# 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.

161 questions
Filter by
Sorted by
Tagged with
29 views

### 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 ...
1 vote
51 views

### 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 ...
• 231
36 views

### 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-...
• 11.6k
1 vote
99 views

### 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 ...
• 370
91 views

### 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 ...
• 665
94 views

• 834
1 vote
77 views

### 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 ...
• 566
165 views

### 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 ...
223 views

### 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 ...
• 471
158 views

### 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 ...
58 views

### 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 ...
1 vote
162 views

### 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 ...
• 53
193 views

### 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 ...
• 447
188 views

### 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 ...
251 views

119 views

### 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?
• 911
1 vote
156 views

### 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 $M(\epsilon)$ ...
• 1,057
174 views

### 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 ...
• 197
92 views

### 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 ...
• 43
681 views

### 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 ...
• 3,442
231 views

### 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 ...
135 views

### 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 ...
1 vote
176 views

### 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 ...
• 2,912
1 vote
85 views

### 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 ...
• 2,912
187 views

### 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 ...
271 views

### 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 ...
• 23.3k
1 vote
284 views

• 4,629
1 vote
106 views

### 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 ...
• 415
263 views

### 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 ...
61 views

### 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 ...
1 vote
49 views

• 125
1 vote