View source for Möbius strip

{{good article}}
{{Short description|Non-orientable surface with one edge}}
{{Use mdy dates|cs1-dates=ly|date=March 2022}}
{{Use list-defined references|date=March 2022}}
[[File:Möbius Strip.jpg|thumb|right|A paper Möbius strip]]
In [[mathematics]], a '''Möbius strip''', '''Möbius band''', or '''Möbius loop'''{{efn|1=Pronounced {{IPAc-en|US|ˈ|m|oʊ|b|i|ə|s|,_|ˈ|m|eɪ|-}} {{respell|MOH|bee|əs|,_|MAY|-}}, {{IPAc-en|UK|ˈ|m|ɜː|b|i|ə|s}};<ref>{{cite LPD|3}}</ref> {{IPA|de|ˈmøːbi̯ʊs|lang}}. As is common for words containing an [[umlaut (diacritic)|umlaut]], it is also often spelled ''Mobius'' or ''Moebius''.}} is a [[Surface (topology)|surface]] 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]] and [[August Ferdinand Möbius]] in 1858, but it had already appeared in [[Ancient Rome|Roman]] mosaics from the third century [[Common Era|CE]]. The Möbius strip is a [[orientability|non-orientable]] surface, meaning that within it one cannot consistently distinguish [[clockwise]] from counterclockwise turns. Every non-orientable surface contains a Möbius strip.

As an abstract [[topological space]], the Möbius strip can be embedded into three-dimensional [[Euclidean space]] 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 [[Knot (mathematics)|knotted]] centerline. Any two embeddings with the same knot for the centerline and the same number and direction of twists are [[Ambient isotopy|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 (topology)|boundary curve]].

Several geometric constructions of the Möbius strip provide it with additional structure. It can be swept as a [[ruled surface]] 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 [[Mathematics of paper folding#Flat folding|folded flat]]; the flattened Möbius strips include the [[trihexaflexagon]]. The Sudanese Möbius strip is a [[minimal surface]] in a [[hypersphere]], 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 [[Uniformization theorem|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 [[Belt (mechanical)|mechanical belts]] that wear evenly on both sides, dual-track [[roller coaster]]s whose carriages alternate between the two tracks, and [[world map]]s printed so that [[antipodes]] 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]]. In popular culture, Möbius strips appear in artworks by [[M. C. Escher]], [[Max Bill]], and others, and in the design of the [[recycling symbol]]. Many architectural concepts have been inspired by the Möbius strip, including the building design for the [[NASCAR Hall of Fame]]. Performers including [[Harry Blackstone Sr.]] and [[Thomas Nelson Downs]] have based stage magic tricks on the properties of the Möbius strip. The [[Canon (music)|canons]] of [[J. S. Bach]] have been analyzed using Möbius strips. Many works of [[speculative fiction]] 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==
{{multiple image|total_width=480
|image1=Aion mosaic Glyptothek Munich W504.jpg|caption1=Mosaic from ancient [[Sentinum]] depicting [[Aion (deity)|Aion]] holding a Möbius strip
|image2=Al-Jazari Automata 1205.jpg|caption2=[[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 {{nowrap|1858.{{r|pickover}}}} 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 {{nowrap|third century CE.{{r|roman|ancient}}}} 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 [[Annulus (mathematics)|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 {{nowrap|match up.{{r|roman}}}} Another mosaic from the town of [[Sentinum]] (depicted) shows the [[zodiac]], held by the god [[Aion (deity)|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 [[lemniscate|figure-eight]]-shaped decorations are also alleged to depict Möbius strips, but whether they were intended to depict flat strips of any type is {{nowrap|unclear.{{r|ancient}}}}

Independently of the mathematical tradition, machinists have long known that [[Belt (mechanical)|mechanical belt]]s 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.{{r|roman}} 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 {{nowrap|chain.{{r|ancient}}}} 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 {{nowrap|garment.{{r|roman}}}}

==Properties==
[[File:Fiddler crab mobius strip.gif|thumb|upright|left|A 2D object traversing once around the Möbius strip returns in mirrored form]]
The Möbius strip has several curious properties. It is a [[orientability|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 {{nowrap|subset.{{r|chirality}}}} 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 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 surface]]s 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.{{sfnp|Pickover|2005|pp=8–9}} 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 {{nowrap|sides.{{r|woll}}}} 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 {{nowrap|strip.{{efn|Essentially this example, but for a [[Klein bottle]] rather than a Möbius strip, is given by {{harvtxt|Blackett|1982}}.{{r|blackett}}}}}} In contrast to disks, spheres, and cylinders, for which it is possible to simultaneously embed an [[uncountable set]] of [[disjoint sets|disjoint]] copies into three-dimensional space, only a countable number of Möbius strips can be simultaneously {{nowrap|embedded.{{r|frolkina|defy|melikhov}}}}

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 {{nowrap|boundary.{{sfnp|Pickover|2005|pp=8–9}}}} A Möbius strip in Euclidean space cannot be moved or stretched into its mirror image; it is a [[chirality (mathematics)|chiral]] object with right- or {{nowrap|left-handedness.{{sfnp|Pickover|2005|p=52}}}} 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 {{nowrap|surfaces.{{sfnp|Pickover|2005|p=12}}}} 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 {{nowrap|other.{{r|kyle}}}} With an even number of twists, however, one obtains a different topological surface, called the {{nowrap|[[Annulus (mathematics)|annulus]].{{sfnp|Pickover|2005|p=11}}}}

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]] is the same as the fundamental group of a circle, an [[infinite cyclic group]]. Therefore, paths on the Möbius strip that start and end at the same point can be distinguished topologically (up to [[homotopy]]) only by the number of times they loop around the strip.{{r|massey}}

{{multiple image|total_width=480
|image1=Moebiusband-1s.svg|caption1=Cutting the centerline produces a double-length two-sided (non-Möbius) strip
|image2=Moebiusband-2s.svg|caption2=A single off-center cut produces a Möbius strip (purple) linked with a double-length two-sided 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 {{nowrap|half-twists.{{sfnp|Pickover|2005|pp=8–9}}}} These interlinked shapes, formed by lengthwise slices of Möbius strips with varying widths, are sometimes called ''paradromic'' {{nowrap|''rings''.{{r|rouseball|paradromic}}}}

{{multiple image|total_width=480
|image1=Tietze-Moebius.svg|caption1=Subdivision into six mutually adjacent regions, bounded by [[Tietze's graph]]
|image2=3 utilities problem moebius.svg|caption2=Solution to the [[three utilities problem]] on a Möbius 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 {{nowrap|plane.{{r|tietze}}}} Six colors are always enough. This result is part of the [[Ringel–Youngs theorem]], which states how many colors each topological surface {{nowrap|needs.{{r|ringel-youngs}}}} The edges and vertices of these six regions form [[Tietze's graph]], which is a [[dual graph]] on this surface for the six-vertex [[complete graph]] but cannot be [[planar graph|drawn without crossings on a plane]]. Another family of graphs that can be [[graph embedding|embedded]] on the Möbius strip, but not on the plane, are the [[Möbius ladder]]s, the boundaries of subdivisions of the Möbius strip into rectangles meeting {{nowrap|end-to-end.{{r|jab-rad-saz}}}} These include the utility graph, a six-vertex [[complete bipartite graph]] whose embedding into the Möbius strip shows that, unlike in the plane, the [[three utilities problem]] can be solved on a transparent Möbius {{nowrap|strip.{{r|larsen}}}} 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 <math>V</math>, <math>E</math>, and <math>F</math> of vertices, edges, and regions satisfy <math>V-E+F=0</math>. For instance, Tietze's graph has <math>12</math> vertices, <math>18</math> edges, and <math>6</math> regions; {{nowrap|<math>12-18+6=0</math>.{{r|tietze}}}}
{{-}}

==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===
{{CSS image crop
|Image = Mobius strip.gif
|bSize = 400
|cWidth = 185
|cHeight = 150
|oTop = 115
|oLeft = 105
|Description=A Möbius strip swept out by a rotating line segment in a rotating plane}}
{{CSS image crop
|Image=Plucker's conoid (n=2).gif
|bSize=360
|cWidth=240
|cHeight=240
|oTop=60
|oLeft=60
|Description=[[Plücker's conoid]] swept out by a different motion of 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 {{nowrap|lines.{{r|maschke}}}} 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]] defined by equations for the [[Cartesian coordinates]] of its points,
<math display=block>
\begin{align}
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{align}</math>
for <math>0 \le u< 2\pi</math> and {{nowrap|<math>-1 \le v\le 1</math>,}}
where one parameter <math>u</math> describes the rotation angle of the plane around its central axis and the other parameter {{nowrap|<math>v</math>}} 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 <math>xy</math>-plane and is centered at {{nowrap|<math>(0, 0, 0)</math>.{{r|parameterization}}}} 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]] swept out by this disk. Because of the one-sidedness of this slice, the sliced torus remains {{nowrap|connected.{{r|split-tori}}}}

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 the form of a self-crossing Möbius {{nowrap|strip.{{r|francis}}}} It has applications in the design of {{nowrap|[[gear]]s.{{r|dooner-seirig}}}}

===Polyhedral surfaces and flat foldings===
[[File:Flexagon.gif|thumb|left|upright=0.6|Trihexaflexagon being flexed]]
A strip of paper can form a [[Mathematics of paper folding#Flat folding|flattened]] Möbius strip in the plane by folding it at <math>60^\circ</math> angles so that its center line lies along an [[equilateral triangle]], 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]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle}} to its width{{snd}}is {{nowrap|<math>\sqrt 3\approx 1.73</math>,}} and the same folding method works for any larger aspect {{nowrap|ratio.{{r|barr|fuchs-tabachnikov}}}} For a strip of nine equilateral triangles, the result is a [[trihexaflexagon]], which can be flexed to reveal different parts of its {{nowrap|surface.{{r|pook}}}} 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 <math>1\times 1</math> strip would become a <math>1\times \tfrac{1}{3}</math> folded strip whose [[Cross section (geometry)|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 {{nowrap|would be.{{r|barr|fuchs-tabachnikov}}}}

{{Multiple image|total_width=400|image1=5-vertex polyhedral Möbius strip.svg|image2=Pentagonal Möbius strip.svg|footer=Five-vertex polyhedral and flat-folded Möbius strips}}
The Möbius strip can also be embedded as a [[polyhedron|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]], which requires six triangles and six vertices, even when represented more abstractly as a [[Abstract simplicial complex|simplicial complex]].{{r|9vertex}}{{efn|The flat-folded Möbius strip formed from three equilateral triangles does not come from an [[abstract simplicial complex]], because all three triangles share the same three vertices, while abstract simplicial complexes require each triangle to have a different set of vertices.}} A five-triangle Möbius strip can be represented most symmetrically by five of the ten equilateral triangles of a [[5-cell|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]]s separate it into two parts that are topologically equivalent to disks or {{nowrap|circles.{{r|kuiper}}}}

Other polyhedral embeddings of Möbius strips include one with four convex [[quadrilateral]]s as faces, another with three non-convex quadrilateral {{nowrap|faces,{{r|szilassi}}}} and one using the vertices and center point of a regular [[octahedron]], with a triangular {{nowrap|boundary.{{r|tuckerman}}}} Every abstract triangulation of the [[projective plane]] can be embedded into 3D as a polyhedral Möbius strip with a triangular boundary after removing one of its {{nowrap|faces;{{r|bon-nak}}}} 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 {{nowrap|edges.{{r|9vertex}}}} However, not every abstract triangulation of the Möbius strip can be represented geometrically, as a polyhedral {{nowrap|surface.{{r|brehm}}}} To be realizable, it is necessary and sufficient that there be no two disjoint non-contractible 3-cycles in the {{nowrap|triangulation.{{r|nak-tsu}}}}

===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 {{nowrap|<math>\sqrt 3\approx 1.73</math>,}} the same ratio as for the flat-folded equilateral-triangle version of the Möbius {{nowrap|strip.{{r|sadowsky-translation}}}} This flat triangular embedding can lift to a smooth{{efn|This piecewise planar and cylindrical embedding has [[smoothness]] class <math>C^2</math>, and can be approximated arbitrarily accurately by [[infinitely differentiable]] {{nowrap|(class <math>C^\infty</math>)}} embeddings.{{r|bartels-hornung}}}} embedding in three dimensions, in which the strip lies flat in three parallel planes between three cylindrical rollers, each tangent to two of the {{nowrap|planes.{{r|sadowsky-translation}}}} Mathematically, a smoothly embedded sheet of paper can be modeled as a [[developable surface]], that can bend but cannot {{nowrap|stretch.{{r|bartels-hornung|starostin-vdh}}}} As its aspect ratio decreases toward <math>\sqrt 3</math>, all smooth embeddings seem to approach the same triangular {{nowrap|form.{{r|darkside}}}}

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 {{nowrap|folds.{{r|fuchs-tabachnikov}}}} 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 {{nowrap|<math>\pi/2\approx 1.57</math>,}} even if self-intersections are allowed. Self-intersecting smooth Möbius strips exist for any aspect ratio above this {{nowrap|bound.{{r|fuchs-tabachnikov|halpern-weaver}}}} Without self-intersections, the aspect ratio must be at {{nowrap|least{{r|schwartz}}}}
<math display=block>\frac{2}{3}\sqrt{3+2\sqrt3}\approx 1.695.</math>

{{unsolved|mathematics|Can a <math>12\times 7</math> paper rectangle be glued end-to-end to form a smooth Möbius strip embedded in space?&thinsp;{{efn|12/7 is the simplest rational number in the range of aspect ratios, between 1.695 and 1.73, for which the existence of a smooth embedding is unknown.}}}}
For aspect ratios between this bound {{nowrap|and <math>\sqrt 3</math>,}} it has been an open problem whether smooth embeddings, without self-intersection, {{nowrap|exist.{{r|fuchs-tabachnikov|halpern-weaver|schwartz}}}} In 2023, [[Richard Schwartz (mathematician)|Richard Schwartz]] announced a proof that they do not exist, but this result still awaits peer review and publication.{{r|optimal|crowell}} If the requirement of smoothness is relaxed to allow [[continuously differentiable]] 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 {{nowrap|becomes.{{efn|These surfaces have smoothness class <math>C^1</math>. For a more fine-grained analysis of the smoothness assumptions that force an embedding to be developable versus the assumptions under which the [[Nash–Kuiper theorem]] allows arbitrarily flexible embeddings, see remarks by {{harvtxt|Bartels|Hornung|2015}}, p. 116, following Theorem 2.2.{{r|bartels-hornung}}}}}} 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]].{{r|dundas}} Although it has no smooth closed embedding into three-dimensional space, it can be embedded smoothly as a closed subset of four-dimensional Euclidean {{nowrap|space.{{r|blanusa}}}}

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]].{{r|bartels-hornung|starostin-vdh}} It is also possible to find [[algebraic surface]]s that contain rectangular developable Möbius {{nowrap|strips.{{r|wunderlich|schwarz}}}}

===Making the boundary circular===
{{multiple image|total_width=480
|image1=Mobius to Klein.gif|caption1=Gluing two Möbius strips to form a Klein bottle
|image2=MobiusStrip-02.png|caption2=A projection of the Sudanese Möbius strip}}
The edge, or [[boundary (topology)|boundary]], of a Möbius strip is [[homeomorphic|topologically equivalent]] to a [[circle]]. In common forms of the Möbius strip, it has a different shape from a circle, but it is [[unknot]]ted, and therefore the whole strip can be stretched without crossing itself to make the edge perfectly {{nowrap|circular.{{r|hilbert-cohn-vossen}}}} One such example 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 [[Immersion (mathematics)|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, and{{snd}}reversing that process{{snd}}a Klein bottle can be sliced along a carefully chosen cut to produce two Möbius {{nowrap|strips.{{r|spivak}}}} 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 {{nowrap|edges.{{r|ddg}}}}

Lawson's Klein bottle is a self-crossing [[minimal surface]] in the [[unit hypersphere]] of 4-dimensional space, the set of points of the form <math display=block>(\cos\theta\cos\phi,\sin\theta\cos\phi,\cos2\theta\sin\phi,\sin2\theta\sin \phi)</math> for {{nowrap|<math>0\le\theta<\pi,0\le\phi<2\pi</math>.{{r|lawson}}}} Half of this Klein bottle, the subset with <math>0\le\phi<\pi</math>, gives a Möbius strip embedded in the hypersphere as a minimal surface with a [[great circle]] as its {{nowrap|boundary.{{r|schleimer-segerman}}}} This embedding is sometimes called the "Sudanese Möbius strip" after topologists Sue Goodman and Daniel Asimov, who discovered it in the {{nowrap|1970s.{{r|sudanese}}}} 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 {{nowrap|circles.{{r|ddg|franzoni}}}} [[Stereographic projection]] transforms this shape from a three-dimensional spherical space into three-dimensional Euclidean space, preserving the circularity of its {{nowrap|boundary.{{r|ddg}}}} 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 {{nowrap|centerline.{{r|schleimer-segerman}}}} Instead, leaving the Sudanese Möbius strip unprojected, in the 3-sphere, leaves it with an infinite group of symmetries isomorphic to the [[orthogonal group]] {{nowrap|<math>\mathrm{O}(2)</math>,}} the group of symmetries of a {{nowrap|circle.{{r|lawson}}}}

[[File:Cross-cap level sets.svg|thumb|upright=0.8|Schematic depiction of a cross-cap with an open bottom, showing its [[level set]]s. This surface crosses itself along the vertical line segment.]]
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 {{nowrap|circle,{{r|huggett-jordan}}}} 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]] 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 {{nowrap|orientation.{{r|flapan}}}} The two parts of the surface formed by the two glued pairs of edges cross each other with a [[Pinch point (mathematics)|pinch point]] like that of a [[Whitney umbrella]] at each end of the crossing {{nowrap|segment,{{r|richeson}}}} the same topological structure seen in Plücker's {{nowrap|conoid.{{r|francis}}}}

===Surfaces of constant curvature===
The open Möbius strip is the [[relative interior]] of a standard Möbius strip, formed by omitting the points on its boundary edge. It may be given a [[Riemannian geometry]] of constant positive, negative, or zero [[Gaussian curvature]]. The cases of negative and zero curvature form geodesically complete surfaces, which means that all [[geodesic]]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 {{nowrap|bundle.{{r|dundas}}}} 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 (topology)|quotient space]] of a plane by a [[glide reflection]], and (together with the plane, [[cylinder]], [[torus]], and [[Klein bottle]]) is one of only five two-dimensional complete {{nowrap|[[flat manifold]]s.{{r|godinho-natario}}}}
;Negative curvature
:The open Möbius strip also admits complete metrics of constant negative curvature. One way to see this is to begin with the [[Poincaré half-plane model|upper half plane (Poincaré) model]] of the [[Hyperbolic geometry|hyperbolic plane]], a geometry of constant curvature whose lines are represented in the model by semicircles that meet the <math>x</math>-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]] (actually two, on opposite sides of the axis of glide-reflection), and is one of only 13 nonstandard {{nowrap|surfaces.{{r|cantwell-conlon}}}} Again, this can be understood as the quotient of the hyperbolic plane by a glide {{nowrap|reflection.{{r|stillwell}}}}
;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 [[Real projective plane|projective plane]].{{r|godinho-natario}} 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 {{nowrap|plane.{{r|seifert-threlfall}}}}

The [[minimal surface]]s 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 {{nowrap|strip,{{r|lopez-martin}}}} after its 1982 description by [[William Hamilton Meeks, III]].{{r|meeks}} 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 {{nowrap|surfaces.{{r|systolic}}}} 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 {{nowrap|curve.{{r|bjorling}}}}

===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 [[diffeomorphic|topologically equivalent]] to the open Möbius {{nowrap|strip.{{r|parker}}}} One way to see this is to extend the Euclidean plane to the [[real projective plane]] by adding one more line, the [[line at infinity]]. By [[projective duality]] 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 {{nowrap|lines.{{r|bickel}}}} Therefore, the space of Euclidean lines is a punctured projective plane, which is one of the forms of the open Möbius {{nowrap|strip.{{r|seifert-threlfall}}}} The space of lines in the [[hyperbolic plane]] can be parameterized by [[unordered pair]]s 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 {{nowrap|strip.{{r|mangahas}}}}

These spaces of lines are highly symmetric. The symmetries of Euclidean lines include the [[affine transformation]]s, and the symmetries of hyperbolic lines include the {{nowrap|[[Möbius transformation]]s.{{r|ramirez-seade}}}} The affine transformations and Möbius transformations both form {{nowrap|6-dimensional}} [[Lie group]]s, topological spaces having a compatible [[Symmetry group|algebraic structure]] describing the composition of {{nowrap|symmetries.{{r|fomenko-kunii|isham}}}} Because every line in the plane is symmetric to every other line, the open Möbius strip is a [[homogeneous space]], a space with symmetries that take every point to every other point. Homogeneous spaces of Lie groups are called [[solvmanifold]]s, and the Möbius strip can be used as a [[counterexample]], showing that not every solvmanifold is a [[nilmanifold]], and that not every solvmanifold can be factored into a [[Direct product of groups|direct product]] of a [[compact space|compact]] solvmanifold {{nowrap|with <math>\mathbb{R}^n</math>.}} 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]] of its action; contracting the [[coset]]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 {{nowrap|<math>x</math>-axis}} consists of all symmetries that take the axis to itself. Each line <math>\ell</math> corresponds to a coset, the set of symmetries that map <math>\ell</math> to the {{nowrap|<math>x</math>-axis.}} Therefore, the [[Quotient space (topology)|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 {{nowrap|strip.{{r|gor-oni-vin}}}}

==Applications==
[[File:Möbius resistor.svg|thumb|upright=0.8|Electrical flow in a [[Möbius resistor]]]]
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]] ribbons twisted to form Möbius strips with new electronic characteristics including helical magnetism{{r|graphene}}
*[[Möbius aromaticity]], a property of [[organic chemical]]s whose molecular structure forms a cycle, with [[molecular orbital]]s aligned along the cycle in the pattern of a Möbius strip{{r|aromaticity1|aromaticity2}}
*The [[Möbius resistor]], a strip of conductive material covering the single side of a [[dielectric]] Möbius strip, in a way that cancels its own [[self-inductance]]{{r|resistor}}{{sfnp|Pickover|2005|pp=45–46}}
*[[Resonator]]s with a compact design and a resonant frequency that is half that of identically constructed linear coils{{r|resonator|resonator2}}
*[[Polarization (waves)|Polarization]] patterns in light emerging from a [[Q-plate|''q''-plate]]{{r|polarization}}
*A proof of the impossibility of continuous, anonymous, and unanimous two-party aggregation rules in [[social choice theory]]{{r|can-ind}}
*[[Möbius loop roller coaster]]s, 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{{r|coaster1|coaster2}}
*[[World map]]s projected onto a Möbius strip with the convenient properties that there are no east–west boundaries, and that the [[antipodes|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{{r|maps1|maps2}}

Scientists have also studied the energetics of [[soap film]]s shaped as Möbius strips,{{r|courant|soap}} the [[chemical synthesis]] of [[molecule]]s with a Möbius strip shape,{{r|synthesis}}{{sfnp|Pickover|2005|pp=52–58}} and the formation of larger [[nanoscale]] Möbius strips using [[DNA origami]].{{r|dna}}

==In popular culture==
[[File:Middelheim Max Bill Eindeloze kronkel 1956 03 Cropped.jpg|thumb|''Endless Twist'', [[Max Bill]], 1956, from the [[Middelheim Open Air Sculpture Museum]]]]
Two-dimensional artworks featuring the Möbius strip include an untitled 1947 painting by [[Corrado Cagli]] (memorialized in a poem by [[Charles Olson]]),{{r|emmer|olson}} and two prints by [[M. C. Escher]]: ''Möbius Band I'' (1961), depicting three folded [[flatfish]] biting each others' tails; and ''Möbius Band II'' (1963), depicting ants crawling around a [[lemniscate]]-shaped Möbius strip.{{r|escher1|escher2}} It is also a popular subject of [[mathematical sculpture]], including works by [[Max Bill]] (''Endless Ribbon'', 1953), [[José de Rivera]] (''[[Infinity (de Rivera)|Infinity]]'', 1967), and [[Sebastián (sculptor)|Sebastián]].{{r|emmer}} A [[trefoil knot|trefoil-knotted]] Möbius strip was used in [[John Robinson (sculptor)|John Robinson]]{{'}}s ''Immortality'' (1982).{{sfnp|Pickover|2005|p=13}} [[Charles O. Perry]]'s ''[[Continuum (sculpture)|Continuum]]'' (1976) is one of several pieces by Perry exploring variations of the Möbius strip.{{r|brecher}}

{{multiple image|total_width=400
|image1=Recycle001.svg|caption1=[[Recycling symbol]]
|image2=Logo of Google Drive (2012-2014).svg|caption2=[[Google Drive]] logo (2012–2014)
|image3=Stamp of Brazil - 1967 - Colnect 263101 - Mobius Symbol.jpeg|caption3=[[Instituto Nacional de Matemática Pura e Aplicada|IMPA]] logo on stamp}}

Because of their easily recognized form, Möbius strips are a common element of [[graphic design]].{{sfnp|Pickover|2005|page=13}} The familiar [[recycling symbol|three-arrow logo]] for [[recycling]], designed in 1970, is based on the smooth triangular form of the Möbius {{nowrap|strip,{{r|peterson}}}} as was the logo for the environmentally-themed [[Expo '74]].{{r|expo74}} Some variations of the recycling symbol use a different embedding with three half-twists instead of {{nowrap|one,{{r|peterson}}}} and the original version of the [[Google Drive]] logo used a flat-folded three-twist Möbius strip, as have other similar designs.{{r|gdrive}} 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.{{r|impa}} The Möbius strip has also featured in the artwork for [[postage stamp]]s from countries including Brazil, Belgium, the Netherlands, and {{nowrap|Switzerland.{{sfnp|Pickover|2005|pp=156–157}}{{r|briefmarken}}}}

[[File:NASCAR Hall of Fame (7553589908).jpg|thumb|upright|[[NASCAR Hall of Fame]] entrance]]
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.{{r|architecture|bridges}} 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.{{r|kazakh}} One notable building incorporating a Möbius strip is the [[NASCAR Hall of Fame]], 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.{{r|nascar}} On a smaller scale, ''Moebius Chair'' (2006) by [[Pedro Reyes (artist)|Pedro Reyes]] is a [[courting bench]] whose base and sides have the form of a Möbius strip.{{r|reyes}} As a form of [[mathematics and fiber arts]], [[scarf|scarves]] have been [[knitting|knit]] into Möbius strips since the work of [[Elizabeth Zimmermann]] in the early 1980s.{{r|zimmermann}} In [[food styling]], Möbius strips have been used for slicing [[bagel]]s,{{r|bagel}} making loops out of [[bacon]],{{r|bacon}} and creating new shapes for [[pasta]].{{r|pasta}}

Although mathematically the Möbius strip and the fourth dimension are both purely spatial concepts, they have often been invoked in [[speculative fiction]] as the basis for a [[time loop]] into which unwary victims may become trapped. Examples of this trope include [[Martin Gardner]]{{'}}s "No-Sided Professor" (1946), [[Armin Joseph Deutsch]]{{'}}s "[[A Subway Named Mobius]]" (1950) and the film ''[[Moebius (1996 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.{{sfnp|Pickover|2005|pp=174–177}} 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]]{{'s}} ''[[In Search of Lost Time]]'' (1913–1927), [[Luigi Pirandello]]{{'s}} ''[[Six Characters in Search of an Author]]'' (1921), [[Frank Capra]]{{'}}s ''[[It's a Wonderful Life]]'' (1946), [[John Barth]]{{'s}} ''[[Lost in the Funhouse]]'' (1968), [[Samuel R. Delany]]{{'}}s ''[[Dhalgren]]'' (1975) and the film ''[[Donnie Darko]]'' (2001).{{sfnp|Pickover|2005|pp=179–187}}

One of the [[Canon (music)|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 reflection|glide-reflect]] symmetry in which each voice in the canon repeats, with [[Inversion (music)|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.{{r|phillips}}{{efn|Möbius strips have also been used to analyze many other canons by Bach and others, but in most of these cases other looping surfaces such as a cylinder could have been used equally well.{{r|phillips}}}} In [[music theory]], 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 (mathematics)|configuration space]] of two unordered points on a circle, the space of all [[Dyad (music)|two-note chords]] takes the shape of a Möbius strip. This conception, and generalizations to more points, is a significant [[Orbifold#Music theory|application of orbifolds to music theory]].{{r|music|chords}} Modern musical groups taking their name from the Möbius strip include American electronic rock trio [[Mobius Band (band)|Mobius Band]]{{r|bandband}} and Norwegian progressive rock band [[Ring Van Möbius]].{{r|ringvan}}

Möbius strips and their properties have been used in the design of [[Magic (illusion)|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]].{{r|magic|gardner}}

==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]]

==Notes==
{{Notelist}}

==References==
{{reflist|refs=

<ref name=9vertex>{{cite journal
 | last1 = Kühnel | first1 = W.
 | last2 = Banchoff | first2 = T. F. | author2-link = Thomas Banchoff
 | doi = 10.1007/BF03026567
 | issue = 3
 | journal = [[The Mathematical Intelligencer]]
 | mr = 737686
 | pages = 11–22
 | title = The 9-vertex complex projective plane
 | url = https://www.math.brown.edu/tbanchof/howison/newbanchoff/publications/pdfs/9Vertex.pdf
 | volume = 5
 | year = 1983| s2cid = 120926324
 }}</ref>

<ref name=ancient>{{cite journal
 | last1 = Cartwright | first1 = Julyan H. E. | author1-link = Julyan Cartwright 
 | last2 = González | first2 = Diego L.
 | arxiv = 1609.07779
 | bibcode = 2016arXiv160907779C
 | doi = 10.1007/s00283-016-9631-8
 | issue = 2
 | journal = [[The Mathematical Intelligencer]]
 | mr = 3507121
 | pages = 69–76
 | title = Möbius strips before Möbius: topological hints in ancient representations
 | volume = 38
 | year = 2016| s2cid = 119587191
 }}</ref>

<ref name=architecture>{{cite conference
 | last1 = Thulaseedas | first1 = Jolly
 | last2 = Krawczyk | first2 = Robert J.
 | editor1-last = Barrallo | editor1-first = Javier
 | editor2-last = Friedman | editor2-first = Nathaniel
 | editor3-last = Maldonado | editor3-first = Juan Antonio
 | editor4-last = Mart\'\inez-Aroza | editor4-first = José
 | editor5-last = Sarhangi | editor5-first = Reza
 | editor6-last = Séquin | editor6-first = Carlo
 | contribution = Möbius concepts in architecture
 | contribution-url = https://archive.bridgesmathart.org/2003/bridges2003-353.html
 | isbn = 84-930669-1-5
 | location = Granada, Spain
 | pages = 353–360
 | publisher = University of Granada
 | title = Meeting Alhambra, ISAMA-BRIDGES Conference Proceedings
 | year = 2003}}</ref>

<ref name=aromaticity1>{{cite journal
 | last = Rzepa | first = Henry S.
 | date = September 2005
 | doi = 10.1021/cr030092l
 | issue = 10
 | journal = Chemical Reviews
 | pages = 3697–3715
 | title = Möbius aromaticity and delocalization
 | volume = 105| pmid = 16218564
 }}</ref>

<ref name=aromaticity2>{{cite journal
 | last1 = Yoon | first1 = Zin Seok
 | last2 = Osuka | first2 = Atsuhiro
 | last3 = Kim | first3 = Dongho
 | date = May 2009
 | doi = 10.1038/nchem.172
 | issue = 2
 | journal = Nature Chemistry
 | pages = 113–122
 | title = Möbius aromaticity and antiaromaticity in expanded porphyrins
 | volume = 1| pmid = 21378823
 | bibcode = 2009NatCh...1..113Y
 }}</ref>

<ref name=bacon>{{cite web|url=https://www.theverge.com/2014/9/5/6110563/mobius-bacon-recipe|first=Ross|last=Miller|title=How to make a mathematically-endless strip of bacon|date=September 5, 2014|work=The Verge}}</ref>

<ref name=bagel>{{cite web|url=https://www.npr.org/sections/thesalt/2015/08/06/429437860/cut-your-bagel-the-mathematically-correct-way|publisher=NPR|title=Cut Your Bagel The Mathematically Correct Way|work=The Salt|first=Dan|last=Pashman|date=August 6, 2015}}</ref>

<ref name=bandband>{{cite magazine|url=https://magnetmagazine.com/2007/08/30/mobius-band-friendly-fire/|title=Mobius Band: Friendly Fire|magazine=[[Magnet (magazine)|Magnet]]|date=August 30, 2007|first=Andrew|last=Parks}}</ref>

<ref name=barr>{{cite book
 | last = Barr | first = Stephen
 | location = New York
 | pages = 40–49, 200–201
 | publisher = Thomas Y. Crowell Company
 | title = Experiments in Topology
 | url = https://archive.org/details/experimentsintop00barr
 | year = 1964| isbn = ((9780690278620))
 }}</ref>

<ref name=bartels-hornung>{{cite journal
 | last1 = Bartels | first1 = Sören
 | last2 = Hornung | first2 = Peter
 | doi = 10.1007/s10659-014-9501-6
 | issue = 1–2
 | journal = Journal of Elasticity
 | mr = 3326187
 | pages = 113–136
 | title = Bending paper and the Möbius strip
 | volume = 119
 | year = 2015| s2cid = 119782792
 }} Reprinted in {{harvtxt|Fosdick|Fried|2016}}, pp. 113–136. See in particular Section 5.2, pp. 129–130.</ref>

<ref name=bickel>{{cite journal
 | last = Bickel | first = Holger
 | doi = 10.1007/BF01229209
 | issue = 1–2
 | journal = Journal of Geometry
 | mr = 1675956
 | pages = 8–15
 | title = Duality in stable planes and related closure and kernel operations
 | volume = 64
 | year = 1999| s2cid = 122209943
 }}</ref>

<ref name=bjorling>{{cite journal
 | last = Mira | first = Pablo
 | doi = 10.1016/j.geomphys.2005.08.001
 | issue = 9
 | journal = Journal of Geometry and Physics
 | mr = 2240407
 | pages = 1506–1515
 | title = Complete minimal Möbius strips in <math>\mathbb{R}^n</math> and the Björling problem
 | volume = 56
 | year = 2006| bibcode = 2006JGP....56.1506M
 }}</ref>

<ref name=blackett>{{cite book
 | last = Blackett | first = Donald W.
 | isbn = 9781483262536
 | page = 195
 | publisher = Academic Press
 | title = Elementary Topology: A Combinatorial and Algebraic Approach
 | year = 1982}}</ref>

<ref name=blanusa>{{cite journal
 | last = Blanuša | first = Danilo | author-link = Danilo Blanuša
 | journal = Bulletin International de l'Académie Yougoslave des Sciences et des Beaux-Arts
 | mr = 71060
 | pages = 19–23
 | title = Le plongement isométrique de la bande de Möbius infiniment large euclidienne dans un espace sphérique, parabolique ou hyperbolique à quatre dimensions
 | volume = 12
 | year = 1954}}</ref>

<ref name=bon-nak>{{cite journal
 | last1 = Bonnington | first1 = C. Paul
 | last2 = Nakamoto | first2 = Atsuhiro
 | doi = 10.1007/s00454-007-9035-9
 | issue = 1
 | journal = [[Discrete & Computational Geometry]]
 | mr = 2429652
 | pages = 141–157
 | title = Geometric realization of a triangulation on the projective plane with one face removed
 | volume = 40
 | year = 2008| s2cid = 10887519
 | doi-access = free
 }}</ref>

<ref name=brecher>{{cite conference
 | last = Brecher | first = Kenneth
 | editor1-last = Swart | editor1-first = David
 | editor2-last = Séquin | editor2-first = Carlo H.
 | editor3-last = Fenyvesi | editor3-first = Kristóf
 | contribution = Art of infinity
 | contribution-url = https://archive.bridgesmathart.org/2017/bridges2017-153.html
 | isbn = 978-1-938664-22-9
 | location = Phoenix, Arizona
 | pages = 153–158
 | publisher = Tessellations Publishing
 | title = Proceedings of Bridges 2017: Mathematics, Art, Music, Architecture, Education, Culture
 | year = 2017}}</ref>

<ref name=brehm>{{cite journal
 | last = Brehm | first = Ulrich
 | doi = 10.2307/2045508
 | issue = 3
 | journal = [[Proceedings of the American Mathematical Society]]
 | mr = 715878
 | pages = 519–522
 | title = A nonpolyhedral triangulated Möbius strip
 | volume = 89
 | year = 1983| jstor = 2045508
 }}</ref>

<ref name=bridges>{{cite journal
 | last = Séquin | first = Carlo H. | author-link = Carlo H. Séquin
 | date = January 2018
 | doi = 10.1080/17513472.2017.1419331
 | issue = 2–3
 | journal = [[Journal of Mathematics and the Arts]]
 | pages = 181–194
 | title = Möbius bridges
 | volume = 12| s2cid = 216116708 }}</ref>

<ref name=briefmarken>{{cite journal
 | last1 = Decker | first1 = Heinz
 | last2 = Stark | first2 = Eberhard
 | issue = 7
 | journal = Praxis der Mathematik
 | mr = 720681
 | pages = 207–215
 | title = Möbius-Bänder: ...und natürlich auch auf Briefmarken
 | volume = 25
 | year = 1983}}</ref>

<ref name=can-ind>{{cite journal
 | last1 = Candeal | first1 = Juan Carlos
 | last2 = Induráin | first2 = Esteban
 | date = January 1994
 | doi = 10.1016/0165-1765(94)90045-0
 | issue = 3
 | journal = [[Economics Letters]]
 | pages = 407–412
 | title = The Moebius strip and a social choice paradox
 | volume = 45}}</ref>

<ref name=cantwell-conlon>{{cite journal
 | last1 = Cantwell | first1 = John
 | last2 = Conlon | first2 = Lawrence
 | arxiv = 1305.1379
 | doi = 10.1007/s10711-014-9975-1
 | journal = [[Geometriae Dedicata]]
 | mr = 3370020
 | pages = 27–42
 | title = Hyperbolic geometry and homotopic homeomorphisms of surfaces
 | volume = 177
 | year = 2015| s2cid = 119640200
 }}</ref>

<ref name=chirality>{{cite book
 | last = Flapan | first = Erica | author-link = Erica Flapan
 | doi = 10.1017/CBO9780511626272
 | isbn = 0-521-66254-0
 | mr = 1781912
 | pages = [https://books.google.com/books?id=ytjLCgAAQBAJ&pg=PA82 82–83]
 | publisher = Mathematical Association of America | location = Washington, DC
 | series = Outlooks
 | title = When Topology Meets Chemistry: A Topological Look at Molecular Chirality
 | title-link = When Topology Meets Chemistry
 | year = 2000}}</ref>

<ref name=chords>{{cite journal
 | last = Tymoczko | first = Dmitri | author-link = Dmitri Tymoczko
 | bibcode = 2006Sci...313...72T
 | date = July 7, 2006
 | doi = 10.1126/science.1126287
 | issue = 5783
 | journal = [[Science (journal)|Science]]
 | jstor = 3846592
 | pages = 72–4
 | pmid = 16825563
 | title = The geometry of musical chords
 | url = http://dmitri.mycpanel.princeton.edu/voiceleading.pdf
 | volume = 313| s2cid = 2877171 }}</ref>

<ref name=coaster1>{{cite book
 | last = Easdown | first = Martin
 | isbn = 9781782001522
 | page = 43
 | publisher = Bloomsbury Publishing
 | title = Amusement Park Rides
 | url = https://books.google.com/books?id=jjTDCwAAQBAJ&pg=PA43
 | year = 2012}}</ref>

<ref name=coaster2>{{cite book
 | last = Hook | first = Patrick
 | isbn = 9780785835776
 | page = 20
 | publisher = Chartwell Books
 | title = Ticket To Ride: The Essential Guide to the World's Greatest Roller Coasters and Thrill Rides
 | url = https://books.google.com/books?id=a7vdDwAAQBAJ&pg=PA20
 | year = 2019}}</ref>

<ref name=courant>{{cite journal
 | last = Courant | first = Richard | author-link = Richard Courant
 | doi = 10.1080/00029890.1940.11990957
 | journal = [[The American Mathematical Monthly]]
 | jstor = 2304225
 | mr = 1622
 | pages = 167–174
 | title = Soap film experiments with minimal surfaces
 | volume = 47
 | year = 1940| issue = 3 }}</ref>

<ref name=crowell>{{cite magazine|magazine=Scientific American|last=Crowell|first=Rachel|date=September 12, 2023|title=Mathematicians solve 50-year-old Möbius strip puzzle|url=https://www.scientificamerican.com/article/mathematicians-solve-50-year-old-moebius-strip-puzzle/}}</ref>

<ref name=darkside>{{cite journal
 | last = Schwarz | first = Gideon E.
 | doi = 10.1080/00029890.1990.11995680
 | issue = 10
 | journal = [[The American Mathematical Monthly]]
 | jstor = 2324325
 | mr = 1079975
 | pages = 890–897
 | title = The dark side of the Moebius strip
 | volume = 97
 | year = 1990}}</ref>

<ref name=ddg>{{cite web
 | last = Knöppel | first = Felix
 | date = Summer 2019
 | title = Tutorial 3: Lawson's Minimal Surfaces and the Sudanese Möbius Band
 | url = https://dgd.service.tu-berlin.de/wordpress/ddg2019/2019/04/29/tutorial-3-lawsons-minimal-surfaces-and-the-sudanese-mobius-band/
 | work = DDG2019: Visualization course at TU Berlin}}</ref>

<ref name=defy>{{cite magazine|url=https://www.quantamagazine.org/mobius-strips-defy-a-link-with-infinity-20190220/|title=Möbius strips defy a link with infinity|first=Evelyn|last=Lamb|magazine=[[Quanta Magazine]]|date=February 20, 2019}}</ref>

<ref name=dna>{{cite web|url=https://arstechnica.com/science/2010/10/chemical-origami-used-to-create-a-dna-mobius-strip/|title=Chemical origami used to create a DNA Möbius strip|work=[[Ars Technica]]|first=Diana|last=Gitig|date=October 18, 2010|access-date=2022-03-28}}</ref>

<ref name=dooner-seirig>{{cite book
 | last1 = Dooner | first1 = David B.
 | last2 = Seireg | first2 = Ali
 | contribution = 3.4.2 The cylindroid
 | contribution-url = https://books.google.com/books?id=xcjM2xmxvBAC&pg=PA135
 | isbn = 9780471045977
 | pages = 135–137
 | publisher = John Wiley & Sons
 | series = Wiley Series in Design Engineering
 | title = The Kinematic Geometry of Gearing: A Concurrent Engineering Approach
 | volume = 3
 | year = 1995}}</ref>

<ref name=dundas>{{cite book
 | last = Dundas | first = Bjørn Ian
 | contribution = Example 5.1.3: The unbounded Möbius band
 | doi = 10.1017/9781108349130
 | isbn = 978-1-108-42579-7
 | mr = 3793640
 | page = https://books.google.com/books?id=7a1eDwAAQBAJ&pg=PA101
 | publisher = Cambridge University Press, Cambridge
 | series = Cambridge Mathematical Textbooks
 | title = A Short Course in Differential Topology
 | year = 2018| s2cid = 125997451
 }}</ref>

<ref name=emmer>{{cite journal
 | last = Emmer | first = Michele
 | date = Spring 1980
 | issue = 2
 | journal = [[Leonardo (journal)|Leonardo]]
 | pages = 108–111
 | title = Visual art and mathematics: the Moebius band
 | url = https://muse.jhu.edu/article/599337/summary
 | volume = 13| doi = 10.2307/1577979
 | jstor = 1577979
 | s2cid = 123908555
 }}</ref>

<ref name=escher1>{{cite book
 | last = Crato | first = Nuno
 | contribution = Escher and the Möbius strip
 | doi = 10.1007/978-3-642-04833-3_29
 | pages = 123–126
 | publisher = Springer
 | title = Figuring It Out: Entertaining Encounters with Everyday Math
 | year = 2010| isbn = 978-3-642-04832-6
 }}</ref>

<ref name=escher2>{{cite web|url=https://www.escherinhetpaleis.nl/escher-today/mobius-strip-i/?lang=en|title=Möbius Strip I|publisher=[[Escher in the Palace]]|first=Erik|last=Kersten|date=March 13, 2017|access-date=2022-04-17}}</ref>

<ref name=expo74>{{cite news |url=https://news.google.com/newspapers?id=v9kvAAAAIBAJ&pg=5133%2C4421507 |newspaper=[[The Spokesman-Review]] |title=Expo '74 symbol selected |date=March 12, 1972 |page=1}}</ref>

<ref name=flapan>{{cite book
 | last = Flapan | first = Erica | author-link = Erica Flapan
 | doi = 10.1090/mbk/096
 | isbn = 978-1-4704-2535-7
 | mr = 3443369
 | pages = 99–100
 | publisher = American Mathematical Society | location = Providence, Rhode Island
 | title = Knots, Molecules, and the Universe: An Introduction to Topology
 | url = https://books.google.com/books?id=q4RbCwAAQBAJ&pg=PA99
 | year = 2016}}</ref>

<ref name=fomenko-kunii>{{cite book
 | last1 = Fomenko | first1 = Anatolij T. | author1-link = Anatoly Fomenko
 | last2 = Kunii | first2 = Tosiyasu L.
 | isbn = 9784431669562
 | page = 269
 | publisher = Springer
 | title = Topological Modeling for Visualization
 | url = https://books.google.com/books?id=8bn0CAAAQBAJ&pg=PA269
 | year = 2013}}</ref>

<ref name=fuchs-tabachnikov>{{cite book
 | last1 = Fuchs | first1 = Dmitry | author1-link = Dmitry Fuchs
 | last2 = Tabachnikov | first2 = Serge | author2-link = Sergei Tabachnikov
 | contribution = Lecture 14: Paper Möbius band
 | doi = 10.1090/mbk/046
 | isbn = 978-0-8218-4316-1
 | mr = 2350979
 | pages = 199–206
 | publisher = American Mathematical Society | location = Providence, Rhode Island
 | title = Mathematical Omnibus: Thirty Lectures on Classic Mathematics
 | url = http://www.math.psu.edu/tabachni/Books/taba.pdf
 | archive-url = https://web.archive.org/web/20160424020238/http://www.math.psu.edu/tabachni/Books/taba.pdf
 | url-status = dead | archive-date = 2016-04-24
 | year = 2007}}</ref>

<ref name=francis>{{cite book
 | last = Francis | first = George K.
 | contribution = Plücker conoid
 | isbn = 0-387-96426-6
 | mr = 880519
 | pages = 81–83
 | publisher = Springer-Verlag, New York
 | title = A Topological Picturebook
 | year = 1987}}</ref>

<ref name=franzoni>{{cite journal
 | last = Franzoni | first = Gregorio
 | doi = 10.1090/noti880
 | issue = 8
 | journal = Notices of the American Mathematical Society
 | mr = 2985809
 | pages = 1076–1082
 | title = The Klein bottle: variations on a theme
 | volume = 59
 | year = 2012| doi-access = free
 | hdl = 11584/76593
 | hdl-access = free
 }}</ref>

<ref name=frolkina>{{cite journal
 | last = Frolkina | first = Olga D.
 | doi = 10.1142/S0218216518420051
 | issue = 9
 | journal = [[Journal of Knot Theory and Its Ramifications]]
 | mr = 3848635
 | pages = 1842005, 9
 | title = Pairwise disjoint Moebius bands in space
 | volume = 27
 | year = 2018| arxiv = 2212.02983
 | s2cid = 126421578
 }}</ref>

<ref name=gardner>{{cite book
 | last = Gardner | first = Martin | author-link = Martin Gardner
 | contribution = The Afghan Bands
 | contribution-url = https://books.google.com/books?id=WkS4BQAAQBAJ&pg=PA70
 | location = New York
 | pages = 70–73
 | publisher = Dover Books
 | title = Mathematics, Magic and Mystery
 | year = 1956}}</ref>

<ref name=gdrive>{{cite news|url=https://sg.news.yahoo.com/did-google-drive-copy-icon-091053776.html|title=Did Google Drive Copy its Icon From a Chinese App?|first=Steven|last=Millward|date=April 30, 2012|work=Tech in Asia|via=Yahoo! News|access-date=2022-03-27}}</ref>

<ref name=godinho-natario>{{cite book
 | last1 = Godinho | first1 = Leonor
 | last2 = Natário | first2 = José
 | doi = 10.1007/978-3-319-08666-8
 | isbn = 978-3-319-08665-1
 | mr = 3289090
 | pages = 152–153
 | publisher = Springer, Cham
 | series = Universitext
 | title = An Introduction to Riemannian Geometry: With Applications to Mechanics and Relativity
 | url = https://books.google.com/books?id=oV4qBAAAQBAJ&pg=PA152
 | year = 2014}}</ref>

<ref name=gor-oni-vin>{{cite book
 | last1 = Gorbatsevich | first1 = V. V.
 | last2 = Onishchik | first2 = A. L.
 | last3 = Vinberg | first3 = È. B.
 | doi = 10.1007/978-3-642-57999-8
 | isbn = 3-540-18697-2
 | mr = 1306737
 | pages = 164–166
 | publisher = Springer-Verlag, Berlin
 | series = Encyclopaedia of Mathematical Sciences
 | title = Lie groups and Lie algebras I: Foundations of Lie Theory; Lie Transformation Groups
 | url = https://books.google.com/books?id=9UCOjHdD_hgC&pg=PA164
 | volume = 20
 | year = 1993}}</ref>

<ref name=graphene>{{cite journal
 | last1 = Yamashiro | first1 = Atsushi
 | last2 = Shimoi | first2 = Yukihiro
 | last3 = Harigaya | first3 = Kikuo
 | last4 = Wakabayashi | first4 = Katsunori
 | arxiv = cond-mat/0309636
 | bibcode = 2004PhyE...22..688Y
 | doi = 10.1016/j.physe.2003.12.100
 | issue = 1–3
 | journal = Physica E
 | pages = 688–691
 | title = Novel electronic states in graphene ribbons: competing spin and charge orders
 | volume = 22
 | year = 2004| s2cid = 17102453
 }}</ref>

<ref name=halpern-weaver>{{cite journal
 | last1 = Halpern | first1 = B.
 | last2 = Weaver | first2 = C.
 | doi = 10.2307/1997711
 | journal = Transactions of the American Mathematical Society
 | mr = 474388
 | pages = 41–70
 | title = Inverting a cylinder through isometric immersions and isometric embeddings
 | volume = 230
 | year = 1977| jstor = 1997711
 | doi-access = free
 }}</ref>

<ref name=hilbert-cohn-vossen>{{cite book
 | last1 = Hilbert | first1 = David | author1-link = David Hilbert
 | last2 = Cohn-Vossen | first2 = Stephan
 | edition = 2nd
 | isbn = 978-0-8284-1087-8
 | pages = 315–316
 | publisher = Chelsea
 | title = Geometry and the Imagination
 | title-link = Geometry and the Imagination
 | year = 1990}}</ref>

<ref name=huggett-jordan>{{cite book
 | last1 = Huggett | first1 = Stephen
 | last2 = Jordan | first2 = David
 | edition = Revised
 | isbn = 978-1-84800-912-7
 | mr = 2483686
 | page = 57
 | publisher = Springer-Verlag
 | title = A Topological Aperitif
 | year = 2009}}</ref>

<ref name=impa>{{cite web|url=https://impa.br/en_US/noticias/para-quem-e-fa-do-impa-dez-curiosidades-sobre-o-instituto/|title=Símbolo do IMPA|work=Para quem é fã do IMPA, dez curiosidades sobre o instituto|publisher=IMPA|access-date=2022-03-27|date=May 7, 2020}}</ref>

<ref name=isham>{{cite book
 | last = Isham | first = Chris J.
 | edition = 2nd
 | isbn = 981-02-3555-0
 | mr = 1698234
 | page = 269
 | publisher = World Scientific
 | series = World Scientific lecture notes in physics
 | title = Modern Differential Geometry for Physicists
 | url = https://books.google.com/books?id=8bn0CAAAQBAJ&pg=PA269
 | volume = 61
 | year = 1999}}</ref>

<ref name=jab-rad-saz>{{cite journal
 | last1 = Jablan | first1 = Slavik
 | last2 = Radović | first2 = Ljiljana
 | last3 = Sazdanović | first3 = Radmila
 | doi = 10.1007/s10910-011-9884-6
 | issue = 10
 | journal = Journal of Mathematical Chemistry
 | mr = 2846715
 | pages = 2250–2267
 | title = Nonplanar graphs derived from Gauss codes of virtual knots and links
 | volume = 49
 | year = 2011| s2cid = 121332704
 }}</ref>

<ref name=kazakh>{{cite news
 | last = Wainwright | first = Oliver
 | date = October 17, 2017
 | newspaper = [[The Guardian]]
 | title = 'Norman said the president wants a pyramid': how starchitects built Astana
 | url = https://www.theguardian.com/cities/2017/oct/17/norman-foster-president-pyramid-architects-built-astana}}</ref>

<ref name=kuiper>{{cite journal
 | last = Kuiper | first = Nicolaas H. | author-link = Nicolaas Kuiper
 | doi = 10.4310/jdg/1214430493
 | journal = [[Journal of Differential Geometry]]
 | mr = 314057
 | pages = 271–283
 | title = Tight topological embeddings of the Moebius band
 | volume = 6
 | year = 1972| issue = 3 | doi-access = free
 }}</ref>

<ref name=kyle>{{cite journal
 | last = Kyle | first = R. H.
 | journal = Proceedings of the Royal Irish Academy, Section A
 | jstor = 20488581
 | mr = 0091480
 | pages = 131–136
 | title = Embeddings of Möbius bands in 3-dimensional space
 | volume = 57
 | year = 1955}}</ref>

<ref name=larsen>{{cite conference
 | last = Larsen | first = Mogens Esrom
 | editor1-last = Guy | editor1-first = Richard K. | editor1-link = Richard K. Guy
 | editor2-last = Woodrow | editor2-first = Robert E.
 | contribution = Misunderstanding my mazy mazes may make me miserable
 | isbn = 0-88385-516-X
 | mr = 1303141
 | pages = 289–293
 | publisher = Mathematical Association of America | location = Washington, DC
 | series = MAA Spectrum
 | title = Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History held at the University of Calgary, Calgary, Alberta, August 1986
 | year = 1994}}. See [https://books.google.com/books?id=FsH2DwAAQBAJ&pg=PA292 Figure 7, p. 292].</ref>

<ref name=lawson>{{cite journal
 | last = Lawson | first = H. Blaine Jr. | author-link = H. Blaine Lawson
 | doi = 10.2307/1970625
 | journal = [[Annals of Mathematics]]
 | jstor = 1970625
 | mr = 270280
 | pages = 335–374
 | series = Second Series
 | title = Complete minimal surfaces in <math>S^3</math>
 | volume = 92
 | year = 1970| issue = 3 }} See Section 7, pp. 350–353, where the Klein bottle is denoted <math>\tau_{1,2}</math>.</ref>

<ref name=lopez-martin>{{cite journal
 | last1 = López | first1 = Francisco J.
 | last2 = Martín | first2 = Francisco
 | doi = 10.1353/ajm.1997.0004
 | issue = 1
 | journal = [[American Journal of Mathematics]]
 | mr = 1428058
 | pages = 55–81
 | title = Complete nonorientable minimal surfaces with the highest symmetry group
 | volume = 119
 | year = 1997| s2cid = 121366986
 }}</ref>

<ref name=magic>{{cite book
 | last = Prevos | first = Peter
 | location = Kangaroo Flat
 | publisher = Third Hemisphere
 | title = The Möbius Strip in Magic: A Treatise on the Afghan Bands
 | year = 2018
 | url = https://horizonofreason.com/pdf/afghan-bands-sample.pdf }}</ref>

<ref name=mangahas>{{cite book
 | last = Mangahas | first = Johanna
 | editor1-last = Clay | editor1-first = Matt
 | editor2-last = Margalit | editor2-first = Dan
 | contribution = Office Hour Five: The Ping-Pong Lemma
 | date = July 2017
 | doi = 10.1515/9781400885398
 | pages = 85–105
 | publisher = Princeton University Press
 | title = Office Hours with a Geometric Group Theorist| isbn = 9781400885398
 }} See in particular Project 7, pp. 104–105.</ref>

<ref name=maps1>{{cite journal
 | last = Tobler | first = Waldo R. | author-link = Waldo R. Tobler
 | journal = Surveying & Mapping
 | page = 486
 | title = A world map on a Möbius strip
 | url = https://books.google.com/books?id=2j44AAAAIAAJ&pg=PA486
 | volume = 21
 | year = 1961}}</ref>

<ref name=maps2>{{cite journal
 | last1 = Kumler | first1 = Mark P. | author1-link = Waldo R. Tobler
 | last2 = Tobler | first2 = Waldo R.
 | date = January 1991
 | doi = 10.1559/152304091783786781
 | issue = 4
 | journal = Cartography and Geographic Information Systems
 | pages = 275–276
 | title = Three world maps on a Moebius strip
 | volume = 18| bibcode = 1991CGISy..18..275K }}</ref>

<ref name=maschke>{{cite journal
 | last = Maschke | first = Heinrich | author-link = Heinrich Maschke
 | doi = 10.2307/1986401
 | issue = 1
 | journal = [[Transactions of the American Mathematical Society]]
 | mr = 1500522
 | page = 39
 | title = Note on the unilateral surface of Moebius
 | volume = 1
 | year = 1900| jstor = 1986401 | doi-access = free
 }}</ref>

<ref name=massey>{{cite book
 | last = Massey | first = William S.
 | isbn = 0-387-97430-X
 | mr = 1095046
 | page = 49
 | publisher = Springer-Verlag | location = New York
 | series = Graduate Texts in Mathematics
 | title = A Basic Course in Algebraic Topology
 | url = https://books.google.com/books?id=laSfDwAAQBAJ&pg=PA49
 | volume = 127
 | year = 1991}}</ref>

<ref name=meeks>{{cite journal
 | last = Meeks | first = William H. III | author-link = William Hamilton Meeks, III
 | doi = 10.1215/S0012-7094-81-04829-8
 | issue = 3
 | journal = [[Duke Mathematical Journal]]
 | mr = 630583
 | pages = 523–535
 | title = The classification of complete minimal surfaces in <math>\mathbb{R}^3</math> with total curvature greater than <math>-8\pi</math>
 | volume = 48
 | year = 1981}}</ref>

<ref name=melikhov>{{cite journal
 | last = Melikhov | first = Sergey A.
 | doi = 10.1142/s0218216519710019
 | issue = 7
 | journal = [[Journal of Knot Theory and Its Ramifications]]
 | mr = 3975576
 | pages = 1971001, 3
 | title = A note on O. Frolkina's paper "Pairwise disjoint Moebius bands in space"
 | volume = 28
 | year = 2019| arxiv = 1810.04089
 | s2cid = 119179202
 }}</ref>

<ref name=music>{{cite web
 | last = Moskowitz | first = Clara | author-link = Clara Moskowitz
 | date = May 6, 2008 | access-date = 2022-03-21
 | title = Music reduced to beautiful math
 | url = https://www.livescience.com/strangenews/080507-math-music.html
 | website = [[Live Science]]}}</ref>

<ref name=nak-tsu>{{cite journal
 | last1 = Nakamoto | first1 = Atsuhiro
 | last2 = Tsuchiya | first2 = Shoichi
 | doi = 10.1016/j.disc.2011.06.007
 | issue = 14
 | journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]
 | mr = 2921579
 | pages = 2135–2139
 | title = On geometrically realizable Möbius triangulations
 | volume = 312
 | year = 2012| doi-access = free
 }}</ref>

<ref name=nascar>{{cite magazine
 | last = Muret | first = Don
 | date = May 17, 2010
 | magazine = Sports Business Journal
 | title = NASCAR Hall of Fame 'looks fast sitting still'
 | url = https://www.sportsbusinessjournal.com/Journal/Issues/2010/05/17/This-Weeks-News/NASCAR-Hall-Of-Fame-Looks-Fast-Sitting-Still.aspx}}</ref>

<ref name=olson>{{cite book
 | last = Byers | first = Mark
 | isbn = 9780198813255
 | pages = 77–78
 | publisher = Oxford University Press
 | title = Charles Olson and American Modernism: The Practice of the Self
 | url = https://books.google.com/books?id=U1ZYDwAAQBAJ&pg=PA77
 | year = 2018}}</ref>

<ref name=optimal>{{cite arXiv|first=Richard|last=Schwartz|author-link=Richard Schwartz (mathematician)|title=The optimal paper Moebius band|date=2023 |eprint=2308.12641|class=math.MG}}</ref>

<ref name=paradromic>{{cite journal
 | last = Bennett | first = G. T. | author-link = Geoffrey Thomas Bennett
 | date = June 1923
 | doi = 10.1038/111882b0
 | issue = 2800
 | journal = [[Nature (journal)|Nature]]
 | page = 882
 | title = Paradromic rings
 | volume = 111| bibcode = 1923Natur.111R.882B | s2cid = 4099647 | doi-access = free
 }}</ref>

<ref name=parameterization>{{cite book
 | last = Junghenn | first = Hugo D.
 | isbn = 978-1-4822-1927-2
 | mr = 3309241
 | page = 430
 | publisher = CRC Press | location = Boca Raton, Florida
 | title = A Course in Real Analysis
 | url = https://books.google.com/books?id=nE63BgAAQBAJ&pg=PA430
 | year = 2015}}</ref>
 
<ref name=parker>{{cite book
 | last = Parker
 | first = Phillip E.
 | editor-last = Del Riego
 | editor-first = L.
 | contribution = Spaces of geodesics
 | contribution-url = http://www.math.wichita.edu/~pparker/research/sog/sog1.zip
 | mr = 1304924
 | pages = 67–79
 | publisher = Soc. Mat. Mexicana, México
 | series = Aportaciones Mat. Notas Investigación
 | title = Differential Geometry Workshop on Spaces of Geometry (Guanajuato, 1992)
 | volume = 8
 | year = 1993
 | access-date = 2022-03-21 
 | archive-date = 2016-03-13 
 | archive-url = https://web.archive.org/web/20160313115359/http://www.math.wichita.edu/~pparker/research/sog/sog1.zip
 | url-status = bot: unknown
 }}</ref>

<ref name=pasta>{{cite news|newspaper=[[The New York Times]]|title=Pasta Graduates From Alphabet Soup to Advanced Geometry|url=https://www.nytimes.com/2012/01/10/science/pasta-inspires-scientists-to-use-their-noodle.html|first=Kenneth|last=Chang|date=January 9, 2012}}</ref>

<ref name=peterson>{{cite book
 | last = Peterson | first = Ivars | author-link = Ivars Peterson
 | contribution = Recycling topology
 | contribution-url = https://books.google.com/books?id=RMH2DwAAQBAJ&pg=PA31
 | isbn = 0-88385-537-2
 | mr = 1874198
 | pages = 31–35
 | publisher = Mathematical Association of America, Washington, DC
 | series = MAA Spectrum
 | title = Mathematical Treks: From Surreal Numbers to Magic Circles
 | year = 2002}}</ref>

<ref name=phillips>{{cite magazine
 | last = Phillips | first = Tony
 | date = November 25, 2016
 | magazine = [[Plus Magazine]]
 | title = Bach and the musical Möbius strip
 | url = https://plus.maths.org/content/topology-music-m-bius-strip}} Reprinted from an American Mathematical Society Feature Column.</ref>

<ref name=pickover>{{cite book
 | last = Pickover | first = Clifford A. | author-link = Clifford A. Pickover
 | isbn = 978-1-56025-826-1
 | pages = 28–29
 | publisher = Thunder's Mouth Press
 | title = The Möbius Strip: Dr. August Möbius's Marvelous Band in Mathematics, Games, Literature, Art, Technology, and Cosmology
 | url = https://archive.org/details/mbiusstripdrau00pick
 | year = 2005}}</ref>

<ref name=polarization>{{cite journal
 | last1 = Bauer | first1 = Thomas
 | last2 = Banzer | first2 = Peter
 | last3 = Karimi | first3 = Ebrahim
 | last4 = Orlov | first4 = Sergej
 | last5 = Rubano | first5 = Andrea
 | last6 = Marrucci | first6 = Lorenzo
 | last7 = Santamato | first7 = Enrico
 | last8 = Boyd | first8 = Robert W.
 | last9 = Leuchs | first9 = Gerd
 | date = February 2015
 | doi = 10.1126/science.1260635
 | issue = 6225
 | journal = [[Science (journal)|Science]]
 | pages = 964–966
 | title = Observation of optical polarization Möbius strips
 | volume = 347| pmid = 25636796
 | bibcode = 2015Sci...347..964B
 | s2cid = 206562350
 }}</ref>

<ref name=pook>{{cite book
 | last = Pook | first = Les
 | contribution = 4.2: The trihexaflexagon revisited
 | contribution-url = https://books.google.com/books?id=MlKL1aG781QC&pg=PA33
 | doi = 10.1017/CBO9780511543302
 | isbn = 0-521-81970-9
 | mr = 2008500
 | pages = 33–36
 | publisher = Cambridge University Press | location = Cambridge, UK
 | title = Flexagons Inside Out
 | year = 2003}}</ref>

<ref name=ramirez-seade>{{cite book
 | last1 = Ramírez Galarza | first1 = Ana Irene
 | last2 = Seade | first2 = José
 | isbn = 978-3-7643-7517-1
 | location = Basel
 | mr = 2305055
 | pages = 83–88, 157–163
 | publisher = Birkhäuser Verlag
 | title = Introduction to Classical Geometries
 | year = 2007}}</ref>

<ref name=resistor>{{cite magazine
 | date = September 25, 1964
 | issue = 13
 | magazine = [[Time (magazine)|Time]]
 | title = Making resistors with math
 | url = https://content.time.com/time/subscriber/article/0,33009,876181,00.html
 | volume = 84}}</ref>

<ref name=resonator>{{cite journal
 | last = Pond | first = J. M.
 | bibcode = 2000ITMTT..48.2465P
 | doi = 10.1109/22.898999
 | issue = 12
 | journal = IEEE Transactions on Microwave Theory and Techniques
 | pages = 2465–2471
 | title = Mobius dual-mode resonators and bandpass filters
 | volume = 48
 | year = 2000}}</ref>

<ref name=resonator2>{{cite journal
 | last1 = Rohde | first1 = Ulrich L.
 | last2 = Poddar | first2 = Ajay
 | last3 = Sundararajan | first3 = D.
 | date = November 2013
 | issue = 11
 | journal = Microwave Journal
 | title = Printed resonators: Möbius strip theory and applications
 | url = https://synergymwave.com/articles/2013/11/article.pdf
 | volume = 56}}</ref>

<ref name=reyes>{{cite web|url=https://news.artnet.com/art-world/pedro-reyes-makes-an-infinite-love-seat-133150|work=Artnet News|title=Pedro Reyes Makes an Infinite Love Seat|first=Blake|last=Gopnik|date=October 17, 2014}}</ref>

<ref name=richeson>{{cite book
 | last = Richeson | first = David S. | author-link = David Richeson
 | isbn = 978-0-691-12677-7
 | mr = 2440945
 | page = [https://books.google.com/books?id=BFeXDwAAQBAJ&pg=PA171 171]
 | publisher = Princeton University Press | location = Princeton, New Jersey
 | title = Euler's Gem: The Polyhedron Formula and the Birth of Topology
 | title-link = Euler's Gem
 | year = 2008}}</ref>

<ref name=ringel-youngs>{{cite journal
 | last1 = Ringel | first1 = G. | author1-link = Gerhard Ringel | last2 = Youngs | first2 = J. W. T.
 | bibcode = 1968PNAS...60..438R
 | doi = 10.1073/pnas.60.2.438 | doi-access = free
 | issue = 2
 | journal = [[Proceedings of the National Academy of Sciences of the United States of America]]
 | volume = 60
 | pages = 438–445
 | mr = 0228378
 | pmc = 225066
 | pmid = 16591648
 | title = Solution of the Heawood map-coloring problem
 | year = 1968}}</ref>

<ref name=ringvan>{{cite magazine|url=https://www.pressreader.com/uk/prog/20210209/281535113669133|title=Ring Van Möbius|magazine=[[Prog (magazine)|Prog]]|date=February 9, 2021|first=Dom|last=Lawson}}</ref>

<ref name=roman>{{cite journal
 | last = Larison | first = Lorraine L.
 | bibcode=1973AmSci..61..544L
 | issue = 5
 | journal = [[American Scientist]]
 | jstor = 27843983
 | pages = 544–547
 | title = The Möbius band in Roman mosaics
 | volume = 61
 | year = 1973}}</ref>

<ref name=rouseball>{{cite book|title=Mathematical Recreations and Problems of Past and Present Times|edition=2nd|publisher=Macmillan and co.|location=London & New York|year=1892|last=Rouse Ball|first=W. W.|author-link=W. W. Rouse Ball|contribution=Paradromic rings|pages=53–54|contribution-url=https://books.google.com/books?id=IwDvAAAAMAAJ&pg=PA53}}</ref>

<ref name=sadowsky-translation>{{cite journal
 | last1 = Hinz | first1 = Denis F.
 | last2 = Fried | first2 = Eliot
 | doi = 10.1007/s10659-014-9490-5
 | issue = 1–2
 | journal = Journal of Elasticity
 | mr = 3326180
 | pages = 3–6
 | title = Translation of Michael Sadowsky's paper "An elementary proof for the existence of a developable Möbius band and the attribution of the geometric problem to a variational problem"
 | volume = 119
 | year = 2015| arxiv = 1408.3034
 | s2cid = 119733903
 }} Reprinted in {{cite book
 | last1 = Fosdick | first1 = Roger
 | last2 = Fried | first2 = Eliot
 | doi = 10.1007/978-94-017-7300-3
 | isbn = 978-94-017-7299-0
 | mr = 3381564
 | pages = 3–6
 | publisher = Springer, Dordrecht
 | title = The Mechanics of Ribbons and Möbius Bands
 | year = 2016| url = https://openresearch.lsbu.ac.uk/download/7bba3852893aa5916361a1c98259f2aee64381672fa60a5dcfb7e926228a4197/6644159/Starostin-Heijden2015_Article_EquilibriumShapesWithStressLoc.pdf
 }}</ref>

<ref name=schleimer-segerman>{{cite conference
 | last1 = Schleimer | first1 = Saul
 | last2 = Segerman | first2 = Henry
 | editor1-last = Bosch | editor1-first = Robert
 | editor2-last = McKenna | editor2-first = Douglas
 | editor3-last = Sarhangi | editor3-first = Reza
 | arxiv = 1204.4952
 | contribution = Sculptures in {{math|''S''<sup>3</sup>}}<!-- html math so formula is properly bluelinked -->
 | contribution-url = https://archive.bridgesmathart.org/2012/bridges2012-103.html
 | isbn = 978-1-938664-00-7
 | location = Phoenix, Arizona
 | pages = 103–110
 | publisher = Tessellations Publishing
 | title = Proceedings of Bridges 2012: Mathematics, Music, Art, Architecture, Culture
 | year = 2012}}</ref>

<ref name=schwartz>{{cite journal
 | last = Schwartz | first = Richard Evan | author-link = Richard Schwartz (mathematician)
 | arxiv = 2008.11610
 | doi = 10.1007/s10711-021-00648-5
 | journal = [[Geometriae Dedicata]]
 | mr = 4330341
 | pages = 255–267
 | title = An improved bound on the optimal paper Moebius band
 | volume = 215
 | year = 2021| s2cid = 220279013 }}</ref>

<ref name=schwarz>{{cite journal
 | last = Schwarz | first = Gideon
 | issue = 1
 | journal = [[Pacific Journal of Mathematics]]
 | mr = 1047406
 | pages = 195–200
 | title = A pretender to the title 'canonical Moebius strip'
 | url = https://projecteuclid.org/euclid.pjm/1102646207
 | volume = 143
 | year = 1990| doi = 10.2140/pjm.1990.143.195
 | doi-access = free
 }}</ref>

<ref name=seifert-threlfall>{{cite book
 | last1 = Seifert | first1 = Herbert | author1-link = Herbert Seifert
 | last2 = Threlfall | first2 = William | author2-link = William Threlfall
 | translator-last = Goldman | translator-first = Michael A.
 | isbn = 0-12-634850-2
 | location = New York & London
 | mr = 575168
 | page = 12
 | publisher = Academic Press
 | series = Pure and Applied Mathematics
 | title = A Textbook of Topology
 | url = https://books.google.com/books?id=rsb8zjP0XHoC&pg=PA12
 | volume = 89
 | year = 1980}}</ref>

<ref name=soap>{{cite journal
 | last1 = Goldstein | first1 = Raymond E. | author1-link = Raymond E. Goldstein
 | last2 = Moffatt | first2 = H. Keith | author2-link = Keith Moffatt
 | last3 = Pesci | first3 = Adriana I. | author3-link = Adriana Pesci
 | last4 = Ricca | first4 = Renzo L. | author4-link = Renzo L. Ricca
 | date = December 2010
 | doi = 10.1073/pnas.1015997107
 | issue = 51
 | journal = [[Proceedings of the National Academy of Sciences]]
 | pages = 21979–21984
 | title = Soap-film Möbius strip changes topology with a twist singularity
 | volume = 107| bibcode = 2010PNAS..10721979G | doi-access = free | pmc = 3009808}}</ref>

<ref name=spivak>{{cite book
 | last = Spivak | first = Michael | author-link = Michael Spivak
 | edition = 2nd
 | location = Wilmington, Delaware
 | page = 591
 | publisher = Publish or Perish
 | title = A Comprehensive Introduction to Differential Geometry, Volume I
 | year = 1979}}</ref>
 
<ref name=split-tori>{{cite conference
 | last = Séquin | first = Carlo H. | author-link = Carlo H. Séquin
 | editor1-last = Sarhangi | editor1-first = Reza
 | editor2-last = Moody | editor2-first = Robert V.
 | contribution = Splitting tori, knots, and Moebius bands
 | contribution-url = https://archive.bridgesmathart.org/2005/bridges2005-211.html
 | isbn = 0-9665201-6-5
 | location = Southwestern College, Winfield, Kansas
 | pages = 211–218
 | publisher = Bridges Conference
 | title = Renaissance Banff: Mathematics, Music, Art, Culture
 | year = 2005}}</ref>

<ref name=starostin-vdh>{{cite journal
 | last1 = Starostin | first1 = E. L.
 | last2 = van der Heijden | first2 = G. H. M.
 | doi = 10.1007/s10659-014-9495-0
 | issue = 1–2
 | journal = Journal of Elasticity
 | mr = 3326186
 | pages = 67–112
 | title = Equilibrium shapes with stress localisation for inextensible elastic Möbius and other strips
 | volume = 119
 | year = 2015| s2cid = 53462568
 | doi-access = free
 }} Reprinted in {{harvtxt|Fosdick|Fried|2016}}, pp. 67–112.</ref>

<ref name=stillwell>{{cite book
 | last = Stillwell | first = John | author-link = John Stillwell
 | contribution = 4.6 Classification of isometries
 | doi = 10.1007/978-1-4612-0929-4
 | isbn = 0-387-97743-0
 | mr = 1171453
 | pages = 96–98
 | publisher = Springer | location = Cham
 | series = Universitext
 | title = Geometry of Surfaces
 | year = 1992}}</ref>

<ref name=sudanese>{{cite web
 | last = Gunn | first = Charles
 | access-date=2022-03-17
 | title = Sudanese Möbius Band
 | url = https://vimeo.com/286360639
 | website = Vimeo| date = August 23, 2018
 }}</ref>

<ref name=synthesis>{{cite journal
 | last1 = Walba | first1 = David M.
 | last2 = Richards | first2 = Rodney M.
 | last3 = Haltiwanger | first3 = R. Curtis
 | date = June 1982
 | doi = 10.1021/ja00375a051
 | issue = 11
 | journal = [[Journal of the American Chemical Society]]
 | pages = 3219–3221
 | title = Total synthesis of the first molecular Moebius strip
 | volume = 104| bibcode = 1982JAChS.104.3219W
 }}</ref>

<ref name=systolic>{{cite journal
 | last1 = Pesci | first1 = Adriana I. | author1-link = Adriana Pesci
 | last2 = Goldstein | first2 = Raymond E. | author2-link = Raymond E. Goldstein
 | last3 = Alexander | first3 = Gareth P.
 | last4 = Moffatt | first4 = H. Keith | author4-link = Keith Moffatt
 | doi = 10.1103/PhysRevLett.114.127801
 | issue = 12
 | journal = [[Physical Review Letters]]
 | mr = 3447638
 | page = 127801
 | title = Instability of a Möbius strip minimal surface and a link with systolic geometry
 | volume = 114
 | year = 2015| pmid = 25860771 | bibcode = 2015PhRvL.114l7801P | url = http://wrap.warwick.ac.uk/74286/7/WRAP_PRL114_127801_2015_alexander.pdf }}</ref>

<ref name=szilassi>{{cite journal
 | last = Szilassi | first = Lajos | author-link = Lajos Szilassi
 | doi = 10.1007/s00454-007-9033-y
 | issue = 3
 | journal = [[Discrete & Computational Geometry]]
 | mr = 2443291
 | pages = 395–400
 | title = A polyhedral model in Euclidean 3-space of the six-pentagon map of the projective plane
 | volume = 40
 | year = 2008| s2cid = 38606607 | doi-access = free
 }}</ref>

<ref name=tietze>{{cite journal
 | last = Tietze | first = Heinrich | author-link = Heinrich Tietze
 | journal = Jahresbericht der Deutschen Mathematiker-Vereinigung
 | pages = 155–159
 | title = Einige Bemerkungen zum Problem des Kartenfärbens auf einseitigen Flächen
 | url = https://sites.math.washington.edu/~mathcircle/circle/2014-15/advanced/mc-14a-w8.pdf
 | volume = 19
 | year = 1910}}</ref>

<ref name=tuckerman>{{cite journal
 | last = Tuckerman | first = Bryant | author-link = Bryant Tuckerman
 | doi = 10.2307/2305482
 | journal = American Mathematical Monthly
 | jstor = 2305482
 | mr = 24138
 | pages = 309–311
 | title = A non-singular polyhedral Möbius band whose boundary is a triangle
 | volume = 55
 | year = 1948| issue = 5 }}</ref>

<ref name=woll>{{cite journal
 | last = Woll | first = John W. Jr.
 | date = Spring 1971
 | doi = 10.2307/3026946
 | issue = 1
 | journal = [[The Two-Year College Mathematics Journal]]
 | jstor = 3026946
 | pages = 5–18
 | title = One-sided surfaces and orientability
 | volume = 2}}</ref>

<ref name=wunderlich>{{cite journal
 | last = Wunderlich | first = W.
 | doi = 10.1007/BF01299052
 | journal = [[Monatshefte für Mathematik]]
 | mr = 143115
 | pages = 276–289
 | title = Über ein abwickelbares Möbiusband
 | volume = 66
 | year = 1962| issue = 3
 | s2cid = 122215321
 }}</ref>

<ref name=zimmermann>{{cite news|first=Nancy J. |last=Thomas |title=Making a Mobius a matter of mathematics|newspaper=[[The Times (Trenton)]]|date=October 4, 1998|page=aa3 |url= https://infoweb.newsbank.com/apps/news/openurl?ctx_ver=z39.88-2004&rft_id=info%3Asid/infoweb.newsbank.com&svc_dat=AWNB&req_dat=0F1B56E1B179D300&rft_val_format=info%3Aofi/fmt%3Akev%3Amtx%3Actx&rft_dat=document_id%3Anews%252F11B7A8B696793450 |via=[[NewsBank]]}}</ref>

}}

==External links==
{{Wiktionary|Möbius strip}}
*{{Commonsinline|Moebius Strip}}
*{{MathWorld|urlname=MoebiusStrip|title=Möbius Strip}}

{{Compact topological surfaces}}
{{Mathematics of paper folding}}
{{Authority control}}

{{DEFAULTSORT:Mobius Strip}}
[[Category:Topology]]
[[Category:Recreational mathematics]]
[[Category:Surfaces]]
[[Category:Eponyms in geometry]]