What is mobius strip: Definition and 19 Discussions

In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and August Ferdinand Möbius in 1858, but it had already appeared in Roman mosaics from the third century CE. The Möbius strip is a non-orientable surface, meaning that within it one cannot consistently distinguish clockwise from counterclockwise turns. Every non-orientable surface contains a Möbius strip.
As an abstract topological space, the Möbius strip can be embedded into three-dimensional Euclidean space in many different ways: a clockwise half-twist is different from a counterclockwise half-twist, and it can also be embedded with odd numbers of twists greater than one, or with a knotted centerline. Any two embeddings with the same knot for the centerline and the same number and direction of twists are topologically equivalent. All of these embeddings have only one side, but when embedded in other spaces, the Möbius strip may have two sides. It has only a single boundary curve.
Several geometric constructions of the Möbius strip provide it with additional structure. It can be swept as a ruled surface by a line segment rotating in a rotating plane, with or without self-crossings. A thin paper strip with its ends joined to form a Möbius strip can bend smoothly as a developable surface or be folded flat; the flattened Möbius strips include the trihexaflexagon. The Sudanese Möbius strip is a minimal surface in a hypersphere, and the Meeks Möbius strip is a self-intersecting minimal surface in ordinary Euclidean space. Both the Sudanese Möbius strip and another self-intersecting Möbius strip, the cross-cap, have a circular boundary. A Möbius strip without its boundary, called an open Möbius strip, can form surfaces of constant curvature. Certain highly-symmetric spaces whose points represent lines in the plane have the shape of a Möbius strip.
The many applications of Möbius strips include mechanical belts that wear evenly on both sides, dual-track roller coasters whose carriages alternate between the two tracks, and world maps printed so that antipodes appear opposite each other. Möbius strips appear in molecules and devices with novel electrical and electromechanical properties, and have been used to prove impossibility results in social choice theory. In popular culture, Möbius strips appear in artworks by M. C. Escher, Max Bill, and others, and in the design of the recycling symbol. Many architectural concepts have been inspired by the Möbius strip, including the building design for the NASCAR Hall of Fame. Performers including Harry Blackstone Sr. and Thomas Nelson Downs have based stage magic tricks on the properties of the Möbius strip. The canons of J. S. Bach have been analyzed using Möbius strips. Many works of speculative fiction feature Möbius strips; more generally, a plot structure based on the Möbius strip, of events that repeat with a twist, is common in fiction.

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  1. L

    Is a Mobius Strip Truly a 2D Object in a 3D Space?

    Can anyone explain the meaning behind a mobius strip? Basically just a means to travel on both sides of a flat surface? It's still a 3D object though since it uses 3D space for the twist to be possible?
  2. cianfa72

    I How Do Transition Functions Define a Möbius Strip in Fiber Bundles?

    Hi, starting from this (old) thread I'm a bit confused about the following: the transition function ##ϕ_{12}(b)## is defined just on the intersection ##U_1\cap U_2## and as said in that thread it actually amounts to the 'instructions' to glue together the two charts to obtain the Möbius...
  3. YoungPhysicist

    I Projecting Möbius Strip Edge: Learn How in 2D Plane

    How can the edge of a Möbius strip being projected on a 2 dimensional plane? Precisely the ending of this video: I just can get it since his animation goes by it so fast.
  4. O

    Understanding the Closed Mobius Strip: Get Help Here!

    Why Mobius strip closed?Mobius strip is compact because it is confined and closed. But I can't understand closed of Mobius strip. Help please.
  5. V

    Mobius Strip & 4D: Exploring Relationship & Dimensions

    What is the relationship with the Mobius strip (or loop) and the 4 dimension? Is Mobius strip a four dimensional object?
  6. E

    If there was a region of space-time shaped like a Mobius Strip

    Would the spin of a particle be measured the same by any set of observers upon that strip? Would an observer experience a reversal of spin if the particle traveled far enough along the strip?
  7. Mandelbroth

    Metric Tensor on a Mobius Strip?

    I was bored, so I tried to do something to occupy myself. I started going through withdrawal, so I finally just gave in and tried to do some math. Three months of no school is going to be painful. I think I have problems. MATH problems. :-p Atrocious comedy aside, Spivak provides a parametric...
  8. D

    Fundamental polygon of a Mobius strip

    Hey I am having a little bit of difficulty. The classification theorem for 2 - manifolds tells me that every 2 -manifold has the following representation: 1) connect sum of n-tori 2) connect sum of n-projective planes 3) a sphere Now, using Massey's book there is a very algorithmic...
  9. T

    Continuing mobius strip theory idea

    It does seem a shame if the previously discussed notion of mobius strip theory, as a modification of string theory, remains untenable. Perhaps some 'mobiusness' can still be adopted into the logic of string theory; it could serve as a visualizable explanatory model of one-handedness. Thoughts...
  10. I

    The Universe: A Mobius Strip of Possibilities?

    Hi all, i am not a physicist, but had a random thought the other day and wanted to find out whether there had ever been a similar theory as to the structure of the universe ?? Looking at solar systems and galaxies etc they seem to (in very basic terms) operate similarly to atoms, ie bits...
  11. jfy4

    Exploring the Relationship between Spinors and Mobius Strips in Rotations

    Hi, I was pondering a bit about the mobius strips and I was wondering if there is a relationship between spinors and there transformation under rotations and that, in a manner of speaking, one must go around a mobius strip twice to return to the original position. To me it seems there would be...
  12. J

    Mobius strip computing unit normal field at 2 points

    Homework Statement I am given this parametrization of the mobius band: F(s,t) = (cos(t)+s*cos(t)*cos(t/2), sin(t)+s*sin(t)*cos(t/2), s*sin(t/2)) Let F1 be F restricted to (0,2*pi) X (-1,1). Let F2 be F restricted to (-pi,pi) X (-1,1). let N1 be the unit normal field determined by F1 let...
  13. Spinnor

    Points on opposite sides a paper Mobius strip.

    Say I pierce a paper Mobius strip with a pin and call the point on the side the pin entered the paper the point A and call the point where the pin comes through the paper the point B. In an idealized Mobius strip are these points different? Can they be the same? I would like a closed surface...
  14. S

    Mobius Strip? What's all the fuss about?

    I don't really get what makes mobius strip so special? Yeah, sure you can get from one from of the strip to another without touching its boundary but so what? BTW, I am not saying the mobius strip is useless. I just want to know how it helps you get a deeper understanding of other dimentions.
  15. O

    Can a Möbius Strip be embedded in R3 with zero Gaussian curvature?

    I would welcome the parametric equations for an embedding in R3 of a locally Euclidean Möbius Strip without self intersections nor singularities and of Gaussian curvature equal to zero. That it exists in R3 is trivial to prove: just get a strip of paper of appropriate length and width, twist and...
  16. S

    Understanding the Topology of Klein Bottles and Mobius Strips

    Hi Does anyone know what a klein bottle and mobius strip is ??what does embedding a surface in R4 mean??Is there any easy way to understand this??..Can someone enlighten me on this??I have an engg background..So please explain in simple language.. Bye Shankar
  17. S

    Is a mobius strip spin 1/2 object?

    Remember Phiysicist always say electron has a spin of 1/2; I can't remember how it was derived? But I noticed a mobius strip exihibit interesting attribute. Can we consider a mobius strip a spin 1/2 object?
  18. R

    What is the Möbius Strip String Theory?

    I had originaly posted a thread on this theory in another forum, but I realize that it is better suited for this forum. In the attached file is a diagram of the string. In this string theory, strings are made of two opposite charges that exist within two extra dimentions. the dimentions have...