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Encyclopedia > Möbius strip
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A Möbius strip made with a piece of paper and tape.

The Möbius strip or Möbius band is a topological object with only one side (one-sided surface) and only one boundary component. It was co-discovered independently by the German mathematicians August Ferdinand Möbius and Johann Benedict Listing in 1858. A model can easily be created by taking a paper strip and giving it a half-twist, and then merging the ends of the strip together to form a single strip. In Euclidean space there are in fact two types of Möbius strip depending on the direction of the half-twist: if the right hand twists the right end of the strip in a clockwise manner, the result is a right-handed Möbius strip. The Möbius strip therefore exhibits chirality. Download high resolution version (1328x824, 295 KB) Wikipedia does not have an article with this exact name. ... Download high resolution version (1328x824, 295 KB) Wikipedia does not have an article with this exact name. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. ... In mathematics, a surface is a two-dimensional manifold. ... In topology a boundary component of a compact surface is a connected component consisting of boundary points of a surface. ... A mathematician is a person whose area of study and research is mathematics. ... August Ferdinand Möbius (November 17, 1790, Schulpforta, Saxony, Germany - September 26, 1868, Leipzig) was a German mathematician and theoretical astronomer. ... Johann Benedict Listing (born July 25, 1808, died December 24, 1882) was a German mathematician. ... 1858 is a common year starting on Friday. ... In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... A clockwise motion is one that proceeds like the clocks hands: from the top to the right, then down and then to the left, and back to the top. ... A person who is right-handed is more dextrous with their right hand than with their left hand: they will write with their right hand, and probably also use this hand for tasks such as personal care, cooking, and so on. ... In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or more particularly cant be mapped to its mirror images by rotations and translations alone. ...


The Möbius strip has several curious properties. If you cut down the middle of the strip, instead of getting two separate strips, it becomes one long strip with two half-twists in it (not a Möbius strip). If you cut this one down the middle, you get two strips wound around each other. Alternatively, if you cut along a Möbius strip, about a third of the way in from the edge, you will get two strips; one is a thinner Möbius strip, the other is a long strip with two half-twists in it (not a Möbius strip). Other interesting combinations of strips can be obtained by making Möbius strips with two or more flips 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. Categories: Stub | Knot theory ...


The Möbius strip is often cited as the inspiration for the infinity symbol , since if one were to stand on a the surface of a Möbius strip, one could walk along it forever. However, this may be apocryphal since the symbol had been in use to represent infinity even before the Möbius strip was discovered. Infinity has discrete meanings in mathematics, philosophy, theology and everyday life. ... In Judeo-Christian theologies, apocrypha refers to religious Sacred text that have questionable authenticity or are otherwise disputed. ...

Contents

Geometry and topology

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A parametric plot of a Möbius strip

One way to represent the Möbius strip as a subset of R3 is using the parametrization: Mobius strip created with Mathematica. ... Mobius strip created with Mathematica. ...

where and . 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: This article describes some of the common coordinate systems that appear in elementary mathematics. ...

log(r)sin(θ / 2) = zcos(θ / 2).

Topologically, the Möbius strip can be defined as the square [0,1] × [0,1] with sides identified by the relation (x,0) ~ (1-x,1) for 0 ≤ x ≤ 1, as in the following diagram: Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. ... In plane geometry, a square is a polygon with four equal sides and equal angles. ... For quotient spaces in linear algebra, see quotient space (linear algebra). ...

 ----> | | | | <---- 

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. In mathematics, a surface is a two-dimensional manifold. ... This article or section should be merged with Orientability. ... In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ... In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. ...


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. The Klein bottle immersed in three-dimensional space. ... In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...


Another closely related manifold is the real projective plane. If a single hole is punctured in the real projective plane, what is left is a Möbius strip. 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. Such a figure is called a cross-cap (a cross-cap can also mean this figure with the disk glued in, i.e. an immersion of the projective plane in R3). In mathematics, the real projective plane is a two-dimensional manifold, that is, a surface, that has basic applications to geometry, but which cannot be embedded in our usual three-dimensional space. ... In mathematics, a cross-cap is a two-dimensional surface that is topologically equivalent to a Möbius strip. ...


It is a common misconception that a cross-cap cannot be formed in three dimensions without the surface intersecting itself. In fact it is possible to embed a Möbius strip in R3 with boundary a perfect circle. Here is the idea: let C be the unit circle in the xy plane in R3. Now connect antipodal points on C, i.e., points at angles θ and θ + π, by an arc of a circle. For θ between 0 and π / 2 make the arc lie above the xy plane, and for other θ the arc below (with two places where the arc lies in the xy plane). However, if a disk is glued in to the boundary circle, the self-intersection of the resulting projective plane is imminent. Illustration of a unit circle. ... Antipodal points on the surface of a sphere are diametrically opposite; on the other side of a globe. ...


In terms of identifications of the sides of a square, as given above: the real projective plane is made by gluing the remaining two sides with 'consistent' orientation (arrows making an anti-clockwise loop); and the Klein bottle is made the other way. In mathematics, the real projective plane is a two-dimensional manifold, that is, a surface, that has basic applications to geometry, but which cannot be embedded in our usual three-dimensional space. ... The Klein bottle immersed in three-dimensional space. ...


Art and technology

This Octopus card bears a stylized Möbius strip design.

The Möbius strip has provided inspiration both for sculptures and for graphical art. M. C. Escher is one of the artists who was especially fond of it and based several of his lithographs on this mathematical object. One famous one, Möbius Strip II (http://www.mcescher.com/Gallery/recogn-bmp/LW441.jpg), features ants crawling around the surface of a Möbius strip. Download high resolution version (992x671, 95 KB)New picture of front of Octopus card. ... Download high resolution version (992x671, 95 KB)New picture of front of Octopus card. ... The Octopus card is a rechargeable contactless stored value smart card used for electronic payment in on-line or off-line systems in Hong Kong. ... Self portrait, 1943¹ Maurits Cornelis Escher (Leeuwarden, June 17, 1898 - Laren, March 27, 1972) was a Dutch artist most known for his woodcuts, lithographs and mezzotints, which tend to feature impossible constructions, explorations of infinity, and tessellations. ... Lithography is a method for printing on a smooth surface, as well as a method of manufacturing semiconductor and MEMS devices. ...


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 generalised Möbius strip. In the short story "A Subway Named Möbius" (http://math.cofc.edu/faculty/kasman/MATHFICT/mfview.php?callnumber=mf102), by A.J. Deutsch, the Boston subway authority builds a new line; the system becomes so tangled that it turns into a Möbius strip, and trains start to disappear. Science fiction is a form of speculative fiction principally dealing with the impact of imagined science and technology, or both, upon society and persons as individuals. ... Arthur C. Clarke, progenitor of communication satellites, is considered by many to be a grand master of science fiction. ... The deepest visible-light image of the cosmos. ... Alternative meanings: Boston (disambiguation) The 18th-century Old State House in Boston is surrounded by tall buildings of the 19th and 20th centuries. ... This page refers to urban rail mass transit systems. ...


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). A conveyor belt or belt conveyor consists of two end pulleys, with a continuous loop of material that rotates about them. ...


A device called a Möbius resistor is a recently discovered electronic circuit element which has the property of cancelling its own inductive reactance. Nikola Tesla patented similar technology in the early 1900s, US#512,340 "Coil for Electro Magnets" was intended for use with his system of global transmission of electricity without wires. A Möbius resistor is an electrical component made up of two conductive surfaces separated by a dielectric material, twisted 180° and connected to form a Möbius strip. ... Nikola Tesla (July 9/July 10, 1856 - January 7, 1943) was a physicist, inventor, and electrical engineer of unusual intellectual brilliance and practical achievement. ...


See also

The Klein bottle immersed in three-dimensional space. ...

External links

  • Visual Math (http://genius.ucsd.edu/~lpat/findit/math.html) has a nice animation of making a Möbius strip.
  • Eric W. Weisstein, Möbius Strip (http://mathworld.wolfram.com/MoebiusStrip.html) at MathWorld.
  • Möbius strip (http://www.cut-the-knot.org/do_you_know/moebius.shtml) — a web page with movies.

 
 

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