Just to have fun with orientation, try making a Moebius strip. If you have a normal rectangular strip of paper, you can join one short end to the other to make a simple closed strip. IF you twist one end (by 180 degrees) first before joining with the other end, you have a

Moebius Strip. The ants in the picture are actually on the same side of the strip, even though they appear to be on two sides. An ordinary strip has two sides (an inside and an outside), but a Moebius strip has only one side!

Think about that! :)

Take the normal rectangular strip and join the long ends to make a long cylinder. Just as in the case of the rectangular strip above, close the circles in the two ends of the cylinder in a simple way to make a hollow doughnut (a

torus).

*What is the analogue of the Moebius Strip for the long cylinder?* It is a Klein Bottle! A hollow doughnut has an inside and an outside, but a Klein Bottle does not. So it cannot hold any water as inside is also outside :P

Caveat: Klein Bottle does not physically exist in 3-dimensions (but in 4 dimensions!), so the picture is an illusion. I don't want you to get all cross-eyed looking at it. :P

If you draw upward arrows on both the short ends of the rectangular strip, then in the simple closed strip, both arrows meet pointing upwards. In the Moebius strip the ends meet with one up and the other down.

For the long cylinder case, if you mark both short ends of the rectangular strip from which its made in the upward direction. Then both circular ends of the cyclinder are oriented in the same way too. If you now join the circles to form a torus, the two circles must now join matching the orientation arrow marks. Can you imagine a way to make them meet in opposite orientation? (one circle one-way and the other circle another way) You can convince yourself that this is not possible inside 3 dimensions.

## 2 comments:

Very interesting. Klein Bottle huh!! Knew about the moebius strip but not the klein bottle. Still thinking about your teaser!

as I say in the teaser you need the fourth dimension. If you see why? then you get the rough construction or the analogy. :)

yep, there is a lot of stuff like.

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