Hurkyl said:
That's because you're visualizing it wrong.
A plane doesn't have a front and a back -- it is just a plane.
A sheet of paper isn't a plane -- it is a (really thin) solid rectangular prism. We can draw a point on one side, and draw a point at "the same place" on the other side, and see them as different points.
But if we are using the sheet of paper to help us visualize a plane, we are supposed to think of those two points as really being the same.
The same is true of a Möbius strip. If we cut a strip of paper to make one, we don't really have a Möbius strip -- instead, we have a (rather squished) solid torus, the same sort of shape as a donut. But if we want to use this strip of paper to help us visualize a Möbius strip, we have to think of the opposite sides as being the same -- so once around takes you back where you started, but everything that stayed put is now reflected in orientation as compared to you.
Yes , I can see that a sheet of paper is not a perfect analogy and the 2 points on 2 different sides are distinct if we look at them as being part of a paper .
I also agree with you that if the strip of paper were of 0 thickness and it was transparent also let's say , the 2 points on opposite sides would exactly coincide. But alas , this is true only as long as we view it from a 3 dimensional space. When I move to the 2 dimensional space these points are distinct.
What I am claiming is that 2 points that coincide in 3 dimensions , need not coincide in 2 dimensions.
Lets get back to my sphere example where i squished the sphere such that 2 of its opposite points now coincide ( when viewed from 3 dimensions ) . Let say you have a point Q , 1 cm away from A on same side as A ( left side let's say) Also there is a point R , 1cm away from B on same side as B ( right side ) . Let us say the circumference of the sphere is very large compared to these distances.
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Now if I move into 2 dimensions , you are saying that the distance between point Q and point R is only 1+1 = 2cm ?
But the problem is that there is no way I can go from Q to R covering only 2 cm . The path from Q to A , then A to B ( 0 distance ) then B to R is invalid. There is no way an object of finite size could pass from A to B . There is not even a way of a point object of 0 size could go from A to B in 0 distance . In 2 dimensions points A and B are completely distinct .
Let me conduct an experiment in this 2d world and try to measure the distance between A and B . I would be using light rays to measure distance . Now light emitted from A would travel in a straight line around the circumference of the sphere to reach B ( this is a straight line in my 2d world although it may not look straight from 3dimensions ) . There is no way light travels 0 distance from A to B . Hence any distance measurement I do based on this light would lead me to conclude that A and B are very far apart.
As you can see , it is impossible to prove from within 2 dimensions that A and B are the same point . Hence A and B are distinct in 2 dimensions.
On a similar line of reasoning as above , I also have lots of doubt regarding the curvature of space time and expansion of space time.
For example we say space is expanding because we see galaxies moving apart .I Just don't get this idea. If space were expanding , there is no way we would know for we too are a part of this space.
As regards to bending of light rays due to curvature of space. - If light rays bend due to curved space , we cannot see the light to be bending because we too are a part of this curved space , for us the light rays always moves in a straight line.
What I want to say here is that from a particular N dimensional space / geometry, if it is impossible to prove certain facts regarding the nature of this space , from within this space ; and you have to resort to looking at this space from higher dimensions to prove things about it , then what you may be seeing ( proving ) in higher dimensions are not necessarilly true in the lower N dimensional space from which you perceive things, because their truth is immaterial in our N dimensional space.
So though you think that the 2 sides of mobius strip coincide from a 3 dimensional space , this coincidence has no observable counterpart from within the 2 dimensional realm of the mobius strip.
There are of course things that you could prove from within this mobius strip / my sphere example that also are readily seen from higher dimensions . For example the fact that the geometry is a closed one is evident because I can reach the same point from where i start by moving in a straight line.