A Möbius strip ( Möbius band ) or Möbius loop is a surface that is formed by connecting the ends of a strip of paper together with a half-twist. This shape was discovered by August Ferdinand Möbiusin and Johann BenedictListing the year 1858.

The discovery of the Möbius strip leads to formation of the mathematical topology which is the study of geometric properties that remain unchanged when an object is stretched or deformed ( shape changes not properties ).

Mathematician and artist Henry Segerman show how a coffee mug is deformed into a symmetrical doughnut.

A Möbius strip cannot be distinguished as clockwise or anticlockwise in turning .So this is a non-orientable surface.

A Möbius strip can be placed in 3-dimensional space in different ways. Like a clockwise half twist or counterclockwise half twist.Or this can be represented as odd numbers of twists which is greater than one.Or like a knotted strip.

The first way to create a Möbius strip in three-dimensional Euclidean space is to sweep a line segment rotating in a plane, which in turn rotates around one of its lines.

Another way ,a strip of paper can form a flattened Möbius strip in the plane by folding it at 60 degree angles so that its center line lies along an equilateral triangle, and attaching the ends.

Interestingly we can create a klein – bottle by gluing two Möbius strips

Applications

Where can you see the Möbius structure??

The applications of Möbius strips lie in several areas. One is mechanical belts that wear evenly on both sides on a rotating shaft .Graphene ribbons twisted to form Möbius strips with new electronic characteristics including helical magnetism.

Dual-track roller coasters in an amusement park ,whose carriages alternate between the two tracks also use this structure to balance. In our world map ,this strip is used in which the antipodes appear opposite each other. It appear even in molecules and devices with novel electrical and electromechanical properties, this has been used to prove impossibility results in social choice theory. The Möbius strip also may be seen in architecture, for example, the Wuchazi Bridge in China.

Properties of Möbius strip

The first one obviously is non- orientable surface. if an two-dimensional object which is asymmetric, slides one time around the strip, it returns to its mirror image of starting position.

Möbius strip when embedded in Euclidean space, it has only one side. A 3- dimensional object reaches at the same position of starting point when it takes a round on the strip.This shows the strip has only one side.

The edge, or boundary, of a Möbius strip is topologically equivalent to a circle.

One is mechanical belts that wear evenly on both sides on a rotating shaft .Graphene ribbons twisted to form Möbius strips with new electronic characteristics including helical magnetism.

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