:Möbius strip

{{multiple image|total_width=480

|image1=Aion mosaic Glyptothek Munich W504.jpg|caption1=Mosaic from ancient Sentinum depicting 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 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, 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 {{nowrap|unclear.{{r|ancient}}}}

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.{{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}}}}

File:Fiddler crab mobius strip.gif

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 {{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 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.{{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 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 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.{{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 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 ladders, 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 $V$, $E$, 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; {{nowrap|$12-18+6=0$.{{r|tietze}}}}

{{-}}

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

{{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,

\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}

for and {{nowrap|$-1\; \backslash le\; v\backslash le\; 1$,}}

where one parameter $u$ describes the rotation angle of the plane around its central axis and the other parameter {{nowrap|$v$}} 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 {{nowrap|$(0,\; 0,\; 0)$.{{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|gears.{{r|dooner-seirig}}}}

File:Flexagon.gif

A strip of paper can form a flattened Möbius strip in the plane by folding it at $60^\backslash circ$ 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|$\backslash sqrt\; 3\backslash approx\; 1.73$,}} 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 $1\backslash times\; 1$ strip would become a $1\backslash times\; \backslash tfrac\{1\}\{3\}$ 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 {{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 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 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 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 hyperplanes 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 quadrilaterals 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}}}}

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|$\backslash sqrt\; 3\backslash approx\; 1.73$,}} 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 $C^2$, and can be approximated arbitrarily accurately by infinitely differentiable {{nowrap|(class $C^\backslash infty$)}} 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 $\backslash sqrt\; 3$, 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|$\backslash pi/2\backslash approx\; 1.57$,}} 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}}}}

$$\backslash frac\{2\}\{3\}\backslash sqrt\{3+2\backslash sqrt3\}\backslash approx\; 1.695.$$

{{unsolved|mathematics|Can a $12\backslash times\; 7$ paper rectangle be glued end-to-end to form a smooth Möbius strip embedded in space? {{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 $\backslash sqrt\; 3$,}} it has been an open problem whether smooth embeddings, without self-intersection, {{nowrap|exist.{{r|fuchs-tabachnikov|halpern-weaver|schwartz}}}} In 2023, 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 $C^1$. 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 surfaces that contain rectangular developable Möbius {{nowrap|strips.{{r|wunderlich|schwarz}}}}

{{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, of a Möbius strip is topologically equivalent to a circle. In common forms of the Möbius strip, it has a different shape from a circle, but it is unknotted, 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 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 $$(\backslash cos\backslash theta\backslash cos\backslash phi,\backslash sin\backslash theta\backslash cos\backslash phi,\backslash cos2\backslash theta\backslash sin\backslash phi,\backslash sin2\backslash theta\backslash sin\; \backslash phi)$$ for {{nowrap|.{{r|lawson}}}} Half of this Klein bottle, the subset with , 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|$\backslash mathrm\{O\}(2)$,}} the group of symmetries of a {{nowrap|circle.{{r|lawson}}}}

File:Cross-cap level sets.svgs. 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 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}}}}

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

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 {{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 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 {{nowrap|strip.{{r|mangahas}}}}

These spaces of lines are highly symmetric. The symmetries of Euclidean lines include the affine transformations, and the symmetries of hyperbolic lines include the {{nowrap|Möbius transformations.{{r|ramirez-seade}}}} The affine transformations and Möbius transformations both form {{nowrap|6-dimensional}} Lie groups, topological spaces having a compatible 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 solvmanifolds, 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 a compact solvmanifold {{nowrap|with $\backslash mathbb\{R\}^n$.}} 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 cosets 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|$x$-axis}} consists of all symmetries that take the axis to itself. Each line $\backslash ell$ corresponds to a coset, the set of symmetries that map $\backslash ell$ to the {{nowrap|$x$-axis.}} 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 {{nowrap|strip.{{r|gor-oni-vin}}}}

File:Möbius resistor.svg]]

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 chemicals whose molecular structure forms a cycle, with molecular orbitals 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}}
• Resonators with a compact design and a resonant frequency that is half that of identically constructed linear coils{{r|resonator|resonator2}}
• Polarization patterns in light emerging from a 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 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{{r|coaster1|coaster2}}
• World maps 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{{r|maps1|maps2}}

Scientists have also studied the energetics of soap films shaped as Möbius strips,{{r|courant|soap}} the chemical synthesis of molecules 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}}

File:Middelheim Max Bill Eindeloze kronkel 1956 03 Cropped.jpg, 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, 1967), and Sebastián.{{r|emmer}} A trefoil-knotted Möbius strip was used in John Robinson{{'}}s Immortality (1982).{{sfnp|Pickover|2005|p=13}} Charles O. Perry's 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=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 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 stamps 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 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 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, scarves have been 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 bagels,{{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) 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 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.{{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 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.{{r|music|chords}} Modern musical groups taking their name from the Möbius strip include American electronic rock trio 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 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}}

• 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

{{Notelist}}

{{reflist|refs=

{{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

}}

{{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

}}

{{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}}

{{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

}}

{{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

}}

{{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}}

{{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}}

{{cite magazine|url=https://magnetmagazine.com/2007/08/30/mobius-band-friendly-fire/|title=Mobius Band: Friendly Fire|magazine=Magnet|date=August 30, 2007|first=Andrew|last=Parks}}

{{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))

}}

{{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.

{{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

}}

{{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 $\backslash mathbb\{R\}^n$ and the Björling problem

| volume = 56

| year = 2006| bibcode = 2006JGP....56.1506M

}}

{{cite book

| last = Blackett | first = Donald W.

| isbn = 9781483262536

| page = 195

| publisher = Academic Press

| title = Elementary Topology: A Combinatorial and Algebraic Approach

| year = 1982}}

{{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}}

{{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

}}

{{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}}

{{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

}}

{{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 }}

{{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}}

{{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}}

{{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

}}

{{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}}

{{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

| 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 }}

{{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}}

{{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}}

{{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 }}

{{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/}}

{{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}}

{{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}}

{{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}}

{{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}}

{{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}}

{{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

}}

{{cite journal

| last = Emmer | first = Michele

| date = Spring 1980

| issue = 2

| 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

}}

{{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

}}

{{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}}

{{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}}

{{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}}

{{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}}

{{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}}

{{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}}

{{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

}}

{{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

}}

{{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}}

{{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}}

{{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}}

{{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}}

{{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

}}

{{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

}}

{{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}}

{{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}}

{{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}}

{{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}}

{{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

}}

{{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}}

{{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

}}

{{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}}

{{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].

{{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 $S^3$

| volume = 92

| year = 1970| issue = 3 }} See Section 7, pp. 350–353, where the Klein bottle is denoted $\backslash tau\_\{1,2\}$.

{{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

}}

{{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 }}

{{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.

{{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}}

{{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 }}

{{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

}}

{{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}}

{{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 $\backslash mathbb\{R\}^3$ with total curvature greater than $-8\backslash pi$

| volume = 48

| year = 1981}}

{{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

}}

{{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}}

{{cite journal

| last1 = Nakamoto | first1 = Atsuhiro

| last2 = Tsuchiya | first2 = Shoichi

| doi = 10.1016/j.disc.2011.06.007

| issue = 14

| journal = Discrete Mathematics

| mr = 2921579

| pages = 2135–2139

| title = On geometrically realizable Möbius triangulations

| volume = 312

| year = 2012| doi-access = free

}}

{{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}}

{{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}}

{{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}}

{{cite journal

| last = Bennett | first = G. T. | author-link = Geoffrey Thomas Bennett

| date = June 1923

| doi = 10.1038/111882b0

| issue = 2800

| journal = Nature

| page = 882

| title = Paradromic rings

| volume = 111| bibcode = 1923Natur.111R.882B | s2cid = 4099647 | doi-access = free

}}

{{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}}

{{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

}}

{{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}}

{{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}}

{{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.

{{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}}

{{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

| pages = 964–966

| title = Observation of optical polarization Möbius strips

| volume = 347| pmid = 25636796

| bibcode = 2015Sci...347..964B

| s2cid = 206562350

}}

{{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}}

{{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}}

{{cite magazine

| date = September 25, 1964

| issue = 13

| magazine = Time

| title = Making resistors with math

| url = https://content.time.com/time/subscriber/article/0,33009,876181,00.html

| volume = 84}}

{{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}}

{{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}}

{{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}}

{{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}}

{{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}}

{{cite magazine|url=https://www.pressreader.com/uk/prog/20210209/281535113669133|title=Ring Van Möbius|magazine=Prog|date=February 9, 2021|first=Dom|last=Lawson}}

{{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}}

{{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}}

{{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

}}

{{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|S³}}

| 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}}

{{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 }}

{{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

}}

{{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}}

{{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}}

{{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}}

{{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}}

{{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.

{{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}}

{{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

}}

{{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

}}

{{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 }}

{{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

}}

{{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}}

{{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 }}

{{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}}

{{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

}}

{{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}}

}}

{{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