Möbius strip

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The Möbius strip or Möbius band (pronounced /ˈmoʊbiːəs/ or /ˈmeɪbiːəs/, i.e. beginning with "Moe" or "may"; German IPA: [ˈmøbiʊs]) is a surface with only one side and only one boundary component. It has the mathematical property of being non-orientable. It is also a ruled surface. It was discovered independently by the German mathematicians August Ferdinand Möbius and Johann Benedict Listing in 1858.^[1][2][3]

A model can easily be created by taking a paper strip and giving it a half-twist, and then joining the ends of the strip together to form a single strip. In Euclidean space there are in fact two types of Möbius strips depending on the direction of the half-twist: clockwise and counterclockwise. The Möbius strip is therefore chiral, which is to say that it is "handed".

It is straightforward to find algebraic equations the solutions of which have the topology of a Möbius strip, but in general these equations do not describe the same geometric shape that one gets from the twisted paper model described above. In particular, the twisted paper model is a developable surface (it has zero Gaussian curvature). A system of differential-algebraic equations that describes models of this type was published in 2007 together with its numerical solution.^[4]

[edit] Properties

The Möbius strip has several curious properties.

A model of a Möbius strip can be constructed by joining the ends of a strip of paper with a single half-twist. A line drawn starting from the seam down the middle will meet back at the seam but at the "other side". If continued the line will meet the starting point and will be double the length of the original strip of paper. This single continuous curve demonstrates that the Möbius strip has only one boundary.

If the strip is cut along about a third of the way in from the edge, creates two strips: One is a thinner Möbius strip - it is the center third of the original strip. The other is a long strip with two full twists in it - this is a neighborhood of the edge of the original strip.

Alternatively, cutting a Möbius strip along the above line, instead of getting two separate strips, it becomes one long strip with two full twists in it, which is not a Möbius strip. This happens because the original strip only has one edge which is twice as long as the original strip of paper. Cutting creates a second independent edge, half of which was on each side of the knife or scissors. Cutting this new, longer, strip down the middle creates two strips wound around each other.

Other interesting combinations of strips can be obtained by making Möbius strips with two or more half-twists in them instead of one. For example, a strip with three half-twists, when divided lengthwise, becomes a strip tied in a trefoil knot. Cutting a Möbius strip, giving it extra twists, and reconnecting the ends produces unexpected figures called paradromic rings.

[edit] Geometry and topology

One way to represent the Möbius strip as a subset of R³ is using the parametrization:

<math>x(u,v)=\left(1+\frac{v}{2}\cos\frac{u}{2}\right)\cos(u)</math>

<math>y(u,v)=\left(1+\frac{v}{2}\cos\frac{u}{2}\right)\sin(u)</math>

<math>z(u,v)=\frac{v}{2}\sin\frac{u}{2}</math>

where <math>0\leq u < 2\pi</math> and <math>-1\leq v\leq 1</math>. This creates a Möbius strip of width 1 whose center circle has radius 1, lies in the x-y plane and is centered at (0,0,0). The parameter u runs around the strip while v moves from one edge to the other.

In cylindrical polar coordinates (r,θ,z), an unbounded version of the Möbius strip can be represented by the equation:

<math>\log(r)\sin\left(\frac{\theta}{2}\right)=z\cos\left(\frac{\theta}{2}\right).</math>

Topologically, the Möbius strip can be defined as the square [0,1] × [0,1] with its top and bottom sides identified by the relation (x,0) ~ (1-x,1) for 0 ≤ x ≤ 1, as in the diagram on the right.

The Möbius strip is a two-dimensional compact manifold (i.e. a surface) with boundary. It is a standard example of a surface which is not orientable. The Möbius strip is also a standard example used to illustrate the mathematical concept of a fiber bundle. Specifically, it is a nontrivial bundle over the circle S¹ with a fiber the unit interval, I = [0,1]. Looking only at the edge of the Möbius strip gives a nontrivial two point (or Z₂) bundle over S¹.

A simple construction of the Möbius strip which can be used to portray it in computer graphics or modeling packages is as follows :

Take a rectangular strip. Rotate it around a fixed point not in its plane. At every step also rotate the strip along a line in its plane (the line which divides the strip in two) and perpendicular to the main orbital radius. The surface generated on 1 complete revolution is the Möbius strip.

[edit] Möbius band with flat edge

The edge of a Möbius strip is topologically equivalent to the circle. Under the usual embeddings of the strip in Euclidean space, as above, this edge is not an ordinary (flat) circle. It is possible to embed a Möbius strip in three-dimensions so that the edge is a circle, and the resulting figure is called the Sudanese Möbius Band.

To see this, first consider such an embedding into the 3-sphere S³ regarded as a subset of R⁴. A parametrization for this embedding is given by

<math>z_1 = \sin\eta\,e^{i\phi}</math>

<math>z_2 = \cos\eta\,e^{i\phi/2}.</math>

Here we have used complex notation and regarded R⁴ as C². The parameter <math>\eta</math> runs from <math>0</math> to <math>\pi</math> and <math>\phi</math> runs from <math>0</math> to <math>2\pi</math>. Since <math>|z_1|^2 + |z_2|^2 = 1</math> the embedded surface lies entirely on S³. The boundary of the strip is given by <math>|z_2| = 1</math> (corresponding to <math>\eta = 0,\pi</math>), which is clearly a circle on the 3-sphere.

To obtain an embedding of the Möbius strip in R³ one maps S³ to R³ via a stereographic projection. The projection point can be any point on S³ which does not lie on the embedded Möbius strip (this rules out all the usual projection points). Stereographic projections map circles to circles and will preserve the circular boundary of the strip. The result is a smooth embedding of the Möbius strip into R³ with a circular edge and no self-intersections.

[edit] Related objects

A closely related 'strange' geometrical object is the Klein bottle. A Klein bottle can be produced by gluing two Möbius strips together along their edges; this cannot be done in ordinary three-dimensional Euclidean space without creating self-intersections.^[5]

Another closely related manifold is the real projective plane. If a circular disk is cut out of the real projective plane, what is left is a Möbius strip.^[6] Going in the other direction, if one glues a disk to a Möbius strip by identifying their boundaries, the result is the projective plane. In order to visualize this, it is helpful to deform the Möbius strip so that its boundary is an ordinary circle (see above). The real projective plane, like the Klein bottle, cannot be embedded in three-dimensions without self-intersections.

In graph theory, the Möbius ladder is a cubic graph closely related to the Möbius strip.

[edit] Art and popular culture

The Möbius strip has provided inspiration both for sculptures and for graphical art. The artist M. C. Escher was especially fond of it and based several of his lithographs on it. One famous example, Möbius Strip II, features ants crawling around the surface of a Möbius strip. It is also a recurrent feature in science fiction stories, such as Arthur C. Clarke's The Wall of Darkness. Science fiction stories sometimes suggest that our universe might be some kind of generalized Möbius strip. This is especially prominent in the Perry Rhodan-series. In the short story "A Subway Named Moebius", by A.J. Deutsch, the Boston subway authority builds a new line, but the system becomes so tangled that it turns into a Möbius strip, and trains start to disappear. The Möbius strip also features prominently in Brian Lumley's Necroscope series of novels.

A popular limerick is often associated with this design which reads ^[7]

"A mathematician confided

That a Möbius band is one-sided,

And you'll get quite a laugh,

If you cut one in half,

For it stays in one piece when divided"

In the popular episodic MMORPG "Guild Wars", there exists an attack called "Moebius Strike"; it is so called, because it gives the player the power to continually use their attack chain (it goes on forever, much like a Möbius Strip).

In "A. Botts and the Moebius Strip", a short story by William Hazlett Upson first published in 1945 in the Saturday Evening Post, the protagonist secretly restitches a conveyor belt to form a Möbius strip to frustrate a superior's attempt to "paint the outside, but not the inside" of the belt as a safety measure.

In the seaQuest DSV episode "Playtime" a Möbius strip is used to show a theory involving time travel and the space/time continuum.

"Moebius" is the title of a Stargate SG-1 episode involving time travel where the alternative timelines shown wrap around like a möbius strip.

The videogame F-Zero GX (for the Nintendo Gamecube) features a circuit called "Green Plant", which is a giant Möbius strip.

The Nelly Furtado song Hey, Man! contains the following lyrics:

We are a part of a circle

It's like a Möbius strip

And it goes 'round and 'round

Until it loses a link

The Sonic the Hedgehog comic book series, based on the video game series, takes place on the planet Mobius. This stems from a mistranslation of Yuji Naka stating that a Möbius strip was used in the video game Sonic the Hedgehog 2.

The Japanese band Buck-Tick song Bran New Lover contains the following lyrics:

English Lyrics:

Right now, let’s throw open Pandora’s box

Tear and scatter the Möbius ring, and break free

Japanese Lyrics:

Pandora no hako wo ima akehanateyo

Chigire kakete mebiusu ringu tokihanatou

The They Might Be Giants song Operators Are Standing By contains the following lyrics:

Operators are standing by

Smoking cigarettes and drinking coffee

Pass round the picture of a Möbius strip

The Mountain Goats song Source Decay contains the following lyrics:

I wish the West Texas highway was a Möbius strip

I could ride it out for ever

The Darren Hayes song Love and Attraction contains the following lyrics:

This strip is Mobius

It's never ending.

Iain Banks' novel The Wasp Factory contains a reference:

My father once had me believing that the earth was a Möbius strip, not a sphere.(chapter one)

• The logo of the Always brand of sanitary napkins made by Procter & Gamble contains a Möbius strip in the shape of the infinity symbol.

• IBM's Power Architecture logo consists of a Möbius strip in the shape of a 'P'.

• In the Dilbert comic strip from December 14th, 2007, Wally refers to his work list as a Möbius Strip in which each task required him to do another task until it all looped back on itself.[1]

[edit] Occurrence and use in nature and technology

There have been technical applications. Giant Möbius strips have been used as conveyor belts that last longer because the entire surface area of the belt gets the same amount of wear, and as continuous-loop recording tapes (to double the playing time). Möbius strips are common in the manufacture of fabric computer printer and typewriter ribbons, as they allow the ribbon to be twice as wide as the print head whilst using both half-edges evenly.

A device called a Möbius resistor is an electronic circuit element which has the property of canceling its own inductive reactance. Nikola Tesla patented similar technology in the early 1900s: ^[8] "Coil for Electro Magnets" was intended for use with his system of global transmission of electricity without wires.

In physics/electro-technology:

• as compact resonator with the resonance frequency with half of identically constructed linear coils ^[9].
• as inductionless resistance ^[10] .
• as superconductors with high transition temperature ^[11]

In chemistry/nano-technology:

• as “knot molecules” with special characteristics (Knotane [2], Chirality)
• as molecular engines ^[12]
• as Graphene volume (nano-graphite) with new electronic characteristics, like helical magnetism ^[13]
• In a special type of aromaticity: Möbius aromaticity
• Charged particles, which were caught in the magnetic field of the earth, can move on a Möbius band ^[14]
• The cyclic protein Kalata B1, active substance of the plant Oldenlandia. O. affinis, as nature cures e.g. for the birth introduction, has a Möbius topology

[edit] See also

• Cross-cap
• Klein bottle
• List of cycles
• Loop
• Molecular knot
• Paradox
• Real projective plane
• Strange loop

[edit] References

[edit] External links

• The protein with a topological twist
• A virtual walk in the solar wind
• Molecular knot
• Johann Benedict Listing
• Eric W. Weisstein, Möbius Strip at MathWorld.
• Möbius strip at cut-the-knot
• Knitted version
• h2g2 - The Amazing Möbius Strip
• Science News 7/28/07: A Twist on the Möbius Band: Researchers work out the shape of a paper strip
• Louis Buckley (July 15, 2007). Möbius strip unravelled. Nature. DOI:10.1038/news070709-16. Retrieved on 2006-07-16.bg:Лист на Мьобиус

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