted sock" Klein ...
Dhruv Bansal's user avatar
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2 answers
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Does this count as a loop on Möbius strip

Suppose we take the Möbius strip as $X = \frac{[0, 1]\times[0, 1]}{\sim}$ with usual equivalence relation. If I define $\alpha: [0, 1] \rightarrow X$ by $x \rightarrow [(x, 1/2)]$, is this a loop? ...
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