Ribbon of Möbius
ribbon of Möbius is a curiosity very easy to make as the diagram shows it below.
Clothes industry of the famous ribbon
It is enough to use a long paper band, to make him undergo a torsion of a half-turn then to stick the two ends. One then is discovered having two unexpected properties: this surface has only one face and only one edge. In mathematics one speaks about nondirectional.
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.
This object was designed simultaneously in by the German mathematician August Ferdinand Möbius and by its compatriot Johann Benedict Listing, although not working together. The name of the first was retained thanks to a report presented to in Paris. One also finds the denominations of bandage, ring or girdle of Möbius or Moebius, in particular in the translations.
Alternatives of the traditional ribbon can be obtained while subjecting the paper band an odd number of half-turns of direct or retrograde direction. The objects obtained are with the traditional ribbon, but one cannot pass (i.e. by homotopy) from one model to another in the space of dimension 3.
The ribbon of Möbius to a half-turn can also be seen like a part of surface of Möbius. Consequently, it can be generated by a swivelling segment whose center describes a fixed circle.
Mathematically, one can define the ribbon asquotient unit of the unit <maths>\R \times [ - 1,1 ] \, \!</maths> by relation of equivalence defined by: <maths>(x, y) \sim (x', y') \, \!</maths> if and only if <maths>\exists K \in \Z \,: \, (x', y') = (x+k, (-1)^k y) \, \!</maths>. By comparison, a "normal" ribbon (trunk of cylinder) would be defined by the relation <maths>\exists K \in \Z \,: \, (x', y') = (x+k, y) \, \!</maths>.
A schematized version of the ribbon of Möbius is used like logo matters since the first day of the Earth in . The Möbius loop indicates that a product can be recycled and also that it can be made recycled materials. It is acted in fact of a ribbon with three half-turns.
See too
External bonds
Artistic adaptations
Thru the Moebius Strip, 80 minutes entirely numerical film is currently in phase of production by Frank Foster In Global DIGITAL Productions (Hong Kong). Exit envisaged in 2005.
The characteristic of this phenomenon was also used in an episode of.
Möbius (that is to say Moebius without the diereses) is also the name of the double final episode of season 8 of .
