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 A Möbius strip in Euclidean space cannot be moved or stretched into its mirror image; it is a chiral object with right- or 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 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 With an even number of twists, however, one obtains a different topological surface, called the 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. 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 These interlinked shapes, formed