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| {{about|the mathematical object|musical group|Mobius Band (band)}}
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| [[Image:Möbius strip.jpg|thumb|250px|right|A Möbius strip made with a piece of paper and tape. If an ant were to crawl along the length of this strip, it would return to its starting point having traversed the entire length of the strip (on both sides of the original paper) without ever crossing an edge.]]
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| The '''Möbius strip''' or '''Möbius band''' ({{IPAc-en|UK|ˈ|m|ɜː|b|i|ə|s}} or {{IPAc-en|US|ˈ|m|oʊ|b|i|ə|s}}; {{IPA-de|ˈmøːbi̯ʊs|lang}}), also '''Mobius''' or '''Moebius''', is a [[surface]] with only one side and only one [[boundary component]]. The Möbius strip has the [[Mathematics|mathematical]] property of being [[orientability|non-orientable]]. It can be realized as a [[ruled surface]]. It was discovered independently by the German [[mathematician]]s [[August Ferdinand Möbius]] and [[Johann Benedict Listing]] in 1858.<ref>{{cite book
| | Do you envy your friends and neighbors who have magazine quality homes? There is no need to wish your house could be decorated nicely. Great interiors don't just happen; it takes an inspired and dedicated individual to take that first step. The only thing you require is proper advice. If you want a trendy home you should continue reading [http://www.riversidemediagroup.com/uggboots.asp http://www.riversidemediagroup.com/uggboots.asp] this article to learn how to accomplish it.<br><br>Instead of rushing into your next interior design project, take time to develop a good image of what you'd like to [http://Www.Answers.com/topic/achieve achieve]. Moods are as varied as, well, your moods. They can swing from bold to relaxed, and anywhere in between. Having a mood in mind before you start designing will help you decide what the best way of planning should be.<br><br>You can do your interior design projects on a budget by finding deals on high-quality equivalents at mass merchant stores. High-end decor is very expensive, but it is usually easy to find something that is similar for a lower cost. It is your choice, of course, but who does not like to save [http://www.riversidemediagroup.com/uggboots.asp Cheap Ugg Boots] money where ever they can.<br><br>You can lower your design budget by opting for quality pieces from big-box merchants. You can easily spend you life's savings on high-end merchandise while decorating, but you do not have to, as there are things that are made well that do not cost nearly as much. Only buy designer brands if similar items cannot be found at a lower cost.<br><br>When designing a kid's room, look at it as if you were your kid. You might not realize what they can reach and what works for their size. Make sure that everything is easy for them to access. Look at it as your kid would and get rid of possible hazards.<br><br>Consider how a room will be used before you choose decorations. Think about the number of people that will be in the room and what they'll be doing. Keep family and friends in mind while designing your living-room, [http://www.riversidemediagroup.com/uggboots.asp http://www.riversidemediagroup.com/uggboots.asp] however focus on you and your mate when designing the bedroom.<br><br>It is a good idea to invest in paint; a paint that is high-quality is suggested. If you purchase [http://www.riversidemediagroup.com/uggboots.asp Cheap Ugg Boots] paint, it always [http://www.riversidemediagroup.com/uggboots.asp http://www.riversidemediagroup.com/uggboots.asp] ends up chipping off and damaging the walls, which in the end costs you more money. So, invest in good quality paint if you want your design projects to last.<br><br>It's an excellent idea to incorporate some trendy ideas into your home's interior. It adds some style and give a room a modern feeling. But, you should only add modern decorations as accents, not permanent fixtures. If your funky lamp gets old, you can easily [http://www.riversidemediagroup.com/uggboots.asp http://www.riversidemediagroup.com/uggboots.asp] replace it. A cheetah print wallpaper is not quite as easily replaced.<br><br>Balance out the cold look and feel of hard flooring like stone, concrete or tiles, with a few carefully selected area rugs. Remember to rotate your rugs frequently so that the rug will wear evenly. By changing angles or moving the rugs, you can easily extend their lives.<br><br>Look for interesting pieces and themes to inspire a new collection. A group that has more than three items of a similar nature in it is a collection. A collection could consist of three vases. If you've got three pictures, that makes a collection. Group your collection together to create an interesting display, and you will have an easy conversation piece.<br><br>Try putting together a collection of artwork. Any item you have at least three of constitutes a collection. Three vases would be a collection. Three pictures could also constitute a collection. Put them together in a display and it will be a nice touch to talk about with visitors.<br><br>You do not need exclusive design magazines or costly professionals when designing your home. You just need to go with your instinct if you want to create a welcoming space for the entire family. The above article should have provided the needed advice on how you can get started in making your dream design into reality! |
| | author = Clifford A. Pickover
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| |date=March 2005
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| | title = The Möbius Strip : Dr. August Möbius's Marvelous Band in Mathematics, Games, Literature, Art, Technology, and Cosmology
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| | publisher = Thunder's Mouth Press
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| | isbn = 1-56025-826-8
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| | authorlink = Clifford A. Pickover
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| }}</ref><ref>{{cite book
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| | author = Rainer Herges
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| | year = 2005
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| | title = Möbius, Escher, Bach – Das unendliche Band in Kunst und Wissenschaft ''. In: Naturwissenschaftliche Rundschau 6/58/2005
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| | pages = 301–310
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| | issn = 0028-1050
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| }}</ref><ref>{{cite book
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| | author = Chris Rodley (ed.)
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| | title = Lynch on Lynch
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| | place =London, Boston
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| | year =1997
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| | pages=231
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| }}</ref>
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| 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 loop. In [[Euclidean space]] there are two types of Möbius strips depending on the direction of the half-twist: [[clockwise|clockwise and counterclockwise]]. That is to say, it is a [[chirality (mathematics)|chiral]] object with "handedness" (right-handed or left-handed).
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| The Möbius band (equally known as the Möbius strip) is not a surface of only one geometry (i.e., of only one exact size and shape), such as the half-twisted paper strip depicted in the illustration to the right. Rather, mathematicians refer to the (closed) Möbius band as any surface that is [[homeomorphic]] to this strip. Its boundary is a simple closed curve, i.e., homeomorphic to a circle. This allows for a very wide variety of geometric versions of the Möbius band as surfaces each having a definite size and shape. For example, any closed rectangle with length L and width W can be glued to itself (by identifying one edge with the opposite edge after a reversal of orientation) to make a Möbius band. Some of these can be smoothly modeled in 3-dimensional space, and others cannot (see section [[Möbius strip#Fattest rectangular Möbius strip in 3-space|''Fattest rectangular Möbius strip in 3-space'']] below). Yet another example is the complete open Möbius band (see section [[Möbius strip#Open Möbius band|''Open Möbius band'']] below). Topologically, this is slightly different from the more usual — closed — Möbius band, in that any open Möbius band has no boundary.
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| It is straightforward to find [[algebraic variety|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]]).
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| A system of [[differential-algebraic equation]]s that describes models of this type
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| was published in 2007 together with its numerical solution.<ref>{{cite journal | author=Starostin E.L., van der Heijden G.H.M. | title=The shape of a Möbius strip | journal=[[Nature Materials]] | url=http://www.nature.com/nmat/journal/v6/n8/abs/nmat1929.html | year=2007 | doi=10.1038/nmat1929 | volume = 6 | pages = 563–7 | pmid=17632519 | issue=8}}</ref>
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| The [[Euler characteristic]] of the Möbius strip is [[zero]].
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| ==Properties==
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| [[File:August Ferdinand Möbius.png|200px|thumbnail|right|August Ferdinand Möbius]]
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| The Möbius strip has several curious properties. 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. This single continuous curve demonstrates that the Möbius strip has only one [[Boundary (topology)|boundary]].
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| Cutting a Möbius strip along the center line with a pair of scissors yields one long strip with two full twists in it, rather than two separate strips; the result is not a Möbius strip. This happens because the original strip only has one edge that is twice as long as the original strip. Cutting creates a second independent edge, half of which was on each side of the scissors. Cutting this new, longer, strip down the middle creates two strips wound around each other, each with two full twists.
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| If the strip is cut along about a third of the way in from the edge, it creates two strips: One is a thinner Möbius strip — it is the center third of the original strip, comprising 1/3 of the width and the same length as the original strip. The other is a longer but thin strip with two full twists in it — this is a [[neighbourhood (mathematics)|neighborhood]] of the edge of the original strip, and it comprises 1/3 of the width and twice the length of the original strip.
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| Other analogous strips can be obtained by similarly joining 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]]. (If this knot is unravelled, the strip is made with eight half-twists in addition to an [[overhand knot]].) A strip with ''N'' half-twists, when bisected, becomes a strip with ''N'' + 1 full twists. Giving it extra twists and reconnecting the ends produces figures called paradromic rings.
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| A strip with an odd-number of half-twists, such as the Möbius strip, will have only one surface and one boundary. A strip twisted an even number of times will have two surfaces and two boundaries.
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| If a strip with an odd number of half-twists is cut in half along its length, it will result in a single, longer strip, with twice as many full twists as the number of half-twists in the original. Alternatively, if a strip with an even number of half-twists is cut in half along its length, it will result in two linked strips, each with the same number of twists as the original.
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| == Geometry and topology ==
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| [[Image:MobiusJoshDif.jpg|thumb|right|A ray-traced parametric plot of a Möbius strip]]
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| [[Image:Moebius strip.svg|thumb|right|A parametric plot of a Möbius strip]]
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| [[Image:MöbiusStripAsSquare.svg|thumb|right|To turn a [[rectangle]] into a Möbius strip, join the edges labelled ''A'' so that the directions of the arrows match.]]
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| One way to represent the Möbius strip as a subset of [[Euclidean space|'''R'''<sup>3</sup>]] is using the parametrization:
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| :<math>x(u,v)= \left(1+\frac{v}{2} \cos \frac{u}{2}\right)\cos u</math>
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| :<math>y(u,v)= \left(1+\frac{v}{2} \cos\frac{u}{2}\right)\sin u</math>
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| :<math>z(u,v)= \frac{v}{2}\sin \frac{u}{2}</math>
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| where {{nowrap|0 ≤ ''u'' < 2π}} and {{nowrap|−1 ≤ ''v'' ≤ 1}}. This creates a Möbius strip of width 1 whose center circle has radius 1, lies in the ''xy'' plane and is centered at {{nowrap|(0, 0, 0)}}. The parameter ''u'' runs around the strip while ''v'' moves from one edge to the other.
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| In [[cylindrical coordinates|cylindrical polar coordinates]] {{nowrap|(''r'', θ, ''z'')}}, an unbounded version of the Möbius strip can be represented by the equation:
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| :<math>\log(r)\sin\left(\frac{1}{2}\theta\right)=z\cos\left(\frac{1}{2}\theta\right).</math>
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| === Fattest rectangular Möbius strip in 3-space ===
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| If a smooth Möbius strip in 3-space is a rectangular one—that is, created from identifying two opposite sides of a geometrical rectangle—then it is known to be possible if the aspect ratio of the rectangle is greater than the square root of 3. (Note that it is the shorter sides of the rectangle that are identified to obtain the Möbius strip.) For an aspect ratio less than or equal to the square root of 3, however, a smooth embedding of a rectangular Möbius strip into 3-space may be impossible.
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| As the aspect ratio approaches the limiting ratio of <math>\sqrt{3}</math> from above, any such rectangular Möbius strip in 3-space seems to approach a shape that in the limit can be thought of as a strip of three equilateral triangles, folded on top of one another so that they occupy just one equilateral triangle in 3-space.
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| If the Möbius strip in 3-space is only once continuously differentiable (in symbols: C<sup>1</sup>), however, then the theorem of Nash-Kuiper shows that there is no lower bound.
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| A rectangular strip too wide to be simply twisted and joined can always be crimped into one narrow enough that can. This is achieved with a sufficient even number of equally spaced parallel folds, in alternate directions, at right angles to the edges to be joined.<ref>
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| {{cite book |last = Barr |first = Stephen |title = Experiments in Topology |year = 1964
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| |publisher = Thomas Y. Crowell Company |location = New York |pages = 48,200-201 }}</ref>
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| With two folds, for example, the [[Cross section (geometry)|cross section]] of the strip is an N which, being still an N after a half-twist, is compatible with joining the edges.
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| === Topology ===
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| [[Topology|Topologically]], the Möbius strip can be defined as the [[square (geometry)|square]] {{nowrap|[0,1] × [0,1]}} with its top and bottom sides [[quotient space|identified]] by the relation {{nowrap|(''x'', 0) ~ (1 − ''x'', 1)}} for {{nowrap|0 ≤ ''x'' ≤ 1,}} as in the diagram on the right.
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| A less used presentation of the Möbius strip is as the topological quotient of a torus.<ref>Tony Phillips, ''[http://www.ams.org/mathmedia/archive/10-2006-media.html Tony Phillips' Take on Math in the Media],'' [[American Mathematical Society]], October 2006</ref> A torus can be constructed as the square {{nowrap|[0,1] × [0,1]}} with the edges identified as {{nowrap|(0,y) ~ (1,y)}} (glue left to right) and {{nowrap|(x,0) ~ (x,1)}} (glue bottom to top). If one then also identified {{nowrap|(x,y) ~ (y,x)}}, then one obtains the Möbius strip. The diagonal of the square (the points (x,x) where both coordinates agree) becomes the boundary of the Möbius strip, and carries an orbifold structure, which geometrically corresponds to "reflection" – [[geodesic]]s (straight lines) in the Möbius strip reflect off the edge back into the strip. Notationally, this is written as T<sup>2</sup>/S<sub>2</sub> – the 2-torus quotiented by the [[group action]] of the [[symmetric group]] on two letters (switching coordinates), and it can be thought of as the [[configuration space]] of two unordered points on the circle, possibly the same (the edge corresponds to the points being the same), with the torus corresponding to two ordered points on the circle.
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| 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 manifold|orientable]]. In fact, the Möbius strip is the epitome of the topological phenomenon of [[Orientability|nonorientability]]. This is because 1) two-dimensional shapes (surfaces) are the lowest-dimensional shapes for which nonorientability is possible, and 2) the Möbius strip is the '''only''' surface that is topologically a subspace of '''every''' non-orientable surface. As a result, any surface is non-orientable if and only if it contains a Möbius band as a subspace.
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| 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 ''S''<sup>1</sup> with a fiber the [[unit interval]], ''I'' = [0,1]. Looking only at the edge of the Möbius strip gives a nontrivial two point (or '''Z'''<sub>2</sub><!-- What is Z2??-->) bundle over ''S''<sup>1</sup>.
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| === Computer graphics ===
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| A simple construction of the Möbius strip which can be used to portray it in computer graphics or modeling packages is as follows :
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| *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 one complete revolution is the Möbius strip.
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| *Take a Möbius strip and cut it along the middle of the strip. This will form a new strip, which is a rectangle joined by rotating one end a whole turn. By cutting it down the middle again, this forms two interlocking whole-turn strips.
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| == Open Möbius band ==
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| The '''open Möbius band''' is formed by deleting the [[boundary (topology)|boundary]] of the standard Möbius band. It is constructed from the set {{nowrap|1=''S'' = { (''x'',''y'') ∈ '''R'''<sup>2</sup> : 0 ≤ ''x'' ≤ 1 and 0 < ''y'' < 1 }}} by identifying (glueing) the points (0,''y'') and (1,1−''y'') for all {{nowrap|1=0 < ''y'' < 1.}}
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| Alternatively, it may also be constructed as a complete surface, by starting with portion of the plane R<sup>2</sup> defined by 0 ≤ y ≤ 1 and identifying (x,0) with (-x,1) for all x in R (the reals). The resulting metric makes the open Möbius band into a (geodesically) complete flat surface (i.e., having Gaussian curvature equal to 0 everywhere). This is the only metric on the Möbius band, up to uniform scaling, that is both flat and complete.
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| Like the plane and the open cylinder, the open Möbius band admits not only a complete metric of constant curvature 0, but also a complete metric of constant negative curvature, say = -1. One way to see this is to begin with the [[Poincaré half-plane model|upper half plane (Poincaré) model]] of the [[Hyperbolic geometry|hyperbolic plane]] ℍ, namely ℍ = {(x,y) ∈ ℝ<sup>2</sup> | y > 0} with the [[Riemannian metric]] given by (dx<sup>2</sup> + dy<sup>2</sup>) / y<sup>2</sup>. The orientation-preserving isometries of this metric are all the maps f: ℍ → ℍ of the form f(z) := (az + b) / (cz + d), where a, b, c, d are real numbers satisfying ad - bc = 1. Here z is a complex number with Im(z) > 0, and we have identified ℍ with {z ∈ ℂ | Im(z) > 0}. One orientation-reversing isometry g of ℍ given by g(z) := -conj(z), where conj(z) denotes the complex conjugate of z. These facts imply that the mapping h: ℍ → ℍ given by h(z) := -2⋅conj(z) is an orientation-reversing isometry of ℍ that generates an infinite cyclic group G of isometries. The quotient ℍ / G of the action of this group can easily be seen to be topologically a Möbius band. But it is also easy to verify that it is complete and noncompact, with constant negative curvature = -1.
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| The space of unoriented lines in the plane is [[diffeomorphic]] to the open Möbius band.<ref>
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| {{cite journal|last=Parker|first=Phillip|year=1993|title=Spaces of Geodesics|journal=Aportaciones Matemáticas|series=Notas de Investigación|pages=67 − 79|publisher=UASLP|url=http://www.math.wichita.edu/~pparker/research/sog.htm}}</ref>
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| To see why, let L(θ) denote the line through the origin at an angle θ to the positive x-axis. For each L(θ) there is the family P(θ) of all lines in the plane that are perpendicular to L(θ). Topologically, the family P(θ) is just a line (because each line in P(θ) intersects the line L(θ) in just one point). In this way, as θ increases in the range 0° ≤ θ < 180°, the line L(θ) represents a line's worth of distinct lines in the plane. But when θ reaches 180°, L(180°) is identical to L(0), and so the families P(0°) and P(180°) of perpendicular lines are also identical families. The line L(0°), however, has returned to itself as L(180°) '''pointed in the opposite direction'''.
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| Every line in the plane corresponds to exactly one line in some family P(θ), for exactly one θ, for 0° ≤ θ < 180°, and P(180°) is identical to P(0°) but returns pointed in the opposite direction. This ensures that the space of all lines in the plane — the union of all the L(θ) for 0° ≤ θ ≤ 180° — is an open Möbius band.
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| The rigid motions of the plane naturally induce bijections of ''the space of lines in the plane'' to itself, which form a group of self-homeomorphisms of the space of lines. But there is no metric on the space of lines in the plane which is invariant under the action of this group of homeomorphisms. In this sense the space of lines in the plane has no natural metric on it.
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| The upshot of this is that the Möbius band possesses a natural 4-dimensional [[Lie group]] of self-homeomorphisms (those given above by rigid motions of the plane), but this high degree of symmetry cannot be exhibited as the group of isometries of any metric.
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| ==Möbius band with round boundary==
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| The edge, or [[boundary (topology)|boundary]], of a Möbius strip is [[homeomorphic]] (topologically equivalent) to a [[circle]]. Under the usual embeddings of the strip in Euclidean space, as above, the boundary is not a round circle. However, it is possible to [[embedding|embed]] a Möbius strip in three dimensions so that the boundary is round like a circle. For example, see Figures 307, 308, and 309 of "Geometry and the imagination".<ref>
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| {{cite book
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| | author = [[David Hilbert|Hilbert, David]]; [[Stephan Cohn-Vossen|Cohn-Vossen, Stephan]]
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| | title = Geometry and the Imagination
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| | edition = 2nd ed.
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| | year = 1952
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| | publisher = Chelsea
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| | isbn = 0-8284-1087-9}}
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| </ref> | |
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| A much more geometric embedding begins with a minimal [[Klein bottle]] immersed in the 3-sphere, as discovered by Blaine Lawson. We then take half of this Klein bottle to get an embedded Möbius band in 4-space; the result is sometimes called the "Sudanese Möbius Band"
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| .<ref>{{cite web | |
| | author = Dan Asimov, Doug Lerner
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| | title = Issue 17 SIGGRAPH '84 Electronic Theater
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| | year = 1984
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| | url = http://www.siggraph.org/publications/video-review/1_36.php}}
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| </ref> Here "sudanese" is a [[portmanteau]] of the names of two topologists, Sue Goodman and Daniel Asimov. Applying stereographic projection to the Sudanese band places it in 3-dimensional space, as can be seen below—a version due to George Francis can be found [http://www.geom.uiuc.edu/graphics/pix/Special_Topics/Differential_Geometry/illiview.html here].
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| From Lawson's minimal Klein bottle we derive an embedding of the band into the [[3-sphere]] ''S''<sup>3</sup>, regarded as a subset of '''C'''<sup>2</sup>, which in turn is a copy of '''R'''<sup>4</sup>. We map angles <math>\eta, \varphi</math> to complex numbers <math>z_1, z_2</math> via
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| :<math>z_1 = \sin\eta\,e^{i\varphi}</math>
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| :<math>z_2 = \cos\eta\,e^{i\varphi/2}.</math>
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| Here the parameter ''η'' runs from 0 to ''π'' and ''φ'' runs from 0 to 2''π''. Since {{nowrap begin}} | ''z''<sub>1</sub> |<sup>2</sup> + | ''z''<sub>2</sub> |<sup>2</sup> = 1 {{nowrap end}} the embedded surface lies entirely in ''S''<sup>3</sup>. The boundary of the strip is given by | ''z''<sub>2</sub> | = 1 (corresponding to ''η'' = 0, ''π''), which is clearly a circle on the 3-sphere.
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| To obtain an embedding of the Möbius strip in '''R'''<sup>3</sup> one maps ''S''<sup>3</sup> to '''R'''<sup>3</sup> via a [[stereographic projection]]. The projection point can be any point on ''S''<sup>3</sup> which does not lie on the embedded Möbius strip (this rules out all the usual projection points). One possible choice is <math>\left\{1/\sqrt{2},i/\sqrt{2}\right\}</math>. 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 '''R'''<sup>3</sup> with a circular edge and no self-intersections.
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| [[Image:MobiusSnail2B.png|center|580px]]
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| ==Related objects==
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| 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.<ref> | |
| {{cite book
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| | last = Spivak
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| | first = Michael
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| | title = A Comprehensive Introduction to Differential Geometry, Volume I
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| | edition = 2nd
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| | series =
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| | year = 1979
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| | publisher = Publish or Perish
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| | location = Wilmington, Delaware
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| | pages = 591
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| | chapter =
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| | chapterurl =
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| | quote =
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| | ref =
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| }}</ref>
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| 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.<ref>
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| {{cite book
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| | last = Hilbert
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| | first = David
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| | coauthors = S. Cohn-Vossen
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| | edition = 2nd
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| | title = Geometry and the Imagination
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| | year = 1999
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| | publisher = American Mathematical Society
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| | location = Providence, Rhode Island
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| | pages = 316
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| | isbn = 978-0-8218-1998-2
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| }}</ref> 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.
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| In [[graph theory]], the [[Möbius ladder]] is a [[cubic graph]] closely related to the Möbius strip.
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| In 1968, Gonzalo Vélez Jahn (UCV, Caracas, Venezuela) discovered three dimensional bodies with Möbian characteristics, later described by Martin Gardner as prismatic rings that became [[toroidal polyhedron]]s.<ref>
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| {{cite book
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| | last = Gardner
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| | first = Martin
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| | title = Mathematical Games
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| | year = 1978
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| | publisher = Scientific American
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| | location = Providence, Rhode Island
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| | pages = 12–13
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| }}</ref>
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| ==Applications==
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| [[Image:Moebiusstripscarf.jpg|right|thumb|200px|A [[scarf]] designed as a Möbius strip.]]
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| There have been several technical applications for the Möbius strip. Giant Möbius strips have been used as [[conveyor belt]]s 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 ribbon]]s, as they allow the ribbon to be twice as wide as the print head while using both halves evenly.
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| A [[Möbius resistor]] is an electronic circuit element that cancels its own inductive reactance. [[Nikola Tesla]] patented similar technology in 1894:<ref>{{US Patent|512340}}</ref> "Coil for Electro Magnets" was intended for use with his system of global transmission of electricity without wires. | |
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| The Möbius strip is the [[configuration space]] of two unordered points on a circle. Consequently, in [[music theory]], the space of all two-note chords, known as [[Dyad (music)|dyads]], takes the shape of a Möbius strip; this and generalizations to more points is a significant [[Orbifold#Music_theory|application of orbifolds to music theory]].<ref>[http://www.livescience.com/strangenews/080507-math-music.html Clara Moskowitz, Music Reduced to Beautiful Math, LiveScience]</ref><ref>{{cite journal |author= Dmitri Tymoczko |title= The Geometry of Musical Chords |date= 7 July 2006 |volume= 313 |issue= 5783 |pages= 72–4 |journal= [[Science (journal)|Science]] |doi= 10.1126/science.1126287 |pmid= 16825563}}</ref>
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| In [[physics]]/electro-technology:
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| *as a compact resonator with the resonance frequency which is half that of identically constructed linear coils<ref>IEEE of Trans. Microwave Theory and Tech., volume. 48, No. 12, pp. 2465–2471, Dec. 2000</ref>
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| *as an inductionless resistor<ref>{{US patent|3267406}}</ref>
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| *as [[superconductors]] with high transition temperature<ref>{{Cite journal|first=Raul Perez|last=Enriquez|title=A Structural parameter for High Tc Superconductivity from an Octahedral Moebius Strip in RBaCuO: 123 type of perovskite|journal=Rev Mex Fis|volume=48|issue=supplement 1|year=2002|page=262|arxiv=cond-mat/0308019|postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}.</ref>
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| In [[chemistry]]/nano-technology:
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| *as [[molecular knot]]s with special characteristics (Knotane [2], Chirality)
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| *as molecular engines<ref>Angew Chem Int OD English one 2005 February 25; 44 (10): 1456–77.</ref>
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| *as graphene volume (nano-graphite) with new electronic characteristics, like helical magnetism<ref>{{Cite journal|title=Novel Electronic States in Graphene Ribbons -Competing Spin and Charge Orders-|first1=Atsushi|last1=Yamashiro|first2=Yukihiro|last2=Shimoi|first3=Kikuo|last3=Harigaya|first4=Katsunori|last4=Wakabayashi|arxiv=cond-mat/0309636|journal=Physica E|volume=22|issue=1–3|pages=688–691|year=2004|doi=10.1016/j.physe.2003.12.100|postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}.</ref>
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| *in a special type of aromaticity: [[Möbius aromaticity]]
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| *charged particles that have been caught in the magnetic field of the earth can move on a Möbius band<ref>IEEE Transactions on plasma Science, volume. 30, No. 1, February 2002</ref>
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| *the [[cyclotide]] (cyclic protein) Kalata B1, active substance of the plant ''Oldenlandia affinis'', contains Möbius topology for the peptide backbone.
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| ==See also==
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| <div style="-moz-column-count:4; column-count:4;">
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| *[[Cross-cap]]
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| *[[Klein bottle]]
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| *[[List of cycles]]
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| *[[Loop (knot)|Loop]]
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| *[[Möbius transformation]]
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| *[[Molecular knot]]
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| *[[Paradox]]
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| *[[Real projective plane]]
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| *[[Strange loop]]
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| *[[Umbilic torus]]
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| </div>
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| ==References==
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| {{Reflist|30em}}
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| ==External links==
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| {{Wiktionary|Möbius strip}}
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| {{Commons category|Moebius strip|Möbius strip}}
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| *[http://www.bbc.co.uk/dna/h2g2/A337592 h2g2 - The Amazing Möbius Strip]
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| *[http://www.toroidalsnark.net/mkmb.html Knitted version]
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| *[http://mechproto.olin.edu/final_projects/average_jo.html The Möbius Gear — A functional planetary gear model in which one gear is a Möbius strip]
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| *{{MathWorld|urlname=MoebiusStrip|title=Möbius Strip}}
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| *[http://www.slideshare.net/sualeh/beyond-the-mobius-strip Beyond the Mobius Strip]
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| *[http://www.openculture.com/2013/02/the_genius_of_js_bachs_crab_canon_visualized_on_a_mobius_strip.html J.S.Bachs Crab canon visualized on a mobius strip]
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| *[http://www.imagesandmusic.nl/dhtml/CD-The-Infinite-Road.htm An M.C. Escher inspired Möbius ring world]
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| {{DEFAULTSORT:Mobius Strip}}
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| [[Category:Topology]]
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| [[Category:Recreational mathematics]]
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| [[Category:Surfaces]]
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