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http://www.tenthdimension.com/flash2.php
Click "Imagining the 10th Dimension".
*Always thought that a flatlander on a Mobius strip would become a mirror image of herself since she has essentially been lifted through the 3rd dimension and flipped over.
Click "Imagining the 10th Dimension".
*Always thought that a flatlander on a Mobius strip would become a mirror image of herself since she has essentially been lifted through the 3rd dimension and flipped over.
If a Mobius strip were a physical object made of say, paper, you could grasp the object between your index finger and your thumb. Let us call the point that your index finger touches A and the point that your thumb touches A'.
If a physical object was crawling around on the Mobius strip- let's say a black and white Othello chip with its left side notched- then as it went around the strip its black face would alway be touching the strip. It starts at A, with its notch pointing to the left. When it gets to A', its notch is pointing to the right. When it completes the circuit and gets back to A, its notch is pointing the left again- its black face never left the strip.
I am treating the Mobius strip as a physical object, just like the flash animation did. However, I have no idea whether the mathematical abstraction behaves differently (e.g. in the mathematical description, are A and A' distinguishable? The answer changes what we would mean by the word 'circuit', because A and A' distinguishable means that a circuit goes from A to A' and A' to A, whereas only the first such segment would be necessary to complete a circuit if A and A' are not distinguishable). I've shown that with the physical interpretation, with A and A' distinguishable, the flatlander will not flip. But if they are not distinguishable, then the flatlander will flip.
Does that make sense?
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This is true, but it is not the terminal points of the trip that we're interested in. A lot of people will say of the Mobius strip that it has but "one side". This is what I'm focusing on. The journey is a peculiar one. We will not expect that the black face never left the strip, since that would involve the flatlander leaving Flatland. What I'm suggesting is that Flatland itself is doing the twisting. The flatlander is not the one doing weird stuff - it's what the Flatlander considers "space itself" that's becoming inverted.
Take a normal strip of paper and label one terminal end A and the other A', then flip the paper over and label the terminal ends on the opposite side with B and B' such that B is directly "beneath" A and B' is directly beneath A'.
Let's say that a flatlander would exist on this plane on one side (the A side) and a trip across Flatland would mean a trek from A to A'. Since Flatland itself has no third dimension, no flatlander would ever see the B side - it is only a "side" to us 3-dimensioners.
Twist the paper as you normally would to make a Mobius strip, bringing the ends together such that A meets B' and A' meets B. Now, with Flatland twisted through 3-space, a trek across Flatland has essentially doubled in distance. To get from one end of the trek to the other requires travelling first from AB' to A'B, then from A'B to AB' again. A flatlander standing still in a spot somewhere between A and A' would observe our flatspace traveller walk from A to A' and back again while making this trip, only on the way back from A', the flatspace traveller, since they would essentially be travelling on the B side at that point, would appear as a mirror image of itself to the flatlander that stood in place the entire time.
The flatlander traveller's surface never has to leave the two dimensional surface of flatland. Flatland is the thing which is twisting in 3-space. The flatlander is unaware that anything has occurred. That's what I was trying to say, though I'm not sure I explained it very well.
*On a related note, I remember reading something about Riemmann saying that if our world were like flatland, say as on a piece of paper, and you crumpled that piece of paper up making creases and bends in it, then as we trekked along our 2 dimensional world and encountered these creases, we would experience an inexplicable force as we passed through them. Our experience of flatland itself would not change save for these forces.
I think he tried to explain gravity and magnetism, etc., as "wrinkles" in a higher dimension that way. I like that.
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