Möbius strip

This article is about the mathematical object. For musical group, see Mobius Band (band).

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: /ˈmɜːbiəs/ or US: /ˈmoʊbiəs/; German: [ˈmøːbi̯ʊs]), also Mobius or Moebius, is a surface with only one surface 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.

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).

The Möbius band (equally known as the Möbius strip) is not a surface of only one geometry (i.e., of only one exact size and shape), such as the half-twisted paper strip depicted in the illustration to the right. Rather, mathematicians refer to the (closed) Möbius band as any surface that is homeomorphic to this strip. Its boundary is a simple closed curve, i.e., homeomorphic to a circle. This allows for a very wide variety of geometric versions of the Möbius band as surfaces each having a definite size and shape. For example, any closed rectangle with length L and width W can be glued to itself (by identifying one edge with the opposite edge after a reversal of orientation) to make a Möbius band. Some of these can be smoothly modeled in 3-dimensional space, and others cannot (see section Fattest rectangular Möbius strip in 3-space below). Yet another example is the complete open Möbius band (see section Open Möbius band below). Topologically, this is slightly different from the more usual — closed — Möbius band, in that any open Möbius band has no boundary.

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.

The Euler characteristic of the Möbius strip is zero.

Properties

August Ferdinand Möbius

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 with a pair of scissors 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 single, longer strip, with twice as many full twists as the number of 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 linked strips, each with the same number of twists as the original.

Geometry and topology

A ray-traced 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:

$x(u,v)=\left(1+{\frac {v}{2}}\cos {\frac {u}{2}}\right)\cos u$

$y(u,v)=\left(1+{\frac {v}{2}}\cos {\frac {u}{2}}\right)\sin u$

$z(u,v)={\frac {v}{2}}\sin {\frac {u}{2}}$

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:

$\log(r)\sin \left({\frac {1}{2}}\theta \right)=z\cos \left({\frac {1}{2}}\theta \right).$

Fattest rectangular Möbius strip in 3-space

If a smooth Möbius strip in 3-space is a rectangular one—that is, created from identifying two opposite sides of a geometrical rectangle—then it is known to be possible if the aspect ratio of the rectangle is greater than the square root of 3. (Note that it is the shorter sides of the rectangle that are identified to obtain the Möbius strip.) For an aspect ratio less than or equal to the square root of 3, however, a smooth embedding of a rectangular Möbius strip into 3-space may be impossible.

As the aspect ratio approaches the limiting ratio of ${\sqrt {3}}$ from above, any such rectangular Möbius strip in 3-space seems to approach a shape that in the limit can be thought of as a strip of three equilateral triangles, folded on top of one another so that they occupy just one equilateral triangle in 3-space.

If the Möbius strip in 3-space is only once continuously differentiable (in symbols: C¹), however, then the theorem of Nash-Kuiper shows that there is no lower bound.

A rectangular strip too wide to be simply twisted and joined can always be crimped into one narrow enough that can. This is achieved with a sufficient even number of equally spaced parallel folds, in alternate directions, at right angles to the edges to be joined. With two folds, for example, the cross section of the strip is an N which, being still an N after a half-twist, is compatible with joining the edges.

Topology

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 topological quotient of a torus. 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. In fact, the Möbius strip is the epitome of the topological phenomenon of nonorientability. This is because 1) two-dimensional shapes (surfaces) are the lowest-dimensional shapes for which nonorientability is possible, and 2) the Möbius strip is the only surface that is topologically a subspace of every non-orientable surface. As a result, any surface is non-orientable if and only if it contains a Möbius band as a subspace.

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.

Computer graphics

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

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.

Alternatively, it may also be constructed as a complete surface, by starting with portion of the plane R² defined by 0 ≤ y ≤ 1 and identifying (x,0) with (-x,1) for all x in R (the reals). The resulting metric makes the open Möbius band into a (geodesically) complete flat surface (i.e., having Gaussian curvature equal to 0 everywhere). This is the only metric on the Möbius band, up to uniform scaling, that is both flat and complete.

Like the plane and the open cylinder, the open Möbius band admits not only a complete metric of constant curvature 0, but also a complete metric of constant negative curvature, say = -1. One way to see this is to begin with the upper half plane (Poincaré) model of the hyperbolic plane ℍ, namely ℍ = {(x,y) ∈ ℝ² | y > 0} with the Riemannian metric given by (dx² + dy²) / y². The orientation-preserving isometries of this metric are all the maps f: ℍ → ℍ of the form f(z) := (az + b) / (cz + d), where a, b, c, d are real numbers satisfying ad - bc = 1. Here z is a complex number with Im(z) > 0, and we have identified ℍ with {z ∈ ℂ | Im(z) > 0}. One orientation-reversing isometry g of ℍ given by g(z) := -conj(z), where conj(z) denotes the complex conjugate of z. These facts imply that the mapping h: ℍ → ℍ given by h(z) := -2⋅conj(z) is an orientation-reversing isometry of ℍ that generates an infinite cyclic group G of isometries. The quotient ℍ / G of the action of this group can easily be seen to be topologically a Möbius band. But it is also easy to verify that it is complete and noncompact, with constant negative curvature = -1.

The space of unoriented lines in the plane is diffeomorphic to the open Möbius band.

To see why, let L(θ) denote the line through the origin at an angle θ to the positive x-axis. For each L(θ) there is the family P(θ) of all lines in the plane that are perpendicular to L(θ). Topologically, the family P(θ) is just a line (because each line in P(θ) intersects the line L(θ) in just one point). In this way, as θ increases in the range 0° ≤ θ < 180°, the line L(θ) represents a line's worth of distinct lines in the plane. But when θ reaches 180°, L(180°) is identical to L(0), and so the families P(0°) and P(180°) of perpendicular lines are also identical families. The line L(0°), however, has returned to itself as L(180°) pointed in the opposite direction.

Every line in the plane corresponds to exactly one line in some family P(θ), for exactly one θ, for 0° ≤ θ < 180°, and P(180°) is identical to P(0°) but returns pointed in the opposite direction. This ensures that the space of all lines in the plane — the union of all the L(θ) for 0° ≤ θ ≤ 180° — is an open Möbius band.

The rigid motions of the plane naturally induce bijections of the space of lines in the plane to itself, which form a group of self-homeomorphisms of the space of lines. But there is no metric on the space of lines in the plane which is invariant under the action of this group of homeomorphisms. In this sense the space of lines in the plane has no natural metric on it.

The upshot of this is that the Möbius band possesses a natural 4-dimensional Lie group of self-homeomorphisms (those given above by rigid motions of the plane), but this high degree of symmetry cannot be exhibited as the group of isometries of any metric.

Möbius band with round boundary

The edge, or boundary, of a Möbius strip is homeomorphic (topologically equivalent) to a circle. Under the usual embeddings of the strip in Euclidean space, as above, the boundary is not a round circle. However, it is possible to embed a Möbius strip in three dimensions so that the boundary is round like a circle. For example, see Figures 307, 308, and 309 of "Geometry and the imagination".

A much more geometric embedding begins with a minimal Klein bottle immersed in the 3-sphere, as discovered by Blaine Lawson. We then take half of this Klein bottle to get an embedded Möbius band in 4-space; the result is sometimes called the "Sudanese Möbius Band" . Here "sudanese" is a portmanteau of the names of two topologists, Sue Goodman and Daniel Asimov. Applying stereographic projection to the Sudanese band places it in 3-dimensional space, as can be seen below—a version due to George Francis can be found here.

From Lawson's minimal Klein bottle we derive an embedding of the band into the 3-sphere S³, regarded as a subset of C², which in turn is a copy of R⁴. We map angles $\eta ,\varphi$ to complex numbers $z_{1},z_{2}$ via

$z_{1}=\sin \eta \,e^{i\varphi }$

$z_{2}=\cos \eta \,e^{i\varphi /2}.$

Here 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 in 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). One possible choice is $\left\{1/{\sqrt {2}},i/{\sqrt {2}}\right\}$. 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

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.

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. 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.

In 1968, Gonzalo Vélez Jahn (UCV, Caracas, Venezuela) discovered three dimensional bodies with Möbian characteristics, later described by Martin Gardner as prismatic rings that became toroidal polyhedrons.

Applications

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: "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.

In physics/electro-technology:

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

In chemistry/nano-technology:

• as molecular knots with special characteristics (Knotane [2], Chirality)
• 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 that have been 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.

See also

• Cross-cap
• Klein bottle
• List of cycles
• Loop
• Möbius transformation
• Molecular knot
• Paradox
• Real projective plane
• Strange loop
• Umbilic torus

External links

• h2g2 - The Amazing Möbius Strip
• Knitted version
• The Möbius Gear — A functional planetary gear model in which one gear is a Möbius strip
• Weisstein, Eric W. "Möbius Strip". MathWorld.
• Beyond the Mobius Strip
• J.S.Bachs Crab canon visualized on a mobius strip
• An M.C. Escher inspired Möbius ring world