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        riddles >> general problem-solving / chatting / whatever >> Mobius strip
        
(Message started by: BenVitale on Jul 20^th, 2010, 10:41am)

Title: Mobius strip
Post by BenVitale on Jul 20^th, 2010, 10:41am

what is the smallest strip of paper that can be twisted into a Mobius strip?

Title: Re: Mobius strip
Post by towr on Jul 20^th, 2010, 11:55am

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.

Title: Re: Mobius strip
Post by BenVitale on Jul 20^th, 2010, 12:26pm

I'm reading this article: Twisted Thinking (http://www.exo.net/~pauld/activities/mobius/MobiusArticle.html)

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.

Title: Re: Mobius strip
Post by towr on Jul 20^th, 2010, 12:59pm

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.

Title: Re: Mobius strip
Post by Grimbal on Jul 20^th, 2010, 1:30pm

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.

Title: Re: Mobius strip
Post by BenVitale on Jul 20^th, 2010, 7:34pm

Thanks Towr and Grimbal for your insights.

Here's a fun puzzle with the Mobius strip: Mobius chess

http://blog.makezine.com/mobius_chess.jpg

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): [hide]http://blog.makezine.com/archive/2007/04/mobius_chess.html [/hide]

Title: Re: Mobius strip
Post by BenVitale on Jul 25^th, 2010, 12:31pm

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 ?

Title: Re: Mobius strip
Post by towr on Jul 25^th, 2010, 12:39pm

What length would that be? The minimum length for a moebius strip? Grimbal already explained there isn't one.

Title: Re: Mobius strip
Post by BenVitale on Jul 25^th, 2010, 12:47pm

on 07/25/10 at 12:39:38, towr wrote:

The minimum length for a moebius strip?

Yes.

Quote:

Grimbal already explained there isn't one.

Okay. I guess I got confused by the paper in the PDF file

Title: Re: Mobius strip
Post by towr on Jul 25^th, 2010, 1:12pm

Read the paragraph on page 5 under the figure there. They give the same solution Grimbal gave, attributed to Martin Gardner.

Title: Re: Mobius strip
Post by BenVitale on Jul 25^th, 2010, 3:42pm

Thanks, towr. I understand now.