Mobius and Klein Bottle

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Pigeon
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Mobius and Klein Bottle

Post by Pigeon » Sun Oct 14, 2012 4:20 am



The Möbius strip or Möbius band, also Mobius or Moebius, is a surface with only one side and only one boundary component. The Möbius strip has the mathematical property of being non-orientable. It can be realized as a ruled surface. It was discovered independently by the German mathematicians August Ferdinand Möbius and Johann Benedict Listing in 1858.

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 loop. In Euclidean space there are two types of Möbius strips depending on the direction of the half-twist: clockwise and counterclockwise. That is to say, it is a chiral object with "handedness" (right-handed or left-handed).

The Möbius strip has several curious properties. 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. This single continuous curve demonstrates that the Möbius strip has only one boundary.

Cutting a Möbius strip along the center line with a pair of scissors yields one long strip with two full twists in it, rather than two separate strips; the result is not a Möbius strip. This happens because the original strip only has one edge that is twice as long as the original strip. Cutting creates a second independent edge, half of which was on each side of the scissors. Cutting this new, longer, strip down the middle creates two strips wound around each other, each with two full twists.



In mathematics, the Klein bottle is a non-orientable surface, informally, a surface (a two-dimensional manifold) in which notions of left and right cannot be consistently defined.

Other related non-orientable objects include the Möbius strip and the real projective plane. Whereas a Möbius strip is a surface with boundary, a Klein bottle has no boundary. (For comparison, a sphere is an orientable surface with no boundary.)

Klein Bottle


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Pigeon
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Re: Mobius and Klein Bottle

Post by Pigeon » Sun Oct 14, 2012 2:41 pm

In mathematics, a cross-cap is a two-dimensional surface that is a model of a Möbius strip with a single self intersection. This self intersection precludes the cross-cap from being topologically equivalent (i.e., homeomorphic) to a Möbius strip. The term ‘cross-cap’, however, often implies that the surface has been deformed so that its boundary is an ordinary circle.

A cross-cap that has been closed up by gluing a disc to its boundary is an immersion of the real projective plane. Two cross-caps glued together at their boundaries form a Klein bottle. An important theorem of topology, the classification theorem for surfaces, states that all two-dimensional compact manifolds without boundary are homeomorphic to spheres with some number of ‘handles’ and at most two cross-caps.

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Remember the idea of a finite but unbounded universe.

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Royal
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Re: Mobius and Klein Bottle

Post by Royal » Tue Oct 16, 2012 1:14 pm

Image

Is this the mathematical concept on how north and south pole forces may converge to balance a craft?

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Pigeon
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Re: Mobius and Klein Bottle

Post by Pigeon » Tue Oct 16, 2012 3:16 pm

Nope.

Could be a Mobius-Klein Trap used to capture a starship and keep it trapped in an area of space forever.

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