Möbius strip

A Möbius strip made with paper and adhesive tape

In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface with only one side (when embedded in three-dimensional Euclidean space) and only one boundary curve. The Möbius strip is the simplest non-orientable surface. It can be realized as a ruled surface.

An example of a Möbius strip can be made by taking a strip of paper and giving one end a half-twist, then joining the ends to form a loop; its boundary is a simple closed curve which can be traced by a single unknotted string. Any topological space homeomorphic to this example is also called a Möbius strip, allowing for a very wide variety of geometric realizations as surfaces with a definite size and shape. For example any rectangle can be glued left-edge to right-edge with a reversal of orientation. Some, but not all, of these can be modeled as smooth surfaces in Euclidean space. The Möbius strip can also be embedded by twisting the strip any odd number of times, or by knotting and twisting the strip before joining its ends. A closely related surface is the complete open Möbius band, a surface with no boundaries in which the width of the strip is extended infinitely far to become a line.

History

Mosaic from ancient Sentinum depicting Aion holding a Möbius strip

Chain pump with a Möbius drive chain, by Ismail al-Jazari (1206)

The discovery of the Möbius strip as a mathematical object is attributed independently to the German mathematicians Johann Benedict Listing and August Ferdinand Möbius in 1858. However, it had been known long before, both as a physical object and in artistic depictions. In particular, it can be seen in several Roman mosaics from the third century AD. In many cases these merely depict coiled ribbons as boundaries around other objects; these can be either Möbius strips or annuli, depending on whether the number of coils is odd or even, respectively, and the existence of odd ones may be merely coincidental. In at least one case, the ribbon was depicted with different colors on different sides, but intended to have an odd number of coils, forcing its artist to make a clumsy fix at the point where the colors do not match up. In the case of a mosaic from the town of Sentinum (depicted), the Möbius strip depicts the zodiac, held by the god Aion, and has only a single twist. There is no clear evidence that the one-sidedness of this visual representation of celestial time was intentional; it could have been chosen merely as a way to make all of the signs of the zodiac appear on the visible side of the strip. Some other ancient depictions of the ourobouros or of lemniscate-shaped decorations are also alleged to depict Möbius strips, but whether they were intended to depict flat strips of any type is unclear.

Independently of the mathematical tradition, machinists have long known that mechanical belts wear half as quickly when they form Möbius strips, using both sides (or rather the same single side) of their material, and also having another potential advantage in evening out any curvature that might otherwise develop in the belt. An early written description of this technique dates to 1871, but this is after the first mathematical publications regarding the Möbius strip. Much older, an image of a chain pump in a work of Ismail al-Jazari from 1206 depicts what can only be a Möbius strip configuration for its drive chain. There is also documentation of a tradition among Parisian seamstresses of initiating novices into the profession by requiring them to stitch a Möbius strip as a collar onto a garment, but without clear dates for the origin of this task.

Properties

A 2d object traversing once around the Möbius strip returns in mirrored form

Cutting the centerline produces a two-sided (non-Möbius) strip

A single off-center cut separates a Möbius strip (purple) from a two-sided strip

Subdivision into six mutually-adjacent regions, bounded by Tietze's graph

Solution to the three utilities problem on a Möbius strip

The Möbius strip has several curious properties. A line drawn along the edge travels in a full circle to a point opposite the starting point. If continued, the line returns to the starting point, and is double the length of the original strip: this single continuous curve traverses the entire boundary. In this sense, the Möbius strip is different from a cylinder or annulus and like a circular disk in having only one boundary. A Möbius strip made with a clockwise half-twist cannot be moved or stretched to give the half-twist counterclockwise; thus, a Möbius strip embedded in Euclidean space is a chiral object with right- or left-handedness. Similarly, Möbius strips with odd numbers of half-twists greater than one, or that are knotted before gluing, are distinct as embedded subsets of three-dimensional space, even though as two-dimensional topological spaces they are all equivalent. With an even number of twists, however, one obtains a different topological space, the annulus.

The Möbius strip is a non-orientable surface: if an asymmetric two-dimensional object slides one time around the strip, it returns to its starting position as its mirror image. More strongly, any surface is non-orientable if and only if it has a Möbius strip as a subset. Relatedly, when embedded into Euclidean space, the Möbius strip has only one side. A three-dimensional object that slides one time around the surface of the strip is not mirrored, but instead returns to the same position on what appears locally to be the other side of the strip. Equivalently, if an embedded Möbius strip is thickened slightly into a three-dimensional object, the surface of the thickened object forms a single connected set. This behavior is different from familiar orientable surfaces in three dimensions such as those modeled by flat sheets of paper, cylindrical drinking straws, or hollow balls, for which one side of the surface is not connected to the other. However, this is a property of its embedding into space rather than an intrinsic property of the Möbius strip itself: there exist other topological spaces in which the Möbius strip can be embedded so that it has two sides. For instance, if the front and back faces of a cube are glued to each other with a left-right mirror reflection, the result is a three-dimensional topological space (the Cartesian product of a Möbius strip with an interval) in which the top and bottom halves of the cube can be separated from each other by a two-sided Möbius strip.

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, but homeomorphic to a cylinder. This happens because the original strip only has one edge, twice as long as the original strip. Cutting produces a second independent edge of the same length, half on each side of the scissors. Cutting this new, longer, strip down the middle produces two strips wound around each other, each with two full twists. If the strip is cut along about a third in from the edge, it produces two linked strips. The center third is a thinner Möbius strip, the same length as the original strip. The other is a thin strip with two full twists, a neighborhood of the edge of the original strip, with twice the length of the original strip.

The Möbius strip can be cut into six mutually-adjacent regions, showing that maps on the surface of the Möbius strip can sometimes require six colors, in contrast to the four color theorem for the plane. The result that six colors are both necessary and sufficient is part of the Ringel–Youngs theorem. The graph formed by the boundary edges and vertices of the partition into six regions is Tietze's graph, which is a dual graph on this surface for the six-vertex complete graph $K_{6}$. Another family of graphs that can be embedded on the Möbius strip, but not in the plane, are the Möbius ladders, the boundaries of subdivisions of the Möbius strip into rectangles meeting end-to-end. The simplest of the Möbius ladders is the complete bipartite graph $K_{3,3}$, and its embedding into the Möbius strip shows that, unlike in the plane, the three utilities problem can be solved on a transparent Möbius strip. The Euler characteristic of the Möbius strip is zero, meaning that for any subdivision of the strip by vertices and edges into regions, the numbers of vertices, edges, and regions satisfies the equation $V-E+R=0$. For instance, Tietze's graph has $12$ vertices and $18$ edges; $12-18+6=0$.

Constructions

There are many different ways of defining geometric surfaces with the topology of the Möbius strip, depending on the additional geometric properties that are desired for this surface.

Sweeping a line segment

A Möbius strip swept out by a rotating line segment in a rotating plane

One way to represent the Möbius strip embedded in three-dimensional Euclidean space is to sweep it out by a rotating line segment in a rotating plane. In order for the swept surface to meet up with itself after a half-twist, the line segment should rotate around its center at half the angular velocity that the plane rotates around one of its lines. This can be described as a parametric surface whose three Cartesian coordinates are functions of two parameters $u$, the rotation angle of the plane, and $v$, the position along the rotating line segment:

$${\begin{aligned}x(u,v)&=\left(1+{\frac {v}{2}}\cos {\frac {u}{2}}\right)\cos u\\y(u,v)&=\left(1+{\frac {v}{2}}\cos {\frac {u}{2}}\right)\sin u\\z(u,v)&={\frac {v}{2}}\sin {\frac {u}{2}}\\\end{aligned}}$$

for $0\leq u<2\pi$ and $-1\leq v\leq 1$. This produces a Möbius strip of width 1, whose center circle has radius 1, lies in the $xy$-plane and is centered at $(0,0,0)$.

Gluing a rectangle

The recycling symbol is based on a rectangle folded into a triangular Möbius strip

A rectangular Möbius strip, made from identifying two opposite sides of a geometrical rectangle that can be bent but not stretched, can be embedded smoothly into three-dimensional space whenever the ratio of lengths of the unglued sides to the glued sides of the rectangle – its aspect ratio – is greater than ${\sqrt {3}}\approx 1.73$. As the aspect ratio decreases toward ${\sqrt {3}}$, any such embedding seems to approach a shape that can be thought of as a strip of three equilateral triangles, folded on top of one another to occupy an equilateral triangle. This folded embedding can be made smooth (smoothness class $C^{2}$), with the portions of the strip between the folds lying in three close-together parallel planes, and with the folds replaced by patches of thin cylinders, each tangent to two of the three planes. The familiar three-arrow logo for recycling is based on this form of the Möbius strip, although some variations of this symbol instead reverse one of the folds to produce a surface that makes three half-twists before it closes, topologically a Möbius strip but embedded differently. It is also possible to find an infinitely differentiable (class $C^{\infty }$) embedding that stays arbitrarily close to this flat and cylindrical embedding.

For an embedding of a glued rectangle that forms a developable surface, there is a lower limit to the aspect ratio that can be achieved: it is impossible to embed a Möbius strip smoothly with an aspect ratio less than $\pi /2\approx 1.57$, even if self-intersections are allowed. For aspect ratios between $\pi /2$ and ${\sqrt {3}}$, smooth self-intersecting Möbius strips can be made, but it is unknown whether they can be made without self-intersections. For embeddings that are only required to be singly continuously differentiable (class $C^{1}$), but are not so smooth as to have a tangent plane everywhere, the Nash–Kuiper theorem implies that any two opposite faces of any rectangle can be glued to form a Möbius strip, no matter how small the aspect ratio becomes. The limiting case, a surface obtained from an infinite strip of the plane between two parallel lines, glued with the opposite orientation to each other, is called the unbounded Möbius strip or the real tautological line bundle. Although it has no smooth embedding into three-dimensional space, it can be embedded smoothly into four-dimensional Euclidean space.

Another method of making a Möbius strip that is even less smooth, from a rectangular strip too wide to simply twist and join, is to first "accordion fold" the strip in its wide direction back and forth using an even number of folds, so that the folded strip becomes narrow enough that it can be twisted and joined, much as a single long-enough strip can be joined. With two folds, for example, a $1\times 1$ strip would become a $1\times {\tfrac {1}{3}}$ folded strip whose cross section is in the shape of an 'N' and would remain an 'N' after a half-twist. This folded strip, three times as long as it is wide, would be long enough to then join at the ends. This method works in principle, but becomes impractical after sufficiently many folds, if paper is used. Using normal paper, this construction can be folded flat, with all the layers of the paper in a single plane. However, the lengthwise folds used in this construction prevent it from forming a three-dimensional embedding in which the layers are separated from each other and bend smoothly without crumpling or stretching away from the folds.

Making the boundary circular

Gluing two Möbius strips to form a Klein bottle

A projection of the Sudanese Möbius band

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 true circle. However, it is possible to embed a Möbius strip in three dimensions so that the boundary is a perfect circle lying in some plane. One way to do this is based on the topology of the Klein bottle, a one-sided surface with no boundary that cannot be embedded into three-dimensional space, but can be immersed (allowing the surface to cross itself in certain restricted ways). One way to form a Klein bottle is to glue two Möbius strips together along their edges, and reversing that process, a Klein bottle can be sliced along a carefully chosen cut to produce two Möbius strips.

One geometric construction of the Klein bottle, known as Lawson's Klein bottle, immerses it as a minimal surface in the unit 3-sphere of 4-dimensional space, the set of points of the form

$$(\cos \theta \cos \phi ,\sin \theta \cos \phi ,\cos 2\theta \sin \phi ,\sin 2\theta \sin \phi )$$

for $0\leq \theta <\pi ,0\leq \phi <2\pi$. Half of this Klein bottle, the subset with $0\leq \phi <\pi$, gives a Möbius band embedded in the 3-sphere as a minimal surface with a circular boundary.

This embedding is sometimes called the "Sudanese Möbius band" after the names of topologists Sue Goodman and Daniel Asimov, who discovered it in the 1970s. Geometrically Lawson's Klein bottle can be constructed by sweeping a great circle through a great-circular motion in the 3-sphere, and the Sudanese Möbius band is obtained by sweeping a semicircle instead of a circle, or equivalently by slicing the Klein bottle along a circle that is perpendicular to all of the swept circles. Stereographic projection places this into three-dimensional Euclidean space, preserving the circularity of its boundary. The most symmetric projection is obtained by using a projection point that lies on that great circle that runs through the midpoint of each of the semicircles, but this produces an unbounded embedding with the projection point removed from its centerline. Instead, leaving the Sudanese Möbius band unprojected, in the 3-sphere, leaves it with an infinite group of symmetries isomorphic to the orthogonal group $\mathrm {O} (2)$, the group of symmetries of a circle.

Surfaces of constant curvature

The open Möbius strip is the open set formed by deleting the boundary of the standard Möbius strip, in any of its other constructions. As a topological space, independent of its embedding, this surface may be given a Riemannian geometry of constant positive, negative, or zero Gaussian curvature. In the cases of negative and zero curvature, the Möbius strip can be constructed as a geodesically complete surface, which means that all geodesics ("straight lines" on the surface) may be extended indefinitely in either direction.

Negative curvature

Like the plane and the open cylinder, the open Möbius strip admits not only a complete metric of constant curvature 0, but also a complete metric of constant negative curvature. One way to see this is to begin with the upper half plane (Poincaré) model of the hyperbolic plane, a geometry of constant curvature whose lines are represented in the model by semicircles that meet the $x$-axis at right angles. Take the subset of the upper half-plane between any two nested semicircles, and identify the outer semicircle with the left-right reversal of the inner semicircle. The result is topologically a complete and non-compact Möbius strip with constant negative curvature.

Zero curvature

An open strip with zero curvature may be constructed by gluing the opposite sides of a plane strip between two parallel lines, described above as the tautological line bundle. The resulting metric makes the open Möbius strip into a (geodesically) complete flat surface (i.e., having Gaussian curvature equal to 0 everywhere). This is the unique metric on the Möbius strip, up to uniform scaling, that is both flat and complete. It is the quotient space of a plane by a glide reflection, and (together with the plane, cylinder, torus, and Klein bottle) is one of only five possible two-dimensional complete flat manifolds.

Positive curvature

A Möbius strip of constant positive curvature cannot be complete, since it is known that the only complete surfaces of constant positive curvature are the sphere and the projective plane. The projective plane $\mathbb {P} ^{2}$ of constant curvature +1 may be constructed as the quotient of the unit sphere $S^{2}$ in $\mathbb {R} ^{3}$ by the antipodal map $A:S^{2}\to S^{2}$ defined by $A(x,y,z)=(-x,-y,-z)$. The open Möbius strip is homeomorphic to the once-punctured projective plane, that is, $\mathbb {P} ^{2}$ with any one point removed. This may be thought of as the closest that a Möbius strip of constant positive curvature can get to being a complete surface: just one point away.

The minimal surfaces are described as having constant zero mean curvature instead of constant Gaussian curvature. The Sudanese Möbius strip was constructed as a minimal surface bounded by a great circle in a 3-sphere, but there is also a unique complete (boundaryless) minimal surface immersed in Euclidean space that has the topology of an open Möbius strip. It is called the Meeks Möbius strip, after its 1982 description by William Hamilton Meeks, III.

Lines in the plane

The space of unoriented lines in the plane is diffeomorphic to the open Möbius band. If a single line at infinity is added to this space, it becomes the space of lines in the real projective plane, which by projective duality is equivalent to the projective plane itself. Removing the line at infinity punctures this space, and the punctured projective plane is one of the forms of the open Möbius strip.

The group $GL(2,\mathbb {R} )$ of real $2\times 2$ matrices with non-zero determinant, when applied to Cartesian coordinates, maps lines in the plane to other lines, and therefore acts as a transformation group of the open Möbius strip, interpreted as the space of all lines. However, there is no metric on the space of lines in the plane that is invariant under this group action. In this sense, the space of lines in the plane has no natural metric on it. This means that the open Möbius strip has a natural 4-dimensional Lie group of transformations, the group $GL(2,\mathbb {R} )$, but this high degree of symmetry cannot be exhibited as the group of isometries of any metric.

Applications

Mathematical art: 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 let the ribbon 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 and electro-technology as:

• A compact resonator with a resonance frequency that is half that of identically constructed linear coils
• An inductionless resistor
• Superconductors with high transition temperature

In chemistry and nano-technology as:

• Graphene volume (nano-graphite) with new electronic characteristics, like helical magnetism
• A special type of aromaticity: Möbius aromaticity
• Charged particles caught in the magnetic field of Earth that 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.

Mobius strip used as a logo on a van in Bristol, England

Arts and entertainment

The Möbius-strip principle has been used as a method of producing the illusion of magic. The trick, known as the Afghan bands, was very popular in the first half of the twentieth century. Many versions of this trick exist and have been performed by famous illusionists such as Harry Blackstone Sr. and Thomas Nelson Downs.

See also

• Calabi–Yau manifold
• Cross-cap
• Umbilic torus
• Ribbon theory

External links

• The Möbius Gear – A functional planetary gear model in which one gear is a Möbius strip
• Weisstein, Eric W. "Möbius Strip". MathWorld.