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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a Möbius strip, Möbius band, or Möbius loop is a
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by
Johann Benedict Listing Johann Benedict Listing (25 July 1808 – 24 December 1882) was a German mathematician. Early life and education J. B. Listing was born in Frankfurt and died in Göttingen. He finished his studies at the University of Göttingen in 1834, and ...
and
August Ferdinand Möbius August Ferdinand Möbius (, ; ; 17 November 1790 – 26 September 1868) was a German mathematician and theoretical astronomer. Life and education Möbius was born in Schulpforta, Electorate of Saxony, and was descended on his mothe ...
in 1858, but it had already appeared in Roman mosaics from the third century CE. The Möbius strip is a non-orientable surface, meaning that within it one cannot consistently distinguish
clockwise Two-dimensional rotation can occur in two possible directions or senses of rotation. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands relative to the observer: from the top to the right, then down and then to ...
from counterclockwise turns. Every non-orientable surface contains a Möbius strip. As an abstract
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
, the Möbius strip can be embedded into three-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
in many different ways: a clockwise half-twist is different from a counterclockwise half-twist, and it can also be embedded with odd numbers of twists greater than one, or with a knotted centerline. Any two embeddings with the same knot for the centerline and the same number and direction of twists are topologically equivalent. All of these embeddings have only one side, but when embedded in other spaces, the Möbius strip may have two sides. It has only a single boundary curve. Several geometric constructions of the Möbius strip provide it with additional structure. It can be swept as a
ruled surface In geometry, a Differential geometry of surfaces, surface in 3-dimensional Euclidean space is ruled (also called a scroll) if through every Point (geometry), point of , there is a straight line that lies on . Examples include the plane (mathemat ...
by a line segment rotating in a rotating plane, with or without self-crossings. A thin paper strip with its ends joined to form a Möbius strip can bend smoothly as a developable surface or be folded flat; the flattened Möbius strips include the trihexaflexagon. The Sudanese Möbius strip is a
minimal surface In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces that ...
in a
hypersphere In mathematics, an -sphere or hypersphere is an - dimensional generalization of the -dimensional circle and -dimensional sphere to any non-negative integer . The circle is considered 1-dimensional and the sphere 2-dimensional because a point ...
, and the Meeks Möbius strip is a self-intersecting minimal surface in ordinary Euclidean space. Both the Sudanese Möbius strip and another self-intersecting Möbius strip, the cross-cap, have a circular boundary. A Möbius strip without its boundary, called an open Möbius strip, can form surfaces of constant curvature. Certain highly symmetric spaces whose points represent lines in the plane have the shape of a Möbius strip. The many applications of Möbius strips include mechanical belts that wear evenly on both sides, dual-track
roller coaster A roller coaster is a type of list of amusement rides, amusement ride employing a form of elevated Railway track, railroad track that carries passengers on a roller coaster train, train through tight turns, steep slopes, and other elements, usua ...
s whose carriages alternate between the two tracks, and
world map A world map is a map of most or all of the surface of Earth. World maps, because of their scale, must deal with the problem of projection. Maps rendered in two dimensions by necessity distort the display of the three-dimensional surface of t ...
s printed so that
antipodes In geography, the antipode () of any spot on Earth is the point on Earth's surface diametrically opposite to it. A pair of points ''antipodal'' () to each other are situated such that a straight line connecting the two would pass through Ea ...
appear opposite each other. Möbius strips appear in molecules and devices with novel electrical and electromechanical properties, and have been used to prove impossibility results in
social choice theory Social choice theory is a branch of welfare economics that extends the Decision theory, theory of rational choice to collective decision-making. Social choice studies the behavior of different mathematical procedures (social welfare function, soc ...
. In popular culture, Möbius strips appear in artworks by
M. C. Escher Maurits Cornelis Escher (; ; 17 June 1898 – 27 March 1972) was a Dutch graphic artist who made woodcuts, lithography, lithographs, and mezzotints, many of which were Mathematics and art, inspired by mathematics. Despite wide popular int ...
, Max Bill, and others, and in the design of the
recycling symbol The universal recycling symbol ( or in Unicode) is a symbol consisting of three chasing arrows folded in a Möbius strip. It is an internationally recognized symbol for recycling. The symbol originated on the first Earth Day in 1970, created ...
. Many architectural concepts have been inspired by the Möbius strip, including the building design for the
NASCAR Hall of Fame The NASCAR Hall of Fame, is a Hall of Fame and Museum located in Charlotte, North Carolina that honors NASCAR and its history. Inductees to the Hall of Fame are drivers who have shown expert skill at NASCAR driving, all-time great crew chiefs ...
. Performers including Harry Blackstone Sr. and Thomas Nelson Downs have based stage magic tricks on the properties of the Möbius strip. The canons of J. S. Bach have been analyzed using Möbius strips. Many works of
speculative fiction Speculative fiction is an umbrella term, umbrella genre of fiction that encompasses all the subgenres that depart from Realism (arts), realism, or strictly imitating everyday reality, instead presenting fantastical, supernatural, futuristic, or ...
feature Möbius strips; more generally, a plot structure based on the Möbius strip, of events that repeat with a twist, is common in fiction.


History

The discovery of the Möbius strip as a mathematical object is attributed independently to the German mathematicians
Johann Benedict Listing Johann Benedict Listing (25 July 1808 – 24 December 1882) was a German mathematician. Early life and education J. B. Listing was born in Frankfurt and died in Göttingen. He finished his studies at the University of Göttingen in 1834, and ...
and
August Ferdinand Möbius August Ferdinand Möbius (, ; ; 17 November 1790 – 26 September 1868) was a German mathematician and theoretical astronomer. Life and education Möbius was born in Schulpforta, Electorate of Saxony, and was descended on his mothe ...
in 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 In many cases these merely depict coiled ribbons as boundaries. When the number of coils is odd, these ribbons are Möbius strips, but for an even number of coils they are topologically equivalent to untwisted rings. Therefore, whether the ribbon is a Möbius strip may be coincidental, rather than a deliberate choice. In at least one case, a ribbon with different colors on different sides was drawn with an odd number of coils, forcing its artist to make a clumsy fix at the point where the colors did not Another mosaic from the town of Sentinum (depicted) shows the
zodiac The zodiac is a belt-shaped region of the sky that extends approximately 8° north and south celestial latitude of the ecliptic – the apparent path of the Sun across the celestial sphere over the course of the year. Within this zodiac ...
, held by the god Aion, as a band with 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 figure-eight-shaped decorations are also alleged to depict Möbius strips, but whether they were intended to depict flat strips of any type is Independently of the mathematical tradition, machinists have long known that mechanical belts wear half as quickly when they form Möbius strips, because they use the entire surface of the belt rather than only the inner surface of an untwisted belt. Additionally, such a belt may be less prone to curling from side to side. An early written description of this technique dates to 1871, which is after the first mathematical publications regarding the Möbius strip. Much earlier, an image of a chain pump in a work of Ismail al-Jazari from 1206 depicts a Möbius strip configuration for its drive Another use of this surface was made by seamstresses in Paris (at an unspecified date): they initiated novices by requiring them to stitch a Möbius strip as a collar onto a


Properties

The Möbius strip has several curious properties. It 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. In particular, a curved arrow pointing clockwise (↻) would return as an arrow pointing counterclockwise (↺), implying that, within the Möbius strip, it is impossible to consistently define what it means to be clockwise or counterclockwise. It is the simplest non-orientable surface: any other surface is non-orientable if and only if it has a Möbius strip as a Relatedly, when embedded into
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
, 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 point of the strip on what appears locally to be its other side, showing that both positions are really part of a single side. 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 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 In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...
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 In contrast to disks, spheres, and cylinders, for which it is possible to simultaneously embed an
uncountable set In mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger t ...
of disjoint copies into three-dimensional space, only a countable number of Möbius strips can be simultaneously A path along the edge of a Möbius strip, traced until it returns to its starting point on the edge, includes all boundary points of the Möbius strip in a single continuous curve. For a Möbius strip formed by gluing and twisting a rectangle, it has twice the length of the centerline of the strip. In this sense, the Möbius strip is different from an untwisted ring and like a circular disk in having only one A Möbius strip in Euclidean space cannot be moved or stretched into its mirror image; it is a
chiral Chirality () is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek language, Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is dist ...
object with right- or 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 they are all equivalent as two-dimensional topological More precisely, two Möbius strips are equivalently embedded in three-dimensional space when their centerlines determine the same knot and they have the same number of twists as each With an even number of twists, however, one obtains a different topological surface, called the The Möbius strip can be continuously transformed into its centerline, by making it narrower while fixing the points on the centerline. This transformation is an example of a deformation retraction, and its existence means that the Möbius strip has many of the same properties as its centerline, which is topologically a circle. In particular, its
fundamental group In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...
is the same as the fundamental group of a circle, an
infinite cyclic group In abstract algebra, a cyclic group or monogenous group is a group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of -adic numbers), that is generated by a single element. That is, it is a set of invertib ...
. Therefore, paths on the Möbius strip that start and end at the same point can be distinguished topologically (up to
homotopy In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. ...
) only by the number of times they loop around the strip. Cutting a Möbius strip along the centerline with a pair of scissors yields one long strip with four half-twists in it (relative to an untwisted annulus or cylinder) rather than two separate strips. Two of the half-twists come from the fact that this thinner strip goes two times through the half-twist in the original Möbius strip, and the other two come from the way the two halves of the thinner strip wrap around each other. The result is not a Möbius strip, but instead is topologically equivalent to a cylinder. Cutting this double-twisted strip again along its centerline produces two linked double-twisted strips. If, instead, a Möbius strip is cut lengthwise, a third of the way across its width, it produces two linked strips. One of the two is a central, thinner, Möbius strip, while the other has two These interlinked shapes, formed by lengthwise slices of Möbius strips with varying widths, are sometimes called ''paradromic'' 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 In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color. ''Adjacent'' means that two regions shar ...
for the Six colors are always enough. This result is part of the Ringel–Youngs theorem, which states how many colors each topological surface The edges and vertices of these six regions form Tietze's graph, which is a
dual graph In the mathematics, mathematical discipline of graph theory, the dual graph of a planar graph is a graph that has a vertex (graph theory), vertex for each face (graph theory), face of . The dual graph has an edge (graph theory), edge for each p ...
on this surface for the six-vertex
complete graph In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices i ...
but cannot be drawn without crossings on a plane. Another family of graphs that can be embedded on the Möbius strip, but not on the plane, are the
Möbius ladder In graph theory, the Möbius ladder , for even numbers , is formed from an by adding edges (called "rungs") connecting opposite pairs of vertices in the cycle. It is a cubic, circulant graph, so-named because (with the exception of (the util ...
s, the boundaries of subdivisions of the Möbius strip into rectangles meeting These include the utility graph, a six-vertex
complete bipartite graph In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set..Electronic edition page 17. Graph theory ...
whose embedding into the Möbius strip shows that, unlike in the plane, the
three utilities problem The three utilities problem, also known as water, gas and electricity, is a mathematical puzzle that asks for non-crossing connections to be drawn between three houses and three utility companies on a Plane (geometry), plane. When posing it in ...
can be solved on a transparent Möbius The
Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's ...
of the Möbius strip is
zero 0 (zero) is a number representing an empty quantity. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and compl ...
, meaning that for any subdivision of the strip by vertices and edges into regions, the numbers V, E, and F of vertices, edges, and regions satisfy V-E+F=0. For instance, Tietze's graph has 12 vertices, 18 edges, and 6 regions;


Constructions

There are many different ways of defining geometric surfaces with the topology of the Möbius strip, yielding realizations with additional geometric properties.


Sweeping a line segment

One way to embed the Möbius strip in three-dimensional Euclidean space is to sweep it out by a line segment rotating in a plane, which in turn rotates around one of its 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 of the plane's rotation. This can be described as a
parametric surface A parametric surface is a surface in the Euclidean space \R^3 which is defined by a parametric equation with two parameters Parametric representation is a very general way to specify a surface, as well as implicit representation. Surfaces that oc ...
defined by equations for the
Cartesian coordinates In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
of its points, \begin x(u,v)&= \left(1+\frac \cos \frac\right)\cos u\\ y(u,v)&= \left(1+\frac \cos\frac\right)\sin u\\ z(u,v)&= \frac\sin \frac\\ \end for 0 \le u< 2\pi and where one parameter u describes the rotation angle of the plane around its central axis and the other parameter describes the position of a point along the rotating line segment. This produces a Möbius strip of width 1, whose center circle has radius 1, lies in the xy-plane and is centered at The same method can produce Möbius strips with any odd number of half-twists, by rotating the segment more quickly in its plane. The rotating segment sweeps out a circular disk in the plane that it rotates within, and the Möbius strip that it generates forms a slice through the
solid torus In mathematics, a solid torus is the topological space formed by sweeping a disk around a circle. It is homeomorphic to the Cartesian product S^1 \times D^2 of the disk and the circle, endowed with the product topology. A standard way to visual ...
swept out by this disk. Because of the one-sidedness of this slice, the sliced torus remains A line or line segment swept in a different motion, rotating in a horizontal plane around the origin as it moves up and down, forms Plücker's conoid or cylindroid, an algebraic
ruled surface In geometry, a Differential geometry of surfaces, surface in 3-dimensional Euclidean space is ruled (also called a scroll) if through every Point (geometry), point of , there is a straight line that lies on . Examples include the plane (mathemat ...
in the form of a self-crossing Möbius It has applications in the design of


Polyhedral surfaces and flat foldings

A strip of paper can form a flattened Möbius strip in the plane by folding it at 60^\circ angles so that its center line lies along an
equilateral triangle An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...
, and attaching the ends. The shortest strip for which this is possible consists of three equilateral triangles, folded at the edges where two triangles meet. Its
aspect ratio The aspect ratio of a geometry, geometric shape is the ratio of its sizes in different dimensions. For example, the aspect ratio of a rectangle is the ratio of its longer side to its shorter side—the ratio of width to height, when the rectangl ...
the ratio of the strip's length to its widthis and the same folding method works for any larger aspect For a strip of nine equilateral triangles, the result is a trihexaflexagon, which can be flexed to reveal different parts of its For strips too short to apply this method directly, one can first "accordion fold" the strip in its wide direction back and forth using an even number of folds. With two folds, for example, a 1\times 1 strip would become a 1\times \tfrac folded strip whose cross section is in the shape of an "N" and would remain an "N" after a half-twist. The narrower accordion-folded strip can then be folded and joined in the same way that a longer strip The Möbius strip can also be embedded as a polyhedral surface in space or flat-folded in the plane, with only five triangular faces sharing five vertices. In this sense, it is simpler than the
cylinder A cylinder () has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infinite ...
, which requires six triangles and six vertices, even when represented more abstractly as a
simplicial complex In mathematics, a simplicial complex is a structured Set (mathematics), set composed of Point (geometry), points, line segments, triangles, and their ''n''-dimensional counterparts, called Simplex, simplices, such that all the faces and intersec ...
. A five-triangle Möbius strip can be represented most symmetrically by five of the ten equilateral triangles of a four-dimensional regular simplex. This four-dimensional polyhedral Möbius strip is the only ''tight'' Möbius strip, one that is fully four-dimensional and for which all cuts by
hyperplane In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is ...
s separate it into two parts that are topologically equivalent to disks or Other polyhedral embeddings of Möbius strips include one with four convex
quadrilateral In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''l ...
s as faces, another with three non-convex quadrilateral and one using the vertices and center point of a regular
octahedron In geometry, an octahedron (: octahedra or octahedrons) is any polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Many types of i ...
, with a triangular Every abstract triangulation of the
projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, paral ...
can be embedded into 3D as a polyhedral Möbius strip with a triangular boundary after removing one of its an example is the six-vertex projective plane obtained by adding one vertex to the five-vertex Möbius strip, connected by triangles to each of its boundary However, not every abstract triangulation of the Möbius strip can be represented geometrically, as a polyhedral To be realizable, it is necessary and sufficient that there be no two disjoint non-contractible 3-cycles in the


Smoothly embedded rectangles

A rectangular Möbius strip, made by attaching the ends of a paper rectangle, can be embedded smoothly into three-dimensional space whenever its aspect ratio is greater than the same ratio as for the flat-folded equilateral-triangle version of the Möbius This flat triangular embedding can lift to a smooth embedding in three dimensions, in which the strip lies flat in three parallel planes between three cylindrical rollers, each tangent to two of the Mathematically, a smoothly embedded sheet of paper can be modeled as a developable surface, that can bend but cannot As its aspect ratio decreases toward \sqrt 3, all smooth embeddings seem to approach the same triangular The lengthwise folds of an accordion-folded flat Möbius strip 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 Instead, unlike in the flat-folded case, there is a lower limit to the aspect ratio of smooth rectangular Möbius strips. Their aspect ratio cannot be less than even if self-intersections are allowed. Self-intersecting smooth Möbius strips exist for any aspect ratio above this Without self-intersections, the aspect ratio must be at \frac\sqrt\approx 1.695. For aspect ratios between this bound it has been an open problem whether smooth embeddings, without self-intersection, In 2023, Richard Schwartz announced a proof that they do not exist, but this result still awaits peer review and publication. If the requirement of smoothness is relaxed to allow
continuously differentiable In mathematics, a differentiable function of one Real number, real variable is a Function (mathematics), function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differ ...
surfaces, the Nash–Kuiper theorem implies that any two opposite edges of any rectangle can be glued to form an embedded Möbius strip, no matter how small the aspect ratio 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 closed embedding into three-dimensional space, it can be embedded smoothly as a closed subset of four-dimensional Euclidean The minimum-energy shape of a smooth Möbius strip glued from a rectangle does not have a known analytic description, but can be calculated numerically, and has been the subject of much study in plate theory since the initial work on this subject in 1930 by Michael Sadowsky. It is also possible to find
algebraic surface In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of di ...
s that contain rectangular developable Möbius


Making the boundary circular

The edge, or boundary, of a Möbius strip is topologically equivalent to a
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
. In common forms of the Möbius strip, it has a different shape from a circle, but it is
unknot In the knot theory, mathematical theory of knots, the unknot, not knot, or trivial knot, is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a Knot (mathematics), knot tied into it, unknotted. To a knot ...
ted, and therefore the whole strip can be stretched without crossing itself to make the edge perfectly One such example is based on the topology of the
Klein bottle In mathematics, the Klein bottle () is an example of a Orientability, non-orientable Surface (topology), surface; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the ...
, 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). A Klein bottle is the surface that results when two Möbius strips are glued together edge-to-edge, andreversing that processa Klein bottle can be sliced along a carefully chosen cut to produce two Möbius For a form of the Klein bottle known as Lawson's Klein bottle, the curve along which it is sliced can be made circular, resulting in Möbius strips with circular Lawson's Klein bottle is a self-crossing
minimal surface In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces that ...
in the unit hypersphere of 4-dimensional space, the set of points of the form (\cos\theta\cos\phi,\sin\theta\cos\phi,\cos2\theta\sin\phi,\sin2\theta\sin \phi) for Half of this Klein bottle, the subset with 0\le\phi<\pi, gives a Möbius strip embedded in the hypersphere as a minimal surface with a
great circle In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Discussion Any arc of a great circle is a geodesic of the sphere, so that great circles in spher ...
as its This embedding is sometimes called the "Sudanese Möbius strip" after topologists Sue Goodman and Daniel Asimov, who discovered it in the 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 strip 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
Stereographic projection In mathematics, a stereographic projection is a perspective transform, perspective projection of the sphere, through a specific point (geometry), point on the sphere (the ''pole'' or ''center of projection''), onto a plane (geometry), plane (th ...
transforms this shape from a three-dimensional spherical space into three-dimensional Euclidean space, preserving the circularity of its 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 produces an unbounded embedding with the projection point removed from its Instead, leaving the Sudanese Möbius strip unprojected, in the 3-sphere, leaves it with an infinite group of symmetries isomorphic to the
orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
the group of symmetries of a The Sudanese Möbius strip extends on all sides of its boundary circle, unavoidably if the surface is to avoid crossing itself. Another form of the Möbius strip, called the cross-cap or crosscap, also has a circular boundary, but otherwise stays on only one side of the plane of this making it more convenient for attaching onto circular holes in other surfaces. In order to do so, it crosses itself. It can be formed by removing a
quadrilateral In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''l ...
from the top of a hemisphere, orienting the edges of the quadrilateral in alternating directions, and then gluing opposite pairs of these edges consistently with this The two parts of the surface formed by the two glued pairs of edges cross each other with a pinch point like that of a Whitney umbrella at each end of the crossing the same topological structure seen in Plücker's


Surfaces of constant curvature

The open Möbius strip is the
relative interior In mathematics, the relative interior of a set is a refinement of the concept of the interior, which is often more useful when dealing with low-dimensional sets placed in higher-dimensional spaces. Formally, the relative interior of a set S (deno ...
of a standard Möbius strip, formed by omitting the points on its boundary edge. It may be given a
Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...
of constant positive, negative, or zero
Gaussian curvature In differential geometry, the Gaussian curvature or Gauss curvature of a smooth Surface (topology), surface in three-dimensional space at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. For ...
. The cases of negative and zero curvature form geodesically complete surfaces, which means that all
geodesic In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conn ...
s ("straight lines" on the surface) may be extended indefinitely in either direction. ;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 The resulting metric makes the open Möbius strip into a (geodesically) complete flat surface (i.e., having zero Gaussian curvature 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 In geometry, a glide reflection or transflection is a geometric transformation that consists of a reflection across a hyperplane and a translation ("glide") in a direction parallel to that hyperplane, combined into a single transformation. Bec ...
, and (together with the plane,
cylinder A cylinder () has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infinite ...
,
torus In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
, and
Klein bottle In mathematics, the Klein bottle () is an example of a Orientability, non-orientable Surface (topology), surface; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the ...
) is one of only five two-dimensional complete ;Negative curvature :The open Möbius strip also admits complete metrics 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. It is a "nonstandard" complete hyperbolic surface in the sense that it contains a complete hyperbolic
half-plane In mathematics, the upper half-plane, is the set of points in the Cartesian plane with The lower half-plane is the set of points with instead. Arbitrary oriented half-planes can be obtained via a planar rotation. Half-planes are an example ...
(actually two, on opposite sides of the axis of glide-reflection), and is one of only 13 nonstandard Again, this can be understood as the quotient of the hyperbolic plane by a glide ;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 In mathematics, a projective plane is a geometric structure that extends the concept of a plane (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, paral ...
. However, in a sense it is only one point away from being a complete surface, as the open Möbius strip is homeomorphic to the once-punctured projective plane, the surface obtained by removing any one point from the projective The
minimal surface In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces that ...
s are described as having constant zero
mean curvature In mathematics, the mean curvature H of a surface S is an ''extrinsic'' measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space. The ...
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 after its 1982 description by William Hamilton Meeks, III. Although globally unstable as a minimal surface, small patches of it, bounded by non-contractible curves within the surface, can form stable embedded Möbius strips as minimal Both the Meeks Möbius strip, and every higher-dimensional minimal surface with the topology of the Möbius strip, can be constructed using solutions to the Björling problem, which defines a minimal surface uniquely from its boundary curve and tangent planes along this


Spaces of lines

The family of lines in the plane can be given the structure of a smooth space, with each line represented as a point in this space. The resulting space of lines is topologically equivalent to the open Möbius One way to see this is to extend the Euclidean plane to the
real projective plane In mathematics, the real projective plane, denoted or , is a two-dimensional projective space, similar to the familiar Euclidean plane in many respects but without the concepts of distance, circles, angle measure, or parallelism. It is the sett ...
by adding one more line, the
line at infinity In geometry and topology, the line at infinity is a projective line that is added to the affine plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. The line at ...
. By
projective duality In projective geometry, duality or plane duality is a formalization of the striking symmetry of the roles played by points and lines in the definitions and theorems of projective planes. There are two approaches to the subject of duality, one th ...
the space of lines in the projective plane is equivalent to its space of points, the projective plane itself. Removing the line at infinity, to produce the space of Euclidean lines, punctures this space of projective Therefore, the space of Euclidean lines is a punctured projective plane, which is one of the forms of the open Möbius The space of lines in the hyperbolic plane can be parameterized by unordered pairs of distinct points on a circle, the pairs of points at infinity of each line. This space, again, has the topology of an open Möbius These spaces of lines are highly symmetric. The symmetries of Euclidean lines include the
affine transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, '' affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More general ...
s, and the symmetries of hyperbolic lines include the The affine transformations and Möbius transformations both form
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
s, topological spaces having a compatible
algebraic structure In mathematics, an algebraic structure or algebraic system consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplicatio ...
describing the composition of Because every line in the plane is symmetric to every other line, the open Möbius strip is a
homogeneous space In mathematics, a homogeneous space is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the action of a group. Homogeneous spaces occur in the theories of Lie groups, algebraic groups and ...
, a space with symmetries that take every point to every other point. Homogeneous spaces of Lie groups are called solvmanifolds, and the Möbius strip can be used as a
counterexample A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "student John Smith is not lazy" is a c ...
, showing that not every solvmanifold is a nilmanifold, and that not every solvmanifold can be factored into a
direct product In mathematics, a direct product of objects already known can often be defined by giving a new one. That induces a structure on the Cartesian product of the underlying sets from that of the contributing objects. The categorical product is an abs ...
of a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
solvmanifold These symmetries also provide another way to construct the Möbius strip itself, as a ''group model'' of these Lie groups. A group model consists of a Lie group and a
stabilizer subgroup In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under func ...
of its action; contracting the
coset In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
s of the subgroup to points produces a space with the same topology as the underlying homogenous space. In the case of the symmetries of Euclidean lines, the stabilizer of the consists of all symmetries that take the axis to itself. Each line \ell corresponds to a coset, the set of symmetries that map \ell to the Therefore, the quotient space, a space that has one point per coset and inherits its topology from the space of symmetries, is the same as the space of lines, and is again an open Möbius


Applications

Beyond the already-discussed applications of Möbius strips to the design of mechanical belts that wear evenly on their entire surface, and of the Plücker conoid to the design of gears, other applications of Möbius strips include: *
Graphene Graphene () is a carbon allotrope consisting of a Single-layer materials, single layer of atoms arranged in a hexagonal lattice, honeycomb planar nanostructure. The name "graphene" is derived from "graphite" and the suffix -ene, indicating ...
ribbons twisted to form Möbius strips with new electronic characteristics including helical magnetism *
Möbius aromaticity In organic chemistry, Möbius aromaticity is a special type of aromaticity believed to exist in a number of organic molecules. In terms of molecular orbital theory these compounds have in common a monocyclic array of molecular orbitals in which t ...
, a property of
organic chemical Some chemical authorities define an organic compound as a chemical compound that contains a Carbon–hydrogen bond, carbon–hydrogen or carbon–carbon bond; others consider an organic compound to be any chemical compound that contains carbon. F ...
s whose molecular structure forms a cycle, with
molecular orbital In chemistry, a molecular orbital is a mathematical function describing the location and wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of finding ...
s aligned along the cycle in the pattern of a Möbius strip *The Möbius resistor, a strip of conductive material covering the single side of a
dielectric In electromagnetism, a dielectric (or dielectric medium) is an Insulator (electricity), electrical insulator that can be Polarisability, polarised by an applied electric field. When a dielectric material is placed in an electric field, electric ...
Möbius strip, in a way that cancels its own
self-inductance Inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The electric current produces a magnetic field around the conductor. The magnetic field strength depends on the magnitude of the ...
*
Resonator A resonator is a device or system that exhibits resonance or resonant behavior. That is, it naturally oscillates with greater amplitude at some frequencies, called resonant frequencies, than at other frequencies. The oscillations in a reso ...
s with a compact design and a resonant frequency that is half that of identically constructed linear coils * Polarization patterns in light emerging from a ''q''-plate *A proof of the impossibility of continuous, anonymous, and unanimous two-party aggregation rules in
social choice theory Social choice theory is a branch of welfare economics that extends the Decision theory, theory of rational choice to collective decision-making. Social choice studies the behavior of different mathematical procedures (social welfare function, soc ...
* Möbius loop roller coasters, a form of dual-tracked roller coaster in which the two tracks spiral around each other an odd number of times, so that the carriages return to the other track than the one they started on *
World map A world map is a map of most or all of the surface of Earth. World maps, because of their scale, must deal with the problem of projection. Maps rendered in two dimensions by necessity distort the display of the three-dimensional surface of t ...
s projected onto a Möbius strip with the convenient properties that there are no east–west boundaries, and that the antipode of any point on the map can be found on the other printed side of the surface at the same point of the Möbius strip Scientists have also studied the energetics of
soap film Soap films are thin layers of liquid (usually water-based) surrounded by air. For example, if two soap bubbles come into contact, they merge and a thin film is created in between. Thus, foams are composed of a network of films connected by Plat ...
s shaped as Möbius strips, the
chemical synthesis Chemical synthesis (chemical combination) is the artificial execution of chemical reactions to obtain one or several products. This occurs by physical and chemical manipulations usually involving one or more reactions. In modern laboratory uses ...
of
molecule A molecule is a group of two or more atoms that are held together by Force, attractive forces known as chemical bonds; depending on context, the term may or may not include ions that satisfy this criterion. In quantum physics, organic chemi ...
s with a Möbius strip shape, and the formation of larger nanoscale Möbius strips using DNA origami.


In popular culture

Two-dimensional artworks featuring the Möbius strip include an untitled 1947 painting by Corrado Cagli (memorialized in a poem by
Charles Olson Charles John Olson (27 December 1910 – 10 January 1970) was a second generation modernist United States poetry, American poet who was a link between earlier Literary modernism, modernist figures such as Ezra Pound and William Carlos Williams an ...
), and two prints by
M. C. Escher Maurits Cornelis Escher (; ; 17 June 1898 – 27 March 1972) was a Dutch graphic artist who made woodcuts, lithography, lithographs, and mezzotints, many of which were Mathematics and art, inspired by mathematics. Despite wide popular int ...
: ''Möbius Band I'' (1961), depicting three folded
flatfish A flatfish is a member of the Ray-finned fish, ray-finned demersal fish Order (biology), suborder Pleuronectoidei, also called the Heterosomata. In many species, both eyes lie on one side of the head, one or the other migrating through or around ...
biting each others' tails; and ''Möbius Band II'' (1963), depicting ants crawling around a
lemniscate In algebraic geometry, a lemniscate ( or ) is any of several figure-eight or -shaped curves. The word comes from the Latin , meaning "decorated with ribbons", from the Greek (), meaning "ribbon",. or which alternatively may refer to the wool fr ...
-shaped Möbius strip. It is also a popular subject of mathematical sculpture, including works by Max Bill (''Endless Ribbon'', 1953), José de Rivera (''
Infinity Infinity is something which is boundless, endless, or larger than any natural number. It is denoted by \infty, called the infinity symbol. From the time of the Ancient Greek mathematics, ancient Greeks, the Infinity (philosophy), philosophic ...
'', 1967), and Sebastián. A trefoil-knotted Möbius strip was used in John Robinsons ''Immortality'' (1982). Charles O. Perry's '' Continuum'' (1976) is one of several pieces by Perry exploring variations of the Möbius strip. Because of their easily recognized form, Möbius strips are a common element of
graphic design Graphic design is a profession, academic discipline and applied art that involves creating visual communications intended to transmit specific messages to social groups, with specific objectives. Graphic design is an interdisciplinary branch of ...
. The familiar three-arrow logo for
recycling Recycling is the process of converting waste materials into new materials and objects. This concept often includes the recovery of energy from waste materials. The recyclability of a material depends on its ability to reacquire the propert ...
, designed in 1970, is based on the smooth triangular form of the Möbius as was the logo for the environmentally-themed Expo '74. Some variations of the recycling symbol use a different embedding with three half-twists instead of and the original version of the
Google Drive Google Drive is a file-hosting service and synchronization service developed by Google. Launched on April 24, 2012, Google Drive allows users to store files in the cloud (on Google servers), synchronize files across devices, and share files ...
logo used a flat-folded three-twist Möbius strip, as have other similar designs. The Brazilian Instituto Nacional de Matemática Pura e Aplicada (IMPA) uses a stylized smooth Möbius strip as its logo, and has a matching large sculpture of a Möbius strip on display in its building. The Möbius strip has also featured in the artwork for
postage stamp A postage stamp is a small piece of paper issued by a post office, postal administration, or other authorized vendors to customers who pay postage (the cost involved in moving, insuring, or registering mail). Then the stamp is affixed to the f ...
s from countries including Brazil, Belgium, the Netherlands, and Möbius strips have been a frequent inspiration for the architectural design of buildings and bridges. However, many of these are projects or conceptual designs rather than constructed objects, or stretch their interpretation of the Möbius strip beyond its recognizability as a mathematical form or a functional part of the architecture. An example is the National Library of Kazakhstan, for which a building was planned in the shape of a thickened Möbius strip but refinished with a different design after the original architects pulled out of the project. One notable building incorporating a Möbius strip is the
NASCAR Hall of Fame The NASCAR Hall of Fame, is a Hall of Fame and Museum located in Charlotte, North Carolina that honors NASCAR and its history. Inductees to the Hall of Fame are drivers who have shown expert skill at NASCAR driving, all-time great crew chiefs ...
, which is surrounded by a large twisted ribbon of stainless steel acting as a façade and canopy, and evoking the curved shapes of racing tracks. On a smaller scale, ''Moebius Chair'' (2006) by Pedro Reyes is a courting bench whose base and sides have the form of a Möbius strip. As a form of mathematics and fiber arts, scarves have been
knit Knitting is a method for production of textile fabrics by interlacing yarn loops with loops of the same or other yarns. It is used to create many types of garments. Knitting may be done by hand or by machine. Knitting creates stitches: ...
into Möbius strips since the work of Elizabeth Zimmermann in the early 1980s. In
food styling Food photography is a still life photography genre used to create appealing still life photographs of food. As a specialization of commercial photography, its output is used in advertisements, magazines, packaging, menus or cookbooks. Professiona ...
, Möbius strips have been used for slicing
bagel A bagel (; ; also spelled beigel) is a bread roll originating in the Jewish communities of Poland. Bagels are traditionally made from yeasted wheat dough that is shaped by hand into a torus or ring, briefly boiled in water, and then baked. ...
s, making loops out of
bacon Bacon is a type of Curing (food preservation), salt-cured pork made from various cuts of meat, cuts, typically the pork belly, belly or less fatty parts of the back. It is eaten as a side dish (particularly in breakfasts), used as a central in ...
, and creating new shapes for
pasta Pasta (, ; ) is a type of food typically made from an Leavening agent, unleavened dough of wheat flour mixed with water or Eggs as food, eggs, and formed into sheets or other shapes, then cooked by boiling or baking. Pasta was originally on ...
. Although mathematically the Möbius strip and the fourth dimension are both purely spatial concepts, they have often been invoked in
speculative fiction Speculative fiction is an umbrella term, umbrella genre of fiction that encompasses all the subgenres that depart from Realism (arts), realism, or strictly imitating everyday reality, instead presenting fantastical, supernatural, futuristic, or ...
as the basis for a
time loop The time loop or temporal loop is a plot device in fiction whereby Character (arts), characters re-experience a span of time which is repeated, sometimes more than once, with some hope of breaking out of the cycle of repetition. Time loops are co ...
into which unwary victims may become trapped. Examples of this trope include
Martin Gardner Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing magic, scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writin ...
s "No-Sided Professor" (1946), Armin Joseph Deutschs " A Subway Named Mobius" (1950) and the film '' Moebius'' (1996) based on it. An entire world shaped like a Möbius strip is the setting of Arthur C. Clarke's "The Wall of Darkness" (1946), while conventional Möbius strips are used as clever inventions in multiple stories of William Hazlett Upson from the 1940s. Other works of fiction have been analyzed as having a Möbius strip–like structure, in which elements of the plot repeat with a twist; these include
Marcel Proust Valentin Louis Georges Eugène Marcel Proust ( ; ; 10 July 1871 – 18 November 1922) was a French novelist, literary critic, and essayist who wrote the novel (in French – translated in English as ''Remembrance of Things Past'' and more r ...
''
In Search of Lost Time ''In Search of Lost Time'' (), first translated into English as ''Remembrance of Things Past'', and sometimes referred to in French as ''La Recherche'' (''The Search''), is a novel in seven volumes by French author Marcel Proust. This early twen ...
'' (1913–1927),
Luigi Pirandello Luigi Pirandello (; ; 28 June 1867 – 10 December 1936) was an Italians, Italian dramatist, novelist, poet, and short story writer whose greatest contributions were his plays. He was awarded the 1934 Nobel Prize in Literature "for his bold and ...
'' Six Characters in Search of an Author'' (1921),
Frank Capra Frank Russell Capra (born Francesco Rosario Capra; May 18, 1897 – September 3, 1991) was an Italian-American film director, producer, and screenwriter who was the creative force behind Frank Capra filmography#Films that won Academy Award ...
s '' It's a Wonderful Life'' (1946),
John Barth John Simmons Barth (; May 27, 1930 – April 2, 2024) was an American writer best known for his postmodern and metafictional fiction. His most highly regarded and influential works were published in the 1960s, and include '' The Sot-Weed Facto ...
''
Lost in the Funhouse ''Lost in the Funhouse'' (1968) is a short story collection by American author John Barth. The postmodern stories are extremely self-conscious and self-reflexive, and are considered to exemplify metafiction. Though Barth's reputation rests mai ...
'' (1968),
Samuel R. Delany Samuel R. "Chip" Delany (, ; born April 1, 1942) is an American writer and literary critic. His work includes fiction (especially science fiction), memoir, criticism, and essays on science fiction, literature, sexual orientation, sexuality, and ...
s '' Dhalgren'' (1975) and the film ''
Donnie Darko ''Donnie Darko'' is a 2001 American Science fiction film, science fiction psychological thriller film written and directed by Richard Kelly (filmmaker), Richard Kelly in his List of directorial debuts, directorial debut, and produced by Flower ...
'' (2001). One of the musical canons by J. S. Bach, the fifth of 14 canons ( BWV 1087) discovered in 1974 in Bach's copy of the '' Goldberg Variations'', features a glide-reflect symmetry in which each voice in the canon repeats, with inverted notes, the same motif from two measures earlier. Because of this symmetry, this canon can be thought of as having its score written on a Möbius strip. In
music theory Music theory is the study of theoretical frameworks for understanding the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory": The first is the "Elements of music, ...
, tones that differ by an octave are generally considered to be equivalent notes, and the space of possible notes forms a circle, the chromatic circle. Because the Möbius strip is the configuration space of two unordered points on a circle, the space of all two-note chords takes the shape of a Möbius strip. This conception, and generalizations to more points, is a significant application of orbifolds to music theory. Modern musical groups taking their name from the Möbius strip include American electronic rock trio Mobius Band and Norwegian progressive rock band Ring Van Möbius. Möbius strips and their properties have been used in the design of stage magic. One such trick, known as the Afghan bands, uses the fact that the Möbius strip remains in one piece as a single strip when cut lengthwise. It originated in the 1880s, and 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

* Möbius counter, a shift register whose output bit is complemented before being fed back into the input bit * Penrose triangle, an impossible figure whose boundary appears to wrap around it in a Möbius strip * Ribbon theory, the mathematical theory of infinitesimally thin strips that follow knotted space curves * Smale–Williams attractor, a fractal formed by repeatedly thickening a space curve to a Möbius strip and then replacing it with the boundary edge *
Umbilic torus The umbilic torus or umbilic bracelet is a single-edged 3-dimensional shape. The lone edge goes three times around the ring before returning to the starting point. The shape also has a single external face. A cross section (geometry), cross sectio ...


Notes


References


External links

* * {{DEFAULTSORT:Mobius Strip Topology Recreational mathematics Surfaces Eponyms in geometry