Editing Möbius strip (section)

==Properties==
[[File:Fiddler crab mobius strip.gif|thumb|upright|left|A 2D object traversing once around the Möbius strip returns in mirrored form]]
The Möbius strip has several curious properties. It is a [[orientability|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 surface]]s 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 sets|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 [[chirality (mathematics)|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 (mathematics)|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 [[planar graph|drawn without crossings on a plane]]. Another family of graphs that can be [[graph embedding|embedded]] on the Möbius strip, but not on the plane, are the [[Möbius ladder]]s, 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 <math>V</math>, <math>E</math>, and <math>F</math> of vertices, edges, and regions satisfy <math>V-E+F=0</math>. For instance, Tietze's graph has <math>12</math> vertices, <math>18</math> edges, and <math>6</math> regions; {{nowrap|<math>12-18+6=0</math>.{{r|tietze}}}}
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