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The Möbius strip has several curious properties. If you cut down the middle of the strip, instead of getting two separate strips, it becomes one long strip with two half-twists in it. If you cut this one down the middle, you get two strips wound around each other. Alternatively, if you cut along a Möbius strip, about a third of the way in from the edge, you will get two strips; one is a thinner Möbius strip, the other is a long strip with two half-twists in it. Other interesting combinations of strips can be obtained by making Möbius strips with two or more flips in them instead of one. For example, a strip with three half-twists, when divided lengthwise, becomes a strip tied in a trefoil knot. Cutting a Möbius strip, giving it extra twists, and reconnecting the ends produces unexpected figures called paradromic ring s.
All physical Möbius strips have two surfaces and two edges, because they all have thickness, even the strip of paper. If the thickness surface is rounded it ceases to be a separate surface, becoming part of the main surface, and eliminating the edges. However this may make it no longer a mobius band.
One way to represent the Möbius strip as a subset of R3 is using the parametrization:
where and . This creates a Möbius strip of width 1 whose center circle has radius 1, lies in the x-y plane and is centered at (0,0,0). The parameter u runs around the strip while v moves from one edge to the other.
In cylindrical polar coordinates (r,θ, z), an unbounded version of the Möbius strip can be represented by the equation:
The Möbius strip is a two-dimensional smooth manifold (a surface) which is not orientable. The Möbius strip is a standard example used to illustrate the mathematical concept of a fiber bundle. Specifically, it is a nontrivial bundle over the circle S1 with a fiber the unit interval, I = [0,1]. Looking only at the edge of the Möbius strip gives a nontrivial two point (or Z2) bundle over S1.
A closely related "strange" geometrical object is the Klein bottle. A Klein bottle can be produced by gluing two Möbius strips together along their edges; this cannot be done in ordinary three-dimensional Euclidean spaceEuclidean space is the usual n dimensional mathematical space, a generalization of the 2- and 3-dimensional spaces studied by Euclid. Formally, for any non-negative integer n n dimensional Euclidean space is the set R n (where R is the set of real numbers without creating self-intersections.
Another closely related manifold is the real projective planeSurfaces Geometric topology In mathematics, the real projective plane is a two-dimensional manifold, that is, a surface, that has basic applications to geometry, but which cannot be embedded in our usual three-dimensional space. It is often described intu. If a single hole is punctured in the real projective plane, what is left is a Möbius strip. Going in the other direction, if one glues a disk to a Möbius strip by identifying their boundaries, the result is the projective plane. In order to visulazing this it is helpful to deform the Möbius strip so that its boundary is an ordinary circle. Such a figure is called a cross-capSurfaces In mathematics, a cross-cap is a two-dimensional surface that is topologically equivalent to a Mobius strip. The term 'cross-cap', however, often implies that the surface has been deformed so that its boundary is an ordinary circle. This cannot b (a cross-cap can also mean this figure with the disk glued in, i.e. an immersion of the projective plane in R3).
It is a common misconception that a cross-cap cannot be formed in three dimensions without the surface intersecting itself. In fact it is possible to embed a Möbius strip in R3 with boundary a perfect circle. Here is the idea: let C be the unit circleIllustration of a unit circle. t is an angle measure. In mathematics, a unit circle is a circle with unit radius, i. a circle whose radius is 1. Frequently, especially in trigonometry, "the" unit circle is the circle of radius 1 centered at the origin (0, in the xy plane in R3. Now connect antipodal pointsAntipodal points on the surface of a sphere are diametrically opposite; on the other side of a globe. For example, "Norway and New Zealand lie in antipodal regions. More generally, in mathematics, antipodal points on a sphere of any dimension are those op on C, i.e., points at angles and , by an arc of a circle. For between and make the arc lie above the xy plane, and for other the arc below (with two places where the arc lies in the xy plane).
In terms of identifications of the sides of a square, as given above: the real projective planeSurfaces Geometric topology In mathematics, the real projective plane is a two-dimensional manifold, that is, a surface, that has basic applications to geometry, but which cannot be embedded in our usual three-dimensional space. It is often described intu is made by gluing the remaining two sides with 'consistent' orientation (arrows making an anti-clockwise loop); and the Klein bottle is made the other way.
