One of the first pages at this site dealt with gluing and cutting a single strip of paper. Recently Alex Grasser sent me five graphic files with a short note, "I have made the effort of drawing the Moebius strips in 3D Studio. This will illustrate my point better." I liked the files and the activity they demonstrate. Two Moebius strips are involved. But these must have different orientations much like two trefoil knots.

When you glue two ends of a strip together, one is rotated 180^o. To get two different orientations for the two strips, rotate the end of one strip in the direction opposite to that in which you rotated the and of the other. Now glue the two strips together and, then, cut each along the middle line:

Straightening the strips will result in a couple of linked shapes that, from a certain angle, will appear as a couple of hearts.

I could not match Alex's pictures, but felt that the page would be utterly incomplete unless I mention at least one other activity with two paper bands. For there is a less exciting but nonetheless surprising result of cutting two cylinders glued together. In the manner of the diagrams used with a single strip, six steps apply to two cylinders glued together:

1. Cut one cylinder along its middle line.
2. Separate the shape along the cut.
3. Straighten the shape. You get a piece of paper with two rings at each end.
4. Cut the connecting strip along its middle line.
5. Move pieces away from each other.
6. Straighten the shape.

The result is a plain square.

A more dramatic effect can be achieved by preparing long strips of paper in advance. The trick is quite suitable for public demonstration. Unless you explain what you are doing, it would be very hard to notice if you rotate the ends a half revolution. Tell somebody from the public to perform the job. The fellow will get a square. Do this yourself, and the result will be a couple of hearts. Explain the reason afterwards.

Copyright © 1996-2008 Alexander Bogomolny