Editing Möbius strip (section)

==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===
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|Description=A Möbius strip swept out by a rotating line segment in a rotating plane}}
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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}}}}