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The grandfather paradox is a proposed paradox of time travel, first described by the science fiction writer René Barjavel in his 1943 book Le Voyageur Imprudent (The Imprudent Traveller).[1[[http://en.wikipedia.org/wiki/Grandfather_paradox#cite_note-barjavel-0|]]] The paradox is this: suppose a man traveled back in time and killed his biological grandfather before the latter met the traveller's grandmother. As a result, one of the traveller's parents (and by extension, the traveller himself) would never have been conceived. This would imply that he could not have travelled back in time after all, which in turn implies the grandfather would still be alive, and the traveller would have been conceived, allowing him to travel back in time and kill his grandfather. Thus each possibility seems to imply its own negation, a type of logical paradox.
An equivalent paradox is known (in philosophy) as autoinfanticide—that is, going back in time and killing oneself as a baby—though when the word was first coined in a paper by Paul Horwich he used the odd version autofanticide.[2[[http://en.wikipedia.org/wiki/Grandfather_paradox#cite_note-horwich-1|]]]
The grandfather paradox has been used to argue that backwards time travel must be impossible. However, a number of possible ways of avoiding the paradox have been proposed, such as the idea that the timeline is fixed and unchangeable, or the idea that the time traveler will end up in a parallel timeline, while the timeline in which the traveler was born remains independent.
Since quantum mechanics is governed by probabilities, an unmeasured entity (in this case, one's historical grandfather) has numerous probable states. When that entity is measured, the number of its probable states singularises, resulting in a single outcome (in this case, ultimately, oneself). Therefore, since the outcome of one's grandfather is known, one killing one's grandfather would be incompatible with that outcome. Thus, the outcome of one's trip backwards in time must be complementary with the state from which one left.[3[[http://en.wikipedia.org/wiki/Grandfather_paradox#cite_note-complementary-2|]]]
There is no possible way to travel back in time. If one were to have a reason to travel back into time, such as to murderHitler before the Holocaust, then if one were to fix the problem or reason they had to go back in time, there would have been no reason to go back in time in the first place. Thus the traveller would have no reason to go back in time, so he would not have fixed the problem, and would therefore be in his original situation.
See the Novikov self-consistency principle and Kip S. Thorne for one view on how backwards time travel could be possible without a danger of paradoxes. According to this hypothesis, the only possible timelines are those which are entirely self-consistent, so that anything a time traveler does in the past must have been part of history all along, and the time traveler can never do anything to prevent the trip back in time from being made since this would represent an inconsistency. In laymen's terms, this is often called destiny, and it is sometimes unpopular because it contradicts the "common sense" notion that people choose their own fates.
There could be "an ensemble of parallel universes" such that when the traveller kills the grandfather, the act took place in (or resulted in the creation of) a parallel universe in which the traveller's counterpart will never be conceived as a result. However, his prior existence in the original universe is unaltered.
Examples of parallel universes postulated in physics are:
M-theory is put forward as a hypothetical master theory that unifies the five superstring theories, although at present it is largely incomplete. One possible consequence of ideas drawn from M-theory is that multiple universes in the form of 3-dimensional membranes known as branes could exist side-by-side in a fourth large spatial dimension (which is distinct from the concept of time as a fourth dimension) - see Brane cosmology. However, there is currently no argument from physics that there would be one brane for each physically possible version of history as in the many-worlds interpretation, nor is there any argument that time travel would take one to a different brane.
The idea of preventing paradoxes by supposing that the time traveler is taken to a parallel universe while his original history remains intact, which is discussed above in the context of science, is also common in science fiction - see Time travel as a means of creating historical divergences.
See also: Predestination paradoxes in fiction
Another resolution, of which the Novikov self-consistency principle can be taken as an example, holds that if one were to travel back in time, the laws of nature (or other intervening cause) would simply forbid the traveler from doing anything that could later result in their time travel not occurring. For example, a shot fired at the traveler's grandfather will miss, or the gun will jam, or misfire, or the grandfather will be injured but not killed, or the person killed will turn out to be not the real grandfather, or some other event will occur to prevent the attempt from succeeding. No action the traveler takes to effect change will ever succeed, as there will always be some form of "bad luck" or coincidence preventing the outcome. In effect, the traveler will be unable to change history from the state they found it. Very commonly in fiction, the time traveler does not merely fail to prevent the actions he seeks to prevent; he in fact precipitates them (see predestination paradox), usually by accident.
This theory might lead to concerns about the existence of free will (in this model, free will may be an illusion, or at least not unlimited). This theory also assumes that causality must be constant: i.e. that nothing can occur in the absence of cause, whereas some theories hold that an event may remain constant even if its initial cause was subsequently eliminated.
Closely related but distinct is the notion of the time line as self-healing. The time-traveler's actions are like throwing a stone in a large lake; the ripples spread, but are soon swamped by the effect of the existing waves. For instance, a time traveler could assassinate a politician who led his country into a disastrous war, but the politician's followers would then use his murder as a pretext for the war, and the emotional effect of that would cancel out the loss of the politician's charisma. Or the traveler could prevent a car crash from killing a loved one, only to have the loved one killed by a mugger, or fall down the stairs, choke on a meal, killed by a stray bullet, etc. In the 2002 film The Time Machine, this scenario is shown where the main character builds a time machine to save his girlfriend who got killed by a robber, yet she still dies, only from a car crash instead. In some stories it is only the event that precipitated the time traveler's decision to travel back in time that cannot be substantially changed, in others all attempted changes will be "healed" in this way, and in still others the universe can heal most changes but not sufficiently drastic ones. This is also the explanation advanced by the Doctor Whorole-playing game, which supposes that Time is like a stream; you can dam it, divert it, or block it, but the overall direction it is headed will resume after a period of conflict.
It also may not be clear whether the time traveller altered the past or precipitated the future he remembers, such as a time traveller who goes back in time to persuade an artist—whose single surviving work is famous—to hide the rest of the works to protect them. If, on returning to his time, he finds that these works are now well-known, he knows he has changed the past. On the other hand, he may return to a future exactly as he remembers, except that a week after his return, the works are found. Were they actually destroyed, as he believed when he travelled in time, and has he preserved them? Or was their disappearance occasioned by the artist's hiding them at his urging, and the skill with which they were hidden, and so the long time to find them, stemmed from his urgency?
Some science fiction stories suggest that causing any paradox will cause the destruction of the universe, or at least the parts of space and time affected by the paradox. The plots of such stories tend to revolve around preventing paradoxes.
Consideration of the grandfather paradox has led some to the idea that time travel is by its very nature paradoxical and therefore logically impossible, on the same order as round squares. For example, the philosopher Bradley Dowden made this sort of argument in the textbook Logical Reasoning, where he wrote:
“
Nobody has ever built a time machine that could take a person back to an earlier time. Nobody should be seriously trying to build one, either, because a good argument exists for why the machine can never be built. The argument goes like this: suppose you did have a time machine right now, and you could step into it and travel back to some earlier time. Your actions in that time might then prevent your grandparents from ever having met one another. This would make you not born, and thus not step into the time machine. So, the claim that there could be a time machine is self-contradictory.
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.
Cutting a Möbius strip along the center line 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 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, each with two full twists.
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 neighborhood of the edge of the original strip, and it comprises 1/3 of the width and twice the length of the original strip.
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 three half-twists in addition to an overhand knot.) Cutting a Möbius strip, giving it extra twists, and reconnecting the ends produces unexpected figures called paradromic rings.
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.
If a strip with a given number of half-twists is cut in half lengthwise, it will result in a longer strip, with the same number of loops as there are half-twists in the original, if the original strip has an odd number of half-twists, or two conjoined strips, each with the same number of twists as the original, if the original strip has an even number of half-twists.
A parametric plot of a Möbius stripTo turn a rectangle into a Möbius strip, join the edges labelled A so that the directions of the arrows match.
One way to represent the Möbius strip as a subset of R3 is using the parametrization:
x(u,v)= textstyle left(1+frac{1}{2}v cos frac{1}{2}uright)cos u
y(u,v)= textstyle left(1+frac{1}{2}vcosfrac{1}{2}uright)sin u
z(u,v)= textstyle frac{1}{2}vsin frac{1}{2}u
where 0 ≤ u < 2π and −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 (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:
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 one complete revolution is the Möbius strip.
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.
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-sphereS3 regarded as a subset of R4. A parametrization for this embedding is given by
z_1 = sineta,e^{ivarphi}
z_2 = coseta,e^{ivarphi/2}.
Here we have used complex notation and regarded R4 as C2. The parameter η runs from 0 to π and φ runs from 0 to 2π. Since z1 the embedded surface lies entirely on S3. The boundary of the strip is given by | z2 | = 1 (corresponding to η = 0, π), 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.
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[[http://en.wikipedia.org/wiki/M%C3%B6bius_strip#cite_note-4|]]]
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[[http://en.wikipedia.org/wiki/M%C3%B6bius_strip#cite_note-5|]]] 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.
A scarf designed as a Möbius strip.
There have been several technical applications for the Möbius strip. 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 while 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:[7[[http://en.wikipedia.org/wiki/M%C3%B6bius_strip#cite_note-6|]]] "Coil for Electro Magnets" was intended for use with his system of global transmission of electricity without wires.
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 dyads, takes the shape of a Möbius strip.[8[[http://en.wikipedia.org/wiki/M%C3%B6bius_strip#cite_note-7|]]][9[[http://en.wikipedia.org/wiki/M%C3%B6bius_strip#cite_note-8|]]]
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Grandfather paradox
From Wikipedia, the free encyclopedia
Jump to: navigation, searchThe grandfather paradox is a proposed paradox of time travel, first described by the science fiction writer René Barjavel in his 1943 book Le Voyageur Imprudent (The Imprudent Traveller).[1[[http://en.wikipedia.org/wiki/Grandfather_paradox#cite_note-barjavel-0|]]] The paradox is this: suppose a man traveled back in time and killed his biological grandfather before the latter met the traveller's grandmother. As a result, one of the traveller's parents (and by extension, the traveller himself) would never have been conceived. This would imply that he could not have travelled back in time after all, which in turn implies the grandfather would still be alive, and the traveller would have been conceived, allowing him to travel back in time and kill his grandfather. Thus each possibility seems to imply its own negation, a type of logical paradox.
An equivalent paradox is known (in philosophy) as autoinfanticide—that is, going back in time and killing oneself as a baby—though when the word was first coined in a paper by Paul Horwich he used the odd version autofanticide.[2[[http://en.wikipedia.org/wiki/Grandfather_paradox#cite_note-horwich-1|]]]
The grandfather paradox has been used to argue that backwards time travel must be impossible. However, a number of possible ways of avoiding the paradox have been proposed, such as the idea that the timeline is fixed and unchangeable, or the idea that the time traveler will end up in a parallel timeline, while the timeline in which the traveler was born remains independent.
Contents
[hide]* 1 Scientific theories[edit] Scientific theories
[edit] Complementary time travel
Since quantum mechanics is governed by probabilities, an unmeasured entity (in this case, one's historical grandfather) has numerous probable states. When that entity is measured, the number of its probable states singularises, resulting in a single outcome (in this case, ultimately, oneself). Therefore, since the outcome of one's grandfather is known, one killing one's grandfather would be incompatible with that outcome. Thus, the outcome of one's trip backwards in time must be complementary with the state from which one left.[3[[http://en.wikipedia.org/wiki/Grandfather_paradox#cite_note-complementary-2|]]][edit] Hedges Theory
There is no possible way to travel back in time. If one were to have a reason to travel back into time, such as to murder Hitler before the Holocaust, then if one were to fix the problem or reason they had to go back in time, there would have been no reason to go back in time in the first place. Thus the traveller would have no reason to go back in time, so he would not have fixed the problem, and would therefore be in his original situation.[edit] Novikov self-consistency principle
See the Novikov self-consistency principle and Kip S. Thorne for one view on how backwards time travel could be possible without a danger of paradoxes. According to this hypothesis, the only possible timelines are those which are entirely self-consistent, so that anything a time traveler does in the past must have been part of history all along, and the time traveler can never do anything to prevent the trip back in time from being made since this would represent an inconsistency. In laymen's terms, this is often called destiny, and it is sometimes unpopular because it contradicts the "common sense" notion that people choose their own fates.[edit] Parallel universes/alternate timelines
There could be "an ensemble of parallel universes" such that when the traveller kills the grandfather, the act took place in (or resulted in the creation of) a parallel universe in which the traveller's counterpart will never be conceived as a result. However, his prior existence in the original universe is unaltered.Examples of parallel universes postulated in physics are:
[edit] Theories in science fiction
[edit] Parallel universes resolution
The idea of preventing paradoxes by supposing that the time traveler is taken to a parallel universe while his original history remains intact, which is discussed above in the context of science, is also common in science fiction - see Time travel as a means of creating historical divergences.[edit] Restricted action resolution
See also: Predestination paradoxes in fictionAnother resolution, of which the Novikov self-consistency principle can be taken as an example, holds that if one were to travel back in time, the laws of nature (or other intervening cause) would simply forbid the traveler from doing anything that could later result in their time travel not occurring. For example, a shot fired at the traveler's grandfather will miss, or the gun will jam, or misfire, or the grandfather will be injured but not killed, or the person killed will turn out to be not the real grandfather, or some other event will occur to prevent the attempt from succeeding. No action the traveler takes to effect change will ever succeed, as there will always be some form of "bad luck" or coincidence preventing the outcome. In effect, the traveler will be unable to change history from the state they found it. Very commonly in fiction, the time traveler does not merely fail to prevent the actions he seeks to prevent; he in fact precipitates them (see predestination paradox), usually by accident.
This theory might lead to concerns about the existence of free will (in this model, free will may be an illusion, or at least not unlimited). This theory also assumes that causality must be constant: i.e. that nothing can occur in the absence of cause, whereas some theories hold that an event may remain constant even if its initial cause was subsequently eliminated.
Closely related but distinct is the notion of the time line as self-healing. The time-traveler's actions are like throwing a stone in a large lake; the ripples spread, but are soon swamped by the effect of the existing waves. For instance, a time traveler could assassinate a politician who led his country into a disastrous war, but the politician's followers would then use his murder as a pretext for the war, and the emotional effect of that would cancel out the loss of the politician's charisma. Or the traveler could prevent a car crash from killing a loved one, only to have the loved one killed by a mugger, or fall down the stairs, choke on a meal, killed by a stray bullet, etc. In the 2002 film The Time Machine, this scenario is shown where the main character builds a time machine to save his girlfriend who got killed by a robber, yet she still dies, only from a car crash instead. In some stories it is only the event that precipitated the time traveler's decision to travel back in time that cannot be substantially changed, in others all attempted changes will be "healed" in this way, and in still others the universe can heal most changes but not sufficiently drastic ones. This is also the explanation advanced by the Doctor Who role-playing game, which supposes that Time is like a stream; you can dam it, divert it, or block it, but the overall direction it is headed will resume after a period of conflict.
It also may not be clear whether the time traveller altered the past or precipitated the future he remembers, such as a time traveller who goes back in time to persuade an artist—whose single surviving work is famous—to hide the rest of the works to protect them. If, on returning to his time, he finds that these works are now well-known, he knows he has changed the past. On the other hand, he may return to a future exactly as he remembers, except that a week after his return, the works are found. Were they actually destroyed, as he believed when he travelled in time, and has he preserved them? Or was their disappearance occasioned by the artist's hiding them at his urging, and the skill with which they were hidden, and so the long time to find them, stemmed from his urgency?
[edit] Destruction resolution
Some science fiction stories suggest that causing any paradox will cause the destruction of the universe, or at least the parts of space and time affected by the paradox. The plots of such stories tend to revolve around preventing paradoxes.[edit] Other considerations
Consideration of the grandfather paradox has led some to the idea that time travel is by its very nature paradoxical and therefore logically impossible, on the same order as round squares. For example, the philosopher Bradley Dowden made this sort of argument in the textbook Logical Reasoning, where he wrote:Consideration of the possibility of backwards time travel in a hypothetical universe described by a Gödel metric led famed logician Kurt Gödel to assert that time might itself be a sort of illusion.[6[[http://en.wikipedia.org/wiki/Grandfather_paradox#cite_note-yourgrau-5|]]][7[[http://en.wikipedia.org/wiki/Grandfather_paradox#cite_note-holt-6|]]] He seems to have been suggesting something along the lines of the block time view in which time does not really "flow" but is just another dimension like space, with all events at all times being fixed within this 4-dimensional "block".
Möbius strip
From Wikipedia, the free encyclopedia
Jump to: navigation, searchThis article is about the mathematical object. See Mobius Band (music group) for the music group.The Möbius strip or Möbius band (pronounced /ˈmiːbiəs/ or /ˈmoʊbiəs/ in English, IPA: [ˈmøːbiʊs] in German) (alternatively written Mobius or Moebius in English) is a surface with only one side and only one boundary component. The Möbius strip 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[[http://en.wikipedia.org/wiki/M%C3%B6bius_strip#cite_note-0|]]][2[[http://en.wikipedia.org/wiki/M%C3%B6bius_strip#cite_note-1|]]][3[[http://en.wikipedia.org/wiki/M%C3%B6bius_strip#cite_note-2|]]]
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 has "handedness" (as in right-handed or left-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[[http://en.wikipedia.org/wiki/M%C3%B6bius_strip#cite_note-3|]]]
The Euler characteristic of the Möbius strip is zero.
Contents
[hide]* 1 Properties[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.Cutting a Möbius strip along the center line 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 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, each with two full twists.
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 neighborhood of the edge of the original strip, and it comprises 1/3 of the width and twice the length of the original strip.
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 three half-twists in addition to an overhand knot.) Cutting a Möbius strip, giving it extra twists, and reconnecting the ends produces unexpected figures called paradromic rings.
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.
If a strip with a given number of half-twists is cut in half lengthwise, it will result in a longer strip, with the same number of loops as there are half-twists in the original, if the original strip has an odd number of half-twists, or two conjoined strips, each with the same number of twists as the original, if the original strip has an even number of half-twists.
[edit] Geometry and topology
One way to represent the Möbius strip as a subset of R3 is using the parametrization:
where 0 ≤ u < 2π and −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 (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:
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 :
[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
Here we have used complex notation and regarded R4 as C2. The parameter η runs from 0 to π and φ runs from 0 to 2π. Since z1 the embedded surface lies entirely on S3. The boundary of the strip is given by | z2 | = 1 (corresponding to η = 0, π), 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[[http://en.wikipedia.org/wiki/M%C3%B6bius_strip#cite_note-4|]]]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[[http://en.wikipedia.org/wiki/M%C3%B6bius_strip#cite_note-5|]]] 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] Occurrence and use in nature and technology
There have been several technical applications for the Möbius strip. 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 while 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:[7[[http://en.wikipedia.org/wiki/M%C3%B6bius_strip#cite_note-6|]]] "Coil for Electro Magnets" was intended for use with his system of global transmission of electricity without wires.
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 dyads, takes the shape of a Möbius strip.[8[[http://en.wikipedia.org/wiki/M%C3%B6bius_strip#cite_note-7|]]][9[[http://en.wikipedia.org/wiki/M%C3%B6bius_strip#cite_note-8|]]]
In physics/electro-technology:
In chemistry/nano-technology:
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