Mobius Strip

The Mobius Strip is a surface with only one side and only one boundary component. The Mšbius 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 Mšbius and Johann Benedict Listing in 1858.

The shape of the Mšbius Strip dates back to ancient times. An Alexandrian manuscript of early Alchemical diagrams contains an illustration with the visual proportions of the Mšbius 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 Mšbius strips depending on the direction of the half-twist: clockwise and counterclockwise. The Mšbius strip is therefore chiral, which is to say that it has "handedness" (right-handed or left-handed, like amino acids)

A Mšbius 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 Mšbius 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 Mšbius 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 Mšbius strips depending on the direction of the half-twist: clockwise and counterclockwise. The Mšbius strip is therefore chiral, which is to say that it is "handed".

The Mšbius strip has several curious properties.

A model of a Mšbius 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 Mšbius 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 Mšbius 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 Mšbius strip about a third of the way in from the edge, creates two strips: One is a thinner Mšbius 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 Mšbius 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 Mšbius 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 Mšbius 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 Mšbius strip. Going in the other direction, if one glues a disk to a Mšbius strip by identifying their boundaries, the result is the projective plane. In order to visualize this, it is helpful to deform the Mšbius 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 Mšbius ladder is a cubic graph closely related to the Mšbius strip.

### Appearance in Science and Technology

There have been technical applications. Giant Mšbius 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). Mšbius 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 Mšbius strip. Giant Mšbius 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). Mšbius 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 Mšbius 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 Mšbius loop.

### Art and Popular Culture

The Mšbius 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, Mšbius Strip II, features ants crawling around the surface of a Mšbius 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 Mšbius strip. In the short story "A Subway Named Mšbius", by A.J. Deutsch, the Boston subway authority builds a new line, but the system becomes so tangled that it turns into a Mšbius strip, and trains start to disappear. The Mšbius 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 Mšbius strip. This allows the ribbon to fit comfortably around the neck while the medal lies flat on the chest.

### Music

The Mšbius 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 Mšbius 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 Mšbius Symmetry in Metamaterials   Science Daily - December 22, 2010

Mšbius 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 Mšbius. 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 Mšbius strip without crossing over an edge. For years, scientists have been searching for an example of Mšbius symmetry in natural materials without any success. A team of scientists has discovered Mšbius symmetry in metamaterials -- materials engineered from artificial "atoms" and "molecules" with electromagnetic properties that arise from their structure rather than their chemical composition.

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