Mobius Strip



The Mobius Strip is a surface with only one side and only one boundary component. The Mbius strip has the mathematical property of being non-orientable. It can be realized as a ruled surface. It was discovered independently by the German mathematicians August Ferdinand Mbius and Johann Benedict Listing in 1858.

The shape of the Mbius Strip dates back to ancient times. An Alexandrian manuscript of early Alchemical diagrams contains an illustration with the visual proportions of the Mbius Strip. This image, on a page titled "The Chrysopoeia of Cleopatra", has the appearance of an Ouroboros, and is referred to as the "One, All".

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 Mbius strips depending on the direction of the half-twist: clockwise and counterclockwise. The Mbius strip is therefore chiral, which is to say that it has "handedness" (right-handed or left-handed, like amino acids)

A Mbius strip made with a piece of paper and tape.

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.

It is straightforward to find algebraic equations the solutions of which have the topology of a Mbius 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.

The Euler characteristic of the Mbius strip is zero.

A model can easily be created by taking a paper strip and giving it a half-twist, and then merging the ends of the strip together to form a single strip. In Euclidean space there are in fact two types of Mbius strips depending on the direction of the half-twist: clockwise and counterclockwise. The Mbius strip is therefore chiral, which is to say that it is "handed".

The Mbius strip has several curious properties.

A model of a Mbius 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 Mbius strip has only one boundary.

If the strip is cut along the above line, instead of getting two separate strips, it becomes one long strip with two full twists in it, which is not a Mbius 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.

Alternatively, cutting along a Mbius strip about a third of the way in from the edge, creates two strips: One is a thinner Mbius strip - it is the center third of the original strip. The other is a long strip with two full twists in it - this is a neighborhood of the edge of the original strip.

Other interesting combinations of strips can be obtained by making Mbius 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. Cutting a Mbius strip, giving it extra twists, and reconnecting the ends produces unexpected figures called paradromic rings.

A closely related 'strange' geometrical object is the Klein bottle. A Klein bottle can be produced by gluing two Mbius strips together along their edges; 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 Mbius strip. Going in the other direction, if one glues a disk to a Mbius strip by identifying their boundaries, the result is the projective plane. In order to visualize this, it is helpful to deform the Mbius strip so that its boundary is an ordinary circle. The projective plane, like the Klein bottle, cannot be embedded in three-dimensions without self-intersections.

In terms of identifications of the sides of a square, as given above: the real projective plane is made by gluing the remaining two sides with 'consistent' orientation (arrows making an anti-clockwise loop); and the Klein bottle is made the other way. In graph theory, the Mbius ladder is a cubic graph closely related to the Mbius strip.




Appearance in Science and Technology

There have been technical applications. Giant Mbius 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). Mbius 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 whilst using both half-edges evenly.

There have been several technical applications for the Mbius strip. Giant Mbius 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). Mbius 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 Mbius 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: "Coil for Electro Magnets" was intended for use with his system of global transmission of electricity without wires.

The international symbol for recycling is a Mbius loop.




Art and Popular Culture

The Mbius 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 lithographs on it. One famous example, Mbius Strip II, features ants crawling around the surface of a Mbius strip.

It is also a recurrent feature in science fiction stories, such as Arthur C. Clarke's The Wall of Darkness.

Science fiction stories sometimes suggest that our universe might be some kind of generalized Mbius strip. In the short story "A Subway Named Mbius", by A.J. Deutsch, the Boston subway authority builds a new line, but the system becomes so tangled that it turns into a Mbius strip, and trains start to disappear. The Mbius strip also features prominently in Brian Lumley's Necroscope series of novels.

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




Music

The Mbius 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 Mbius strip; this and generalizations to more points is a significant application of orbifolds to music theory. Read more




In the News ...


Mobius strips of light made for the first time   New Scientist - January 29, 2015

Twist a two-dimensional strip of paper then tape its ends together and it transforms into a one-sided loop. It's not magic; it's a Mobius strip. These mathematical structures show up everywhere from M.C. Escher drawings to electrical circuits, but almost never in nature. Now, a team of physicists have shown for the first time that light can be coaxed into a Mobius shape.




Strange New Twist: Researchers Discover Mbius Symmetry in Metamaterials   Science Daily - December 22, 2010

Mbius symmetry, the topological phenomenon that yields a half-twisted strip with two surfaces but only one side, has been a source of fascination since its discovery in 1858 by German mathematician August Mbius. As artist M.C. Escher so vividly demonstrated in his "parade of ants," it is possible to traverse the "inside" and "outside" surfaces of a Mbius strip without crossing over an edge. For years, scientists have been searching for an example of Mbius symmetry in natural materials without any success. A team of scientists has discovered Mbius symmetry in metamaterials -- materials engineered from artificial "atoms" and "molecules" with electromagnetic properties that arise from their structure rather than their chemical composition.





INFINITY




MATHEMATICS INDEX


PHYSICAL SCIENCES INDEX


ALPHABETICAL INDEX


CRYSTALINKS HOME PAGE


PSYCHIC READING WITH ELLIE


2012 THE ALCHEMY OF TIME