Ribbon of Möbius

See also: Moebius

In Topology, the ribbon of Möbius (also called band of Möbius ) is a closed Surface whose edge is reduced to a Cercle. It with the characteristic to be regulated and not-directional. This surface was described independently in 1858 by the mathematicians August Ferdinand Möbius (1790 - 1868) and Johann Benedict Listing (1808 - 1882) . The name of the first was retained thanks to a report presented to the Academy of Science with Paris. One also finds the denominations of band , ring or girdles of Möbius or Moebius, in particular in the translations.

It is easy to visualize the band of Möbius in space: a simple model is carried out while making undergo a torsion of a half-turn to a long band of Papier, then by sticking the two ends. If one divides the ribbon in two in the direction length, one obtains a ring single, twisted, but which has two distinct faces and two distinct edges.
If one recuts it in the direction length, one obtains… two rings distinct, bored and twisted one on the other.

The Ruban of Möbius also feeds, from its characteristic, of the debates in philosophy. The speculations whose it can be the object thus inspired celebrates it psychoanalyst Jacques Lacan.

Definition by torsion of a band in space

Traditional ribbon of Möbius

The ribbon of Möbius can be generated by a swivelling segment whose center describes a fixed circle. A Paramétrage correspondent is

The curves v = v₀ , T varying only, are many segments, connecting at uniform speed the point v = v₀, T = - 1 and the point v = v₀, T = 1 . This segment is thus length 2.

The curve T = 0 is a circle of radius 2 in the horizontal plane; it represents the trajectory of the center of the segments. The angle which the segment with the horizontal direction forms is v₀ . When the center made a full rotation on the horizontal circle (addition of π to varible the v ), the segment made a half-turn only. What causes connection for example point T = 1, v = π with T = - 1, v = 0 .

The edge of the ribbon is given by the curve T = 1 or T = - 1 . But it is the same curve: the edge of the ribbon of Möbius is in only one piece (related).

Other figures obtained by torsion

Alternatives of the traditional ribbon can be obtained while subjecting the paper band an odd number of half-turns of direct or retrograde direction. It is enough to adjust the preceding parameter setting:

Figures obtained for K and - K are énantiomorphes, i.e. images mirrors one of the other.

If one accepts even values of K one obtains ribbons with two faces, more or less twisted.

Comparison of the various ribbons

One can be interested in the curve forming the edge of these ribbons. It has a Entortillement different for each value from K . Twisting is calculated for example in projection (seen top), by counting the number of times where the curve passes above itself. One cannot deform continuously (i.e. by Homotopie) a type of ribbon in another in the space of dimension 3.

However the various ribbons are homeomorphic S with the traditional ribbon of Möbius, i.e. there is no intrinsic difference between them. This one is related to the way in which they are plunged in the space of dimension 3.

The ribbon of Möbius to a half-turn can also be seen like part of the surface of Möbius.

Calculation length of the ribbon

The ribbon of Möbius can be carried out with a flexible ribbon thickness of a sheet of paper of 70 gr. for example. To obtain a ribbon without abrupt folding, it is necessary that, for a width of ribbon equalizes to 1, the length either higher than 1,732 - or the square root of 3. It is possible to go towards smaller in length until making meet, with a helicoid inversion, the sides opposite of a square, but foldings will be abrupt.

Definition by abstract identification

Mathematically, one can define the ribbon as the unit quotient of the unit $\backslash \; R\; \backslash \; times\; \backslash ,\; \backslash !$ by the Relation of equivalence defined by: $(X,\; there)\; \backslash \; sim\; (x\text{'},\; y\text{'})\; \backslash ,\; \backslash !$ if and only if $\backslash \; exists\; K\; \backslash \; in\; \backslash \; Z\; \backslash ;\; \backslash ,\; (x\text{'},\; y\text{'})\; =\; (x+k,\; (-\; 1)\; ^k\; there)\; \backslash ,\; \backslash !$, provided Topology quotient. By comparison, a “normal” ribbon (trunk of cylinder) would be defined by the relation $\backslash \; exists\; K\; \backslash \; in\; \backslash \; Z\; \backslash ;\; \backslash ,\; (x\text{'},\; y\text{'})\; =\; (x+k,\; there)\; \backslash ,\; \backslash !$.

That makes it possible to see mathematically what occurs when the ribbon is cut out: if $p:\backslash R\; \backslash times\; \backslash \; rightarrow\; \backslash \; R\; \backslash \; times/\backslash \; Z$ is the application of passage to the quotient, $p\; (\backslash \; R\; \backslash \; times\; \backslash \; \{0\; \backslash \})$ is a circle of which the complementary one related is .

One can also carry out the ribbon of Möbius like the complementary one to a disc open in the projective Plane real (seen like the sphere after idendification of the points diametrically opposite).

Artistic representations

• a schematized version of the ribbon of Möbius is used like logo matters which can be recycled since the first Jour of the Earth in 1970. The Möbius loop indicates that a product can be recycled, or that it was manufactured starting from recycled materials. It is acted in fact of a ribbon with three half-turns.
• Films:
• Thru the Moebius Strip , 80 minutes entirely numerical film realized by Frank Foster in Global DIGITAL Productions (HongKong). The film left to the United States in 2005.
• Séries:
• the ribbon of Möbius was also used in the drawing of a planet in an episode of the cartoon Ulysses 31.
• Moebius is also the name of the double final episode of season 8 of Stargate SG-1.
• Video games:
• Mobius is also the " codename" of the pilot, and incidentally the hero whom you control in the play Ace Combat 4: Distant Thunder. Its complete code name is " Mobius 1" , and its personal badge a ribbon of Möbius.
• Mobius Ring is a race of the plays F-Zero AX and F-Zero GX left in Arcade and on Gamecube.
• Romance:
• In 2010, Odyssey two , the aberration of behavior of the computer HALL 9000 is indicated like a " buckle Hofstadter - Möbius".

• Representation:
• an enormous ring of Möbius is represented in the form of sculpture with Lille, in the district of Wazemmes. This sculpture, created by Marco Slinckaert occupies the center of a roundabout opposite the CPAM. It is also called snake by its great resemblance.
• Maurits Cornelis Escher, engraver and draftsman Dutch (1898-1972), made many studies on the ribbon of Möbius.
• Escher drew a representation of a band of Möbius with ants.

The band of Möbius tool of reflection

In the vocabulary of Jacques Lacan: 1962/63 - Distresses it - 1/9/63 - What makes that a specular image is distinct from what it represents? it is that the line becomes the left and conversely. - A surface with only one face cannot be turned over. - Thus a band of Moebius , if you turn over one from there on itself, it will be always identical to itself. It is what I invite not to have not of specular image.

See too

• Bottle of Klein
• Flexagone
• Orientation (mathematics)

External bonds

• Equations and representations

Pragmatic

• Realization of the ribbon
• Realization of the ribbon, characteristics, and properties of cuttings

various Reflections

• Vocabulary of Jacques Lacan

Simple: Möbius strip