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Does the concept of a Möbius torus make sense: taking a cylinder (instead of a rectangle as in the case of the Möbius strip) and twisting it before joining its ends? Or will the resulting twisted torus be indistinguishable from the normal torus in any relevant respect?

[This equivalent to the well-known Möbius strip should be called Möbius cylinder but it would have so much in common with a torus that I preferred to call it a Möbius torus.]

Embedded in Euclidean space the twisted and untwisted torus "look" the same - opposed to Möbius strip and cylinder -, the difference would be only in their intrinsic properties. But can there be such differences? And how do I specify them?

PS: I posted a follow-up question here.

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    @Christia$n$ Blatter: Ca$n$ you please help $m$e to understand better: the simply twisted torus (which I tried to ask for) has a locally euclidean metric (just like the Klein bottle), is homeomorphic to the regular torus, but has another conformal type. What exactly is the conformal type? Does it make sense to think of (closed) geodesics in this context?2012-10-23

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As remarked above: What you get is the Klein bottle. Put differently: The result is what you get when you take two Möbius strips (which both have one boundary) and glue both boundaries together (which does not work when embedded in 3d space but works in theory). See this image from http://im-possible.info/english/articles/klein-bottle/: enter image description here

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    They might not meet all the technical details of the definition; but as long as you're willing to overlook the defect where it punches through its side, you can buy glass klein bottles as conversation pieces even if you're trapped in 3-space.2012-10-20
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In the course of experimenting with 3-D models, I developed a method for twisting a toroid of arbitrary cross section. I called these forms Möbioids. Of course, if the cross-section is circular the result will be indistinguishable. What I did was to consider cylinders of non-circular cross-section. However, then the twist angles are quantized. Depending on the particular cross-section and number of twists, you can get forms with one or more surfaces. The figures below show an astroid cross-section with one twist of $\pi/4$ (left) and a pentacuspid cross-section with six twists of $2\pi/5$ (right). Each has a single surface. Again, not all twists lead to a single surface. You can find more images and some animations at A New Twist on Möbius1.

Some rendered images of Möbioids as taken from the link in the text: http://old.nationalcurvebank.org////moebius2/moebius2.htm

1 Archived version in case the link above ever dies

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The question is one of boundary between two and three dimensional space. For example, bisect cut a bicycle tube which lays flat, draw a green line down one side of the tube, draw a red line down the other side of the tube. Then rotate one end 180-degrees and cement it to the other end. The green line meets the red line on both sides of the tube. It looks nothing like a Klein bottle. If you now inflate the tube from a two dimensional flat plane to a three dimensional torus, in effect you are creating an infinite loop where say, an ant travelling in one direction along the inside of the tube has twice the distance to walk to return to his original starting point. This topography is not found in nature, at least not on the surface of a sphere. If the tube is blown up enough, say to the size of a universe, you have finite space WITHIN the tube, although it is quite possible to have infinite space on the outside of the tube, surrounding it. We simply don't know at this point since all we can comprehend is our interior space. All that we do know is that we exist in a two-dimensional closed (finite) space within a three-dimensional open (infinite) space. To say we live in a three dimensional world (height, length, depth) is naive at best.