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Möbius strip a real Mathmatical Paradox
The Möbius strip or Möbius band (pronounced /ˈmøbiʊs/) is a surface with only one side and only one boundary component. It has the mathematical property of being non-orientable. It is also a ruled surface. It was discovered independently by the German mathematicians August Ferdinand Möbius and Johann Benedict Listing in 1858.
To make it simple ,it is three dimensional figure,but actually with only one surface.So I would call it a Mathmatical Paradox. A model can easily be created by taking a paper strip and giving it a half-twist, and then joining the ends of the strip together to form a single strip. In Euclidean space there are in fact two types of Möbius strips depending on the direction of the half-twist: clockwise and counterclockwise. The Möbius strip is therefore chiral, which is to say that it is “handed”.To further simplify,and make it more comprehensible,Möbius strip is a one-sided surface i.e a line can be drawn along the strip of paper that will cover both sides, returning to the starting-point to meet itself,despite of being three dimensional figure.

It is of great interest in "topology" generally known as 'Rubber Geometry.'
Möbius  strip is used as universal recycling symbol and is internationally recognized symbol used to designate recyclable materials. It is composed of three chasing arrows that form a Möbius strip or unending loop.The mobius strip is even the signature architectural feature of the NASCAR Hall of Fame presently under construction in Charlotte, NC.

wears Mobius earring
Courtesy of Sinisa Rancov
The Möbius strip has several curious properties.A model of a Möbius strip can be constructed by joining the ends of a strip of paper with a single half-twist. A line drawn starting from the seam down the middle will meet back at the seam but at the “other side”. If continued the line will meet the starting point and will be double the length of the original strip of paper. This single continuous curve demonstrates that the Möbius strip has only one boundary.
If the strip is cut along about a third of the way in from the edge, it creates two strips: One is a thinner Möbius strip – it is the center third of the original strip. The other is a long strip with two full twists in it – this is a neighborhood of the edge of the original strip.
Alternatively, cutting a Möbius strip along the above line, instead of getting two separate strips, it becomes one long strip with two full twists in it, which is not a Möbius strip. This happens because the original strip only has one edge which is twice as long as the original strip of paper. Cutting creates a second independent edge, half of which was on each side of the knife or scissors. Cutting this new, longer, strip down the middle creates two strips wound around each other.
Other analogous strips can be obtained by similarly joining strips with two or more half-twists 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 rings(Rings produced by cutting a strip that has been given 'm' half twists and been re-attached into 'n' equal  strips ).

Technological Use
 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 allow the ribbon to be twice as wide as the print head whilst using both half-edges evenly.
A device called a Möbius resistor is an electronic circuit element which has the property of canceling its own inductive reactance. Nikola Tesla patented similar technology in the early 1900s: “Coil for Electro Magnets” was intended for use with his system of global transmission of electricity without wires.

This is also used in flim named "Infinity" and in many other teleseries and poems.