# Möbius strip

Möbius strip is a surface with only one side (when embedded in three-dimensional Euclidean space) and only one boundary curve.

The Möbius strip is the simplest non-orientable surface. It can be realized as a ruled surface. Its discovery is attributed independently to the German mathematicians Johann Benedict Listing and August Ferdinand Möbius in 1858, though similar structures can be seen in Roman mosaics c. 200–250 AD.

An example of a Möbius strip can be created by taking a strip of paper and giving one end a half-twist, then joining the ends to form a loop; its boundary is a simple closed curve which can be traced by a single unknotted string.

Any topological space homeomorphic to this example is also called a Möbius strip, allowing for a very wide variety of geometric realizations as surfaces with a definite size and shape. For example, any rectangle can be glued left-edge to right-edge with a reversal of orientation.

Some, but not all, of these can be smoothly modeled as surfaces in Euclidean space. A closely related, but not homeomorphic, surface is the complete open Möbius band, a surface with no boundaries in which the width of the strip is extended infinitely to become a Euclidean line.

A half-twist clockwise gives an embedding of the Möbius strip which cannot be moved or stretched to give the half-twist counterclockwise; thus, a Möbius strip embedded in Euclidean space is a chiral object with right- or left-handedness. The Möbius strip can also be embedded by twisting the strip any odd number of times, or by knotting and twisting the strip before joining its ends.

The Euler characteristic of the Möbius strip is zero.

A closely related ‘strange’ geometrical object is the Klein bottle. A Klein bottle could in theory be produced by gluing two Möbius strips together along their edges; however this cannot be done in ordinary three-dimensional Euclidean space without creating self-intersections.

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.

Going in the other direction, if one glues a disk to a Möbius strip by identifying their boundaries, the result is the projective plane. To visualize this, it is helpful to deform the Möbius strip so that its boundary is an ordinary circle. The real projective plane, like the Klein bottle, cannot be embedded in three-dimensions without self-intersections.

Applications of Möbius strip:

In physics/electro-technology as:

• A compact resonator with a resonance frequency that is half that of identically constructed linear coils
• An inductionless resistor
• Superconductors with high transition temperature
• Möbius resonato

In chemistry/nano-technology as:

• Molecular knots with special characteristics (Knotane [2], Chirality)
• Molecular engines
• Graphene volume (nano-graphite) with new electronic characteristics, like helical magnetism
• A special type of aromaticity: Möbius aromaticity
• Charged particles caught in the magnetic field of the Earth that can move on a Möbius band
• The cyclotide (cyclic protein) kalata B1, active substance of the plant Oldenlandia affinis, contains Möbius topology for the peptide backbone.