Date: Tue, 21 Oct 1997 17:00:34 +0200
From: Alexander Grässer
To: Alexander Bogomolny

Hello, Alexander,

Here is an interesting thing to do with 2 Möbius strips. Firstly, some background:

Years ago I saw a problem posed by Martin Gardener. It went something like this: Take two strips of paper, glue them together without twisting them so that you have two cylinder like structures. Now glue the outsides together at a 90 degree angle. Now you cut the strips down the middle. The resulting shape will be a square. (That is if your strips are of equal length.) Now do the same again, but this time twist one strip. In other words one strip becomes a Möbius of "order" 1. What do you get if you cut the strips down the middle? The answer of course is another square.

Now to the interesting part:

Do the same as above but this time twist both strips. What do you get if you cut the strips down the middle? The answer is not obvious, it depends on the direction of the turns.

  1. if both strips are turned in the same direction then you will get a two separate pieces, one of which is completely twisted and the other is a "biangle", if I may call it that!

  2. If the two strips are turned in different directions , e.g.,. one right and one left then something truly amazing happens. The strips do not separate, the stay linked and when straightened out the appear heart shaped.

    This is a nice gimmick for Valentines day. Just provide two strips of paper and a couple of instructions and voilà an instant love letter!! Maybe this "discovery" have some implications in topology as well.

Thank you for providing a brilliant web page. I truly enjoy all the various puzzles.

Thank you again

Alexander Grässer

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