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One way to represent the Möbius strip embedded in three-dimensional Euclidean space is by the parametrization:.

For a smaller aspect ratio, it is not known whether a smooth embedding is possible. If the Möbius strip in three-space is only once continuously differentiable class C 1 , however, then the theorem of Nash-Kuiper shows that no lower bound exists.

A method of making a Möbius strip from a rectangular strip too wide to simply twist and join e.

This folded strip, three times as long as it is wide, would be long enough to then join at the ends. This method works in principle, but becomes impractical after sufficiently many folds, if paper is used.

Using normal paper, this construction can be folded flat , with all the layers of the paper in a single plane, but mathematically, whether this is possible without stretching the surface of the rectangle is not clear.

A less used presentation of the Möbius strip is as the topological quotient of a torus. The diagonal of the square the points x , x where both coordinates agree becomes the boundary of the Möbius strip, and carries an orbifold structure, which geometrically corresponds to "reflection" — geodesics straight lines in the Möbius strip reflect off the edge back into the strip.

The Möbius strip is a two-dimensional compact manifold i. It is a standard example of a surface that is not orientable.

In fact, the Möbius strip is the epitome of the topological phenomenon of nonorientability. This is because two-dimensional shapes surfaces are the lowest-dimensional shapes for which nonorientability is possible and the Möbius strip is the only surface that is topologically a subspace of every nonorientable surface.

As a result, any surface is nonorientable if and only if it contains a Möbius band as a subspace. The Möbius strip is also a standard example used to illustrate the mathematical concept of a fiber bundle.

Looking only at the edge of the Möbius strip gives a nontrivial two point or Z 2 bundle over S 1.

A simple construction of the Möbius strip that can be used to portray it in computer graphics or modeling packages is:. The open Möbius band is formed by deleting the boundary of the standard Möbius band.

It may be constructed as a surface of constant positive, negative, or zero Gaussian curvature. In the cases of negative and zero curvature, the Möbius band can be constructed as a geodesically complete surface, which means that all geodesics "straight lines" on the surface may be extended indefinitely in either direction.

The group of isometries of this Möbius band is 1-dimensional and is isomorphic to the special orthogonal group SO 2.

The resulting metric makes the open Möbius band into a geodesically complete flat surface i. This is the only metric on the Möbius band, up to uniform scaling, that is both flat and complete.

The group of isometries of this Möbius band is 1-dimensional and is isomorphic to the orthogonal group SO 2. Constant positive curvature: A Möbius band of constant positive curvature cannot be complete, since it is known that the only complete surfaces of constant positive curvature are the sphere and the projective plane.

The open Möbius band is homeomorphic to the once-punctured projective plane, that is, P 2 with any one point removed. This may be thought of as the closest that a Möbius band of constant positive curvature can get to being a complete surface: just one point away.

The group of isometries of this Möbius band is also 1-dimensional and isomorphic to the orthogonal group O 2.

The space of unoriented lines in the plane is diffeomorphic to the open Möbius band. Hence the same group forms a group of self-homeomorphisms of the Möbius band described in the previous paragraph.

But there is no metric on the space of lines in the plane that is invariant under the action of this group of homeomorphisms. In this sense, the space of lines in the plane has no natural metric on it.

This means that the Möbius band possesses a natural 4-dimensional Lie group of self-homeomorphisms, given by GL 2, R , but this high degree of symmetry cannot be exhibited as the group of isometries of any metric.

The edge, or boundary , of a Möbius strip is homeomorphic topologically equivalent to a circle. Under the usual embeddings of the strip in Euclidean space, as above, the boundary is not a true circle.

However, it is possible to embed a Möbius strip in three dimensions so that the boundary is a perfect circle lying in some plane.

For example, see Figures , , and of "Geometry and the imagination". A much more geometric embedding begins with a minimal Klein bottle immersed in the 3-sphere, as discovered by Blaine Lawson.

We then take half of this Klein bottle to get a Möbius band embedded in the 3-sphere the unit sphere in 4-space.

The result is sometimes called the "Sudanese Möbius Band", [14] where "sudanese" refers not to the country Sudan but to the names of two topologists, Sue Goodman and Daniel Asimov.

Applying stereographic projection to the Sudanese band places it in three-dimensional space, as can be seen below — a version due to George Francis can be found here.

From Lawson's minimal Klein bottle we derive an embedding of the band into the 3-sphere S 3 , regarded as a subset of C 2 , which is geometrically the same as R 4.

To obtain an embedding of the Möbius strip in R 3 one maps S 3 to R 3 via a stereographic projection.

The projection point can be any point on S 3 that does not lie on the embedded Möbius strip this rules out all the usual projection points.

Stereographic projections map circles to circles and preserves the circular boundary of the strip. The result is a smooth embedding of the Möbius strip into R 3 with a circular edge and no self-intersections.

The Sudanese Möbius band in the three-sphere S 3 is geometrically a fibre bundle over a great circle, whose fibres are great semicircles.

The most symmetrical image of a stereographic projection of this band into R 3 is obtained by using a projection point that lies on that great circle that runs through the midpoint of each of the semicircles.

Each choice of such a projection point results in an image that is congruent to any other. But because such a projection point lies on the Möbius band itself, two aspects of the image are significantly different from the case illustrated above where the point is not on the band: 1 the image in R 3 is not the full Möbius band, but rather the band with one point removed from its centerline ; and 2 the image is unbounded — and as it gets increasingly far from the origin of R 3 , it increasingly approximates a plane.

Yet this version of the stereographic image has a group of 4 symmetries in R 3 it is isomorphic to the Klein 4-group , as compared with the bounded version illustrated above having its group of symmetries the unique group of order 2.

If all symmetries and not just orientation-preserving isometries of R 3 are allowed, the numbers of symmetries in each case doubles.

But the most geometrically symmetrical version of all is the original Sudanese Möbius band in the three-sphere S 3 , where its full group of symmetries is isomorphic to the Lie group O 2.

Having an infinite cardinality that of the continuum , this is far larger than the symmetry group of any possible embedding of the Möbius band in R 3.

Using projective geometry , an open Möbius band can be described as the set of solutions to a polynomial equation.

Adding a polynomial inequality results in a closed Möbius band. These relate Möbius bands to the geometry of line bundles and the operation of blowing up in algebraic geometry.

This is the case for the Möbius band. Deleting this line gives the set. There is a realization of the closed Möbius band as a similar set, but with an additional inequality to create a boundary:.

This is the same as the union of the lines through the origin, except that it contains one copy of the origin for each line.

The lines themselves describe the ruling of the Möbius band. This change in sign is the algebraic manifestation of the half-twist.

A closely related 'strange' geometrical object is the Klein bottle. A Klein bottle could in theory be produced by gluing two Möbius strips together along their edges; however this cannot be done in ordinary three-dimensional Euclidean space without creating self-intersections.

Another closely related manifold is the real projective plane. If a circular disk is cut out of the real projective plane, what is left is a Möbius strip.

To visualize this, it is helpful to deform the Möbius strip so that its boundary is an ordinary circle see above.

The real projective plane, like the Klein bottle, cannot be embedded in three-dimensions without self-intersections.

In graph theory , the Möbius ladder is a cubic graph closely related to the Möbius strip. There have been several technical applications for the Möbius strip.

Giant Möbius strips have been used as conveyor belts that last longer because the entire surface area of the belt gets the same amount of wear, and as continuous-loop recording tapes to double the playing time.

Möbius strips are common in the manufacture of fabric computer printer and typewriter ribbons , as they let the ribbon be twice as wide as the print head while using both halves evenly.

A Möbius resistor is an electronic circuit element that cancels its own inductive reactance. Nikola Tesla patented similar technology in [20] "Coil for Electro Magnets" was intended for use with his system of global transmission of electricity without wires.

The Möbius strip is the configuration space of two unordered points on a circle. Consequently, in music theory , the space of all two-note chords, known as dyads , takes the shape of a Möbius strip; this and generalizations to more points is a significant application of orbifolds to music theory.

The Möbius strip principle has been used as a method of creating the illusion of magic. The trick, known as the Afghan bands, was very popular in the first half of the twentieth century.

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