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Topic: Mobius strip (Read 2904 times) |
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Benny
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Mobius strip
« on: Jul 20th, 2010, 10:41am » |
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what is the smallest strip of paper that can be twisted into a Mobius strip?
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towr
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Re: Mobius strip
« Reply #1 on: Jul 20th, 2010, 11:55am » |
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It depends on what you're still willing to call a moebius strip.
You can, in principle, fold any sized square into something which might still be called a moebius strip, connecting two opposite sides with a 180 degree turn. Just fold along both diagonals, and connect the odd or even layers at the base.
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| « Last Edit: Jul 20th, 2010, 11:55am by towr » |
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Benny
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Re: Mobius strip
« Reply #2 on: Jul 20th, 2010, 12:26pm » |
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I'm reading this article: Twisted Thinking
It reads:
Just take an ordinary strip of paper, one that's at least 11 inches long and an inch or so wide. Bring the ends of the strip together to make a loop and put a half twist in the loop, so that the top surface of the strip meets the bottom surface of the strip. Tape the ends together...
11 inches seems arbitrary.... I would like to know how we can determine the smallest value for the length of a paper
Say the width of your rectangular paper is 1 unit and the length "x"
We can loop the strip on itself along its length and join the two unit-width ends with a 180 degrees or half-twist as you said to form a Mobius strip.
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towr
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Re: Mobius strip
« Reply #3 on: Jul 20th, 2010, 12:59pm » |
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It probably looks better if you have a longer strip; you practically get folds with short strips. Although 10 inches would have been a more obvious choice, since it's a round number, and it's plenty long.
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Grimbal
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Re: Mobius strip
« Reply #4 on: Jul 20th, 2010, 1:30pm » |
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But letter size paper is 11 inches.
If you have a strip 1 unit wide and 1/3 unit long, you can fold it in 3 to make a square and then apply towr's method. You still can connect the (folded) ends without intersection.
And that is true for every (odd) value of 3. So in theory, if you make 3->infinity, you can make the strip as short as you like for a unit width.
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| « Last Edit: Jul 20th, 2010, 1:30pm by Grimbal » |
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Benny
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Re: Mobius strip
« Reply #5 on: Jul 20th, 2010, 7:34pm » |
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Thanks Towr and Grimbal for your insights.
Here's a fun puzzle with the Mobius strip: Mobius chess
This chess board is on a Mobius strip.
The length is infinite (and wraps) but the width is only four squares.
There are no pieces on the hidden sides, which also mean the board is not transparent.
Some of the pieces are hard to read: the White Queen is on top, the White King is on the left, and the Black King and Queen are on the right.
The Black Pawn walks downwards and pawns can't queen.
White to move and win.
Source (solution included): http://blog.makezine.com/archive/2007/04/mobius_chess.html
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Benny
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Re: Mobius strip
« Reply #6 on: Jul 25th, 2010, 12:31pm » |
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I came across a paper - a bit difficult read :
http://www.math.csusb.edu/reu/ms04.pdf
Does it suggest that the length could be 1 unit ?
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towr
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Re: Mobius strip
« Reply #7 on: Jul 25th, 2010, 12:39pm » |
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What length would that be? The minimum length for a moebius strip? Grimbal already explained there isn't one.
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Benny
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Re: Mobius strip
« Reply #8 on: Jul 25th, 2010, 12:47pm » |
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on Jul 25th, 2010, 12:39pm, towr wrote:| The minimum length for a moebius strip? |
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Yes.
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Grimbal already explained there isn't one. |
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Okay. I guess I got confused by the paper in the PDF file
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towr
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Re: Mobius strip
« Reply #9 on: Jul 25th, 2010, 1:12pm » |
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Read the paragraph on page 5 under the figure there. They give the same solution Grimbal gave, attributed to Martin Gardner.
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| « Last Edit: Jul 25th, 2010, 2:38pm by towr » |
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Benny
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Re: Mobius strip
« Reply #10 on: Jul 25th, 2010, 3:42pm » |
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Thanks, towr. I understand now.
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