MOBIUS STRIP

Raychell M. SantosMAT-Mathematics

Special Topics in Mathematics

What is a Möbius Strip?

The Möbius strip or Möbius band is a surface with only one side and only one boundary component. The Möbius strip has the mathematical property of being non-orientable.

Also, the Mobius strip or band is a loop with a twist.

The Möbius strip or Möbius band is a surface with only one side and only one boundary component. The Möbius strip has the mathematical property of being non-orientable.

Also, the Mobius strip or band is a loop with a twist.

Illustration 1:

An object moving on its surface could reach all points of the surface without crossing over an edge.

An object moving on its surface could reach all points of the surface without crossing over an edge.

Illustration 2:

 If an ant were to crawl along the length of this strip, it would return to its starting point having traversed every part of the strip without ever crossing an edge.

 If an ant were to crawl along the length of this strip, it would return to its starting point having traversed every part of the strip without ever crossing an edge.

What is a Möbius Strip?

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 in fact two types of Möbius strips depending on the direction of the half-twist: clockwise and counter-clockwise. The Möbius strip is therefore chiral, which is to say that it has "handedness" (right-handed or left-handed).

 The twisted paper model, such as the Mobius strip is a developable surface.

What is a Möbius Strip?

In mathematics, a developable surface is a surface with zero Gaussian curvature. That is, it is "surface" that can be flattened onto a plane without distortion (i.e. "stretching" or "compressing").

 A positive Gaussian curvature value means the surface is locally either a peak or a valley. A negative value means the surface locally has a saddle points. And a zero value means the surface is flat in at least one direction (ie, a Mobius strip, a plane and a cylinder have zero Gaussian curvature).

From left to right: a surface of negative Gaussian curvature (hyperboloid), a surface of zero Gaussian curvature (cylinder), and a surface of positive Gaussian curvature (sphere).

History of the Möbius Strip

It was discovered by August Ferdinand Möbius, a pioneer in the field of topology and a German astronomer and Mathematician, in 1858.

It was discovered by August Ferdinand Möbius, a pioneer in the field of topology and a German astronomer and Mathematician, in 1858.

But it was independently discovered by German mathematician Johann Benedict Listing in 1858 who published it while Mobius did not.

But it was independently discovered by German mathematician Johann Benedict Listing in 1858 who published it while Mobius did not.

History of the Möbius Strip

This band was used during the Industrial Revolution, when many all factories had a single source of power (steam or water wheel). Individual pieces were connected to turning shafts by belts and wheels and these belts needed to be replace frequently due to wear. By using the Möbius band, the bands wore evenly and lasted twice as long. These bands are still used today in modern factories. They are also employed in some types of printer bands (last twice as long as a circular band).

This band was used during the Industrial Revolution, when many all factories had a single source of power (steam or water wheel). Individual pieces were connected to turning shafts by belts and wheels and these belts needed to be replace frequently due to wear. By using the Möbius band, the bands wore evenly and lasted twice as long. These bands are still used today in modern factories. They are also employed in some types of printer bands (last twice as long as a circular band).

Properties of Mobius Strip

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.

Properties of Mobius Strip

Cutting a Möbius strip along the center line yields one long strip with two full twists in it, rather than two separate strips.

Properties of Mobius Strip

If the strip is cut along about a third of the way in from the edge, it creates two strips.

Properties of Mobius 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.

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.

As continuous-loop recording tapes (to double the playing time)

Occurrence and Use in Nature and Technology

Möbius strips are common in the manufacture of dot matrix 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.

Occurrence and Use in Nature and Technology

A device called a Mobius resistor is an electronic circuit element which has the property of canceling its own inductive reactance.

A Möbius resistor is an electrical component made up of two conductive surfaces separated by a dielctric material, twisted 180° and connected to form a Mobius strip. It provides a resistor which has no residual self-inductance, meaning that it can resist the flow of electricity without causing magnetic interference at the same time.

Current in a Mobius resistor

Occurrence and Use in Nature and Technology

In physics/electro-technology:

as compact resonator with the resonance frequency which is half that of identically constructed linear coils as inductionless resistance as superconductors with high transition temperature

In chemistry/nano-technology:

as molecular knots with special characteristics as molecular engines as graphene volume (nano-graphite) with new electronic characteristics, like helical magnetism In a special type of aromaticity: Möbius aromaticity Charged particles, which were caught in the magnetic field of the earth, 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.

Occurrence and Use in Art and Popular Culture

The Möbius strip has provided inspiration both for sculptures and for graphical art.

The artist M. C. Escher was especially fond of it and based several of his prints on it. One famous example, Möbius Strip II, features ants crawling around the surface of a Möbius strip.

His flight of swans looks like it might be a Möbius Strip, but it's not.

A granite sculpture by Max Bill entitled Endless Ribbon

Double Mobius Strip Sculpture by Plamen Yordanov

Occurrence and Use in Art and Popular Culture

This science fiction story suggest that our universe might be some kind of generalised Mobius strip

In the short story A Subway Named Möbius, by A.J. Deutch, the Boston subway authority builds a new line, but the system becomes so tangled that it turns into a Möbius strip, and trains start to disappear.

Occurrence and Use in Art and Popular Culture

Jazz pianist Cedar Walton released an album entitled Mobius in 1975, featuring cover art based on the strip.

In 2007, Paul McCartney  included a Möbius strip in the 45 R.P.M version of his single “Ever Present Past”. The single package included a unique "cut out and make your own mobius strip" insert.

Occurrence and Use in Art and Popular Culture

The fascinating shape of the Mobius strip expresses transformation. Thus, it has been chosen as the Universal symbol of recycling since it represents the process of transforming waste materials into useful resources.

Medals, such as those awarded in competitions like the Olympics, often feature a neck ribbon configured as a Möbius strip. This allows the ribbon to fit comfortably around the neck while the medal lies flat on the chest.

Occurrence and Use in Art and Popular Culture

A scarf designed as a Möbius strip. A wedding ring designed as a Möbius strip.

A Gateway to Higher Dimensions by popular science writer Clifford Pickover explores the weird world of the Mobius Strip

Occurrence and Use in Art and Popular Culture

The Mobius Strip and the Infinity Symbol

The symbol itself is properly called a lemniscus, a latin noun which means "pendant ribbon" and was first used in 1694 by Jacob Bernoulli (1654-1705) to describe a planar curve now called the Lemniscate of Bernoulli.

The mathematical symbol for infinity is called the lemniscate.

The lemniscate is patterned after the device known as a mobius strip, forming an 'endless' two- dimensional surface.

"The world is a circle without a beginning,And nobody knows where it really ends.Everything depends on where youAre in the circle that never begins.Nobody knows where the circle ends...”

Excerpts from 'The World Is A Circle‘ by Burt Bacharach and Hal David [1972]