A Möbius 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 the entire length of the strip (on both sides of the original paper) without ever crossing an edge.

The Möbius strip or Möbius band (UK /ˈmɜrbiəs/ or US /ˈmoʊbiəs/; German: [ˈmøːbi̯ʊs]), also Mobius or Moebius, is a surface with only one side and only one boundary component. The Möbius 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 Möbius and Johann Benedict Listing in 1858.^[1][2][3]

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 two types of Möbius strips depending on the direction of the half-twist: clockwise and counterclockwise. That is to say, it is a chiral object with "handedness" (right-handed or left-handed).

It is straightforward to find algebraic equations the solutions of which have the topology of a Möbius 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.^[4]

The Euler characteristic of the Möbius strip is zero.

Properties[link]

The Möbius strip has several curious properties. 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.

Cutting a Möbius strip along the center line yields one long strip with two full twists in it, rather than two separate strips; the result is not a Möbius strip. This happens because the original strip only has one edge that is twice as long as the original strip. Cutting creates a second independent edge, half of which was on each side of the scissors. Cutting this new, longer, strip down the middle creates two strips wound around each other, each with two full twists.

If the strip is cut along about a third of the way in from the edge, it creates two strips: One is a thinner Möbius strip — it is the center third of the original strip, comprising 1/3 of the width and the same length as the original strip. The other is a longer but thin strip with two full twists in it — this is a neighborhood of the edge of the original strip, and it comprises 1/3 of the width and twice the length of the original 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. (If this knot is unravelled, the strip is made with eight half-twists in addition to an overhand knot.) A strip with N half-twists, when bisected, becomes a strip with N+1 full twists. Giving it extra twists and reconnecting the ends produces figures called paradromic rings.

A strip with an odd-number of half-twists, such as the Möbius strip, will have only one surface and one boundary. A strip twisted an even number of times will have two surfaces and two boundaries.

If a strip with an odd number of half-twists is cut in half along its length, it will result in a longer strip, with the same number of loops as there are half-twists in the original. Alternatively, if a strip with an even number of half-twists is cut in half along its length, it will result in two conjoined strips, each with the same number of twists as the original.

Geometry and topology[link]

A ray-tracing parametric plot of a Möbius strip

A parametric plot of a Möbius strip

To turn a rectangle into a Möbius strip, join the edges labelled A so that the directions of the arrows match.

One way to represent the Möbius strip as a subset of R³ is using the parametrization:

Failed to parse (Missing texvc executable; please see math/README to configure.): x(u,v)= \left(1+\frac{1}{2}v \cos \frac{u}{2}\right)\cos u

Failed to parse (Missing texvc executable; please see math/README to configure.): y(u,v)= \left(1+\frac{1}{2}v\cos\frac{u}{2}\right)\sin u

Failed to parse (Missing texvc executable; please see math/README to configure.): z(u,v)= \frac{1}{2}v\sin \frac{u}{2}

where 0 ≤ u < 2π and −1 ≤ v ≤ 1. This creates a Möbius strip of width 1 whose center circle has radius 1, lies in the xy plane and is centered at (0, 0, 0). The parameter u runs around the strip while v moves from one edge to the other.

In cylindrical polar coordinates (r, θ, z), an unbounded version of the Möbius strip can be represented by the equation:

Failed to parse (Missing texvc executable; please see math/README to configure.): \log(r)\sin\left(\frac{1}{2}\theta\right)=z\cos\left(\frac{1}{2}\theta\right).

Topologically, the Möbius strip can be defined as the square [0,1] × [0,1] with its top and bottom sides identified by the relation (x, 0) ~ (1 − x, 1) for 0 ≤ x ≤ 1, as in the diagram on the right.

A less used presentation of the Möbius strip is as the orbifold quotient of a torus.^[5] A torus can be constructed as the square [0,1] × [0,1] with the edges identified as (0,y) ~ (1,y) (glue left to right) and (x,0) ~ (x,1) (glue bottom to top). If one then also identified (x,y) ~ (y,x), then one obtains the Möbius strip. The diagonal of the square (the points (x,x) where both coordinates agree) becomes the boundary of the Möbius strip, and carries an orbifold structure, which geometrically corresponds to "reflection" – geodesics (straight lines) in the Möbius strip reflect off the edge back into the strip. Notationally, this is written as T²/S₂ – the 2-torus quotiented by the group action of the symmetric group on two letters (switching coordinates), and it can be thought of as the configuration space of two unordered points on the circle, possibly the same (the edge corresponds to the points being the same), with the torus corresponding to two ordered points on the circle.

The Möbius strip is a two-dimensional compact manifold (i.e. a surface) with boundary. It is a standard example of a surface which is not orientable. The Möbius strip is also a standard example used to illustrate the mathematical concept of a fiber bundle. Specifically, it is a nontrivial bundle over the circle S¹ with a fiber the unit interval, I = [0,1]. Looking only at the edge of the Möbius strip gives a nontrivial two point (or Z₂) bundle over S¹.

A simple construction of the Möbius strip which can be used to portray it in computer graphics or modeling packages is as follows :

• Take a rectangular strip. Rotate it around a fixed point not in its plane. At every step also rotate the strip along a line in its plane (the line which divides the strip in two) and perpendicular to the main orbital radius. The surface generated on one complete revolution is the Möbius strip.
• Take a Möbius strip and cut it along the middle of the strip. This will form a new strip, which is a rectangle joined by rotating one end a whole turn. By cutting it down the middle again, this forms two interlocking whole-turn strips.

Open Möbius band[link]

The open Möbius band is formed by deleting the boundary of the standard Möbius band. It is constructed from the set S = { (x,y) ∈ R² : 0 ≤ x ≤ 1 and 0 < y < 1} by identifying (glueing) the points (0,y) and (1,1−y) for all 0 < y < 1.

Occurrence and use in mathematics[link]

The space of unoriented lines in the plane is diffeomorphic to the open Möbius band.^[6]

To see why, notice that each line in the plane has an equation ax + by + c = 0 for fixed constants a, b and c. We can identify the equation ax + by + c = 0 with the point (a,b,c). However, the line given by ax + by + c = 0 is also given by λ(ax + by + c) = (λa)x + (λb)y + (λc) = 0 for all λ ≠ 0.

These equations, which give the same line, are identified with the points (λa,λb,λc). Thus, the space of lines in the plane is a (proper) subset of the real projective plane; where the equation ax + by + c = 0 corresponds to the point (a:b:c) in homogeneous coordinates. It is only a subset because some equations of the form ax + by + c = 0 do not give lines. We need to disallow a = b = 0 to be sure that the equation ax + by + c = 0 does indeed give a line. The space of unoriented lines in the plane is given by deleting the point (0:0:c) = (0:0:1) from the real projective plane. This space is exactly the open Möbius band.

Möbius band with flat edge[link]

The edge of a Möbius strip is topologically equivalent to the circle. Under the usual embeddings of the strip in Euclidean space, as above, this edge is not an ordinary (flat) circle. It is possible to embed a Möbius strip in three dimensions so that the edge is a circle. One way to think of this is to begin with a minimal Klein bottle immersed in the 3-sphere and take half of it, which is an embedded Möbius band in 4-space; this figure M has been called the "Sudanese Möbius Band". (The name comes from a combination of the names of two topologists, Sue Goodman and Daniel Asimov). Applying stereographic projection to M puts it in 3-dimensional space, as can be seen here as well as in the pictures below. (Some have incorrectly labeled the stereographic image in 3-space "Sudanese", but this is rather an image of the actual Sudanese one, which has a high degree of symmetry as a Riemannian surface: its isometry group contains SO(2). A well-known parametrization of it follows.)

To see this, first consider such an embedding into the 3-sphere S³ regarded as a subset of R⁴. A parametrization for this embedding is given by {(z₁(η,φ), z₂(η,φ))}, where

Failed to parse (Missing texvc executable; please see math/README to configure.): z_1 = \sin\eta\,e^{i\varphi}

Failed to parse (Missing texvc executable; please see math/README to configure.): z_2 = \cos\eta\,e^{i\varphi/2}.

Here we have used complex notation and regarded R⁴ as C². The parameter η runs from 0 to π and φ runs from 0 to 2π. Since | z₁ |² + | z₂ |² = 1 the embedded surface lies entirely on S³. The boundary of the strip is given by | z₂ | = 1 (corresponding to η = 0, π), which is clearly a circle on the 3-sphere.

To obtain an embedding of the Möbius strip in R³ one maps S³ to R³ via a stereographic projection. The projection point can be any point on S³ which does not lie on the embedded Möbius strip (this rules out all the usual projection points). Stereographic projections map circles to circles and will preserve the circular boundary of the strip. The result is a smooth embedding of the Möbius strip into R³ with a circular edge and no self-intersections.

Related objects[link]

A closely related 'strange' geometrical object is the Klein bottle. A Klein bottle can be produced by gluing two Möbius strips together along their edges; this cannot be done in ordinary three-dimensional Euclidean space without creating self-intersections.^[7]

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.^[8] Going in the other direction, if one glues a disk to a Möbius strip by identifying their boundaries, the result is the projective plane. In order to visualize this, it is helpful to deform the Möbius strip so that its boundary is an ordinary circle (see above). The real projective plane, like the Klein bottle, cannot be embedded in three-dimensions without self-intersections.

In graph theory, the Möbius ladder is a cubic graph closely related to the Möbius strip.

Applications[link]

A scarf designed as a Möbius strip.

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, and as continuous-loop recording tapes (to double the playing time). Möbius 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 halves evenly.

A Möbius resistor is an electronic circuit element that cancels its own inductive reactance. Nikola Tesla patented similar technology in 1894:^[9] "Coil for Electro Magnets" was intended for use with his system of global transmission of electricity without wires.

The Möbius 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 Möbius strip; this and generalizations to more points is a significant application of orbifolds to music theory.^[10][11]

In physics/electro-technology:

• as a compact resonator with the resonance frequency which is half that of identically constructed linear coils^[12]
• as an inductionless resistor^[13]
• as superconductors with high transition temperature^[14]

In chemistry/nano-technology:

• as molecular knots with special characteristics (Knotane [2], Chirality)
• as molecular engines^[15]
• as graphene volume (nano-graphite) with new electronic characteristics, like helical magnetism^[16]
• in a special type of aromaticity: Möbius aromaticity
• charged particles that have been caught in the magnetic field of the earth can move on a Möbius band^[17]
• the cyclotide (cyclic protein) Kalata B1, active substance of the plant Oldenlandia affinis, contains Möbius topology for the peptide backbone.

See also[link]

References[link]

External links[link]

Wikimedia Commons has media related to: Moebius surfaces

• Möbius Strip Video
• A virtual walk in the solar wind
• Animation of a rotating Sudanese Möbius band
• Louis Buckley (2007-07-15). "Möbius strip unravelled". Nature. DOI:10.1038/news070709-16. http://www.nature.com/news/2007/070709/full/070709-16.html. Retrieved 2006-07-16. 
• h2g2 - The Amazing Möbius Strip
• Johann Benedict Listing
• Knitted version
• Möbius Strip Music Box
• Möbius strip at cut-the-knot
• Science News 7/28/07: A Twist on the Möbius Band: Researchers work out the shape of a paper strip
• The Möbius Gear — A functional planetary gear model in which one gear is a Möbius strip
• Visualization of J. S. Bach's crab canon on a Möbius strip
• The protein with a topological twist
• Tridimensional and rotational möbius strip
• Weisstein, Eric W., "Möbius Strip" from MathWorld.
• Huckel spectra of Mobius pi systems
• Beyond the Mobius Strip

Robert Whitman.

Robert Whitman (born 1935 in New York City) is an American artist best known for his seminal theater pieces of the early 1960s combining visual and sound images, actors, film, slides, and evocative props in environments of his own making. Since the late 1960s he has worked with new technologies, and his most recent work incorporates cellphones.

Background[link]

Whitman studied literature at Rutgers University from 1953 to 1957 and art history at Columbia University in 1958. He is represented by The Pace Gallery in New York.

Theater works[link]

He was a member of the group of visual artists - Allan Kaprow, Red Grooms, Jim Dine, and Claes Oldenburg - who in the early 1960s presented theater pieces on the Lower East Side in Manhattan. Whitman has presented more than 40 theater pieces in the United States and abroad, including American Moon, E.G. and Mouth at the Rueben Gallery. Night Time Sky was his contribution to the First New York Theater Rally in New York in 1965; Prune Flat was first presented at the Cinematheque in New York in 1965 and has been performed numerous times since.

In 1966 Whitman was one of the 10 New York artists who worked with Billy Klüver and more than 30 engineers and scientists from Bell Telephone Laboratories to create works that incorporated new technology for 9 Evenings: Theatre & Engineering, a series of performance artworks presented October 13–23 in 1966 at the 69th Regiment Armory in New York City. For this piece, Two Holes of Water- 3, Whitman used seven automobiles on the floor of the Armory, from which were projected film, over-the-air television programs, and closed-circuit television projections of live performances and actions, including images from one of the first fiber-optic miniature video cameras.

A retrospective, Robert Whitman: Theater Works, 1960–1976 held in 1976 sponsored by the Dia Art Foundation and presented six earlier works and the premiere of Light Touch. His theater works have been presented at Galerie Maeght Festival in France, Contemporary Arts Museum, Houston, Texas, Moderna Museet, Stockholm; Walker Art Center, Vera List Art Center, MIT and many more. Ghost, his most recent theater performance, was staged at PaceWildenstein Gallery in New York City in 2002.

In 2003, Dia Art Foundation, New York presented, Playback, a large-scale retrospective exhibition of Whitman’s works. The exhibition traveled to Porto, Portugal, and opened at the Museum of Contemporary Art in Barcelona in September 2005. A major book, Playback, a comprehensive study of his work, accompanies this exhibition.

In the fall of 2004, Whitman presented a theater performance, Antenna, in Leeds, England, sponsored by Lumens, as part of the New Media Festival there.

Sculpture and installations[link]

He has collaborated with engineers on installations and works that incorporate new technology: laser sculptures, including Solid Red Line, in which a red line draws itself around the walls of a room and then erases itself; Pon, a sound-activated metallized PET film mirror installation shown at The Jewish Museum in New York in 1969.

His long collaboration with optics scientist John Forkner began with a mirror, light and sound installation for the Art and Technology exhibition at the Los Angeles County Museum of Art in 1971. They developed an optical system that allowed real images to float in space, to appear and disappear in an environment made up of a wall array of 6-inch corner reflectors in which the visitors saw multiple images of themselves.

Whitman was one of the co-founders of Experiments in Art and Technology along with engineers Billy Klüver and Fred Waldhauer and artist Robert Rauschenberg - a project to provide contemporary artists with access to new technology as it developed in research institutions and laboratories. Whitman was one of the core artists for the Pepsi Pavilion at Expo '70, Osaka Japan, a project administered by E.A.T. One of the main features of the interior of the Pavilion was the central performance space in a 90 ft diameter 120 degree spherical mirror made of aluminized reflective PET film, which produced real images of the visitors hanging upside down in space.

Significant one-person exhibitions of Whitman's sculpture and installation pieces include shows at The Museum of Modern Art in New York, The Hudson River Museum of Contemporary Art, Chicago, and Thielska Galleriet, Stockholm. Whitman has had one person gallery exhibitions at PaceWildenstein in New York, and has been included in many group exhibitions.

Telecommunications projects[link]

Whitman, working with Experiments in Art and Technology E.A.T in the early 1970s, developed and participated in a number of innovative communications projects : - Anand Project: he was part of an interdisciplinary team to develop methods for instructional television programming for rural Indian villages; - Children and Communications, open environments for children to work with a variety of communication equipment; - Telex: Q&A: a worldwide person-to-person question and answer opportunity using telex equipment in New York, Stockholm, Ahmedabad, India, and Tokyo; - Artists and Television, artists’ programming on New York cable channels.

In 1972 Whitman produced his first telephone piece, NEWS, in which participants, using pay phones, called in reports which were broadcast live over radio station WBAI. NEWS was performed later in Houston, Minneapolis, and other cities over a two- or three-year period.

A later performance in Leeds, England in 2002, utilized cell phones, and the calls were broadcast in real time on large speakers in a public square in the town. A recording of the performance was made available by the sponsor, Lumens, at Ubuweb.

In the summer of 2005, Whitman presented Local Report, a video cell phone project.

Awards[link]

Whitman has received many awards: Guggenheim Fellowship (1976); Creative Artists Public Service Grant; Citation of Fine Arts, Brandeis University; Creative Arts Award Xerox Company Grant.

See also[link]

• Happenings
• Expanded Cinema
• Fluxus at Rutgers University
• Experiments in Art and Technology

External links[link]

• The Pace Gallery
• Cellphone Performance (2004) on Ubuweb
• Local Report website
• Essay on Playback exhibition (2003)
• Brooklyn Rail Robert Whitman with Joan Waltemath