A STEM Project: The Magical Mathematical World of the Möbius Strip and the Klein Bottle

The world of mathematics can be magical.  Magicians have performed magic tricks which use mathematical concepts such as the Möbius strip to astound audiences and stimulate the imagination.

Afghan Bands

In the 1800's, there was a magic trick called the "Afghan Bands".  A magician took a long strip of paper and glued the ends together to create a loop.  He repeated the procedure for two more strips of paper. He cut each of the three strips down the center of the strip and the results amazed audiences because they were all different.

What did the magician do to make each result different? The answer was, he put a twist in the paper.  The first strip had no twist and the result was two loops.  The second strip had a half twist and the result was a strip that was twice as long as the original loop.  The third strip had two half twists in the strip and the result was two loops that were intertwined.

To recreate this magic trick, cut six 2 inch wide x 11 inch long strips of paper.  Each Afghan Band needs two of these strips.

Afghan Band #1

Tape two strips of paper together and then tape the ends together to form a loop. Cut the loop lengthwise in half. 

 The result is two loops with an inside and an outside surface.

Afghan Band #2

Tape two strips of paper together and then put a half twist on one end of the paper.  Tape the ends together to form a loop. This is a Möbius strip.

Cut the loop in half lengthwise.

The result is one loop which is twice as long as the original with four half twists or two full twists.

Afghan Band #3

Tape two strips of paper together and then put a half twist on one end of the paper.  

Using the same end, put another half twist on this end of the paper. Tape the ends together to form a loop. 

Cut the loop in half.

The result is two loops that are intertwined with two half twists.

Now that you are amazed by the creation of these Afghan Bands.  Who discovered this mathematical intrigue of twisting a piece of paper? It was a mathematician named Johann Benedict Listing at the University of Leipzig in 1858. He was interested in topology which is the study of surfaces. It was a fellow mathematician, August Ferdinand Möbius, who named the strip of paper with one twist, a Möbius strip. The Afghan Band #2 (see above), before it was cut, was a Möbius strip.

There are many variations of cutting a Möbius strip. Try them for yourself by making Möbius strips with different numbers of half twists. Here is a generalization which can be made about the twist variations that are cut in half down the center line.

Given n is a half twist and bisecting a Möbius strip:

If there are an odd number of half twists, the result will be one band with half the width and twice the length, with (2n + 2) half twists.

If there are an even number of half twists, the result will be two bands with half the width, each band will be of equal length, with n half twists.

For further explorations, try cutting Mobius strips into thirds and varying the number of twists  You will be amazed by the results. Compare them to the results that you achieved when the Mobius strips were cut in half down the center line.

An ordinary sheet of paper has two sides and one edge. A Möbius strip has been described as a "strip without a second side". 

Take a strip of paper, give it a half twist, and tape it together to form a loop. If you take a pen and draw a line along the center of the strip, you will see that the line runs along both sides of the loop. Thus giving the appearance that the Möbius strip has just one side.

However, the second side is still there. The half twist in the paper created a pathway for the opposite side to be connected.

If two bunnies are placed at the edge of a clear Möbius strip and one of the bunnies decides to circumnavigate the edge by hopping. Moving the clear strip to simulate the hopping of the pink bunny until the other white bunny becomes visible through the clear strip. 

 The pink bunny will have hopped half way around the clear strip, one circuit, when the white bunny is visible upside down on the opposite side of the clear strip. The pink bunny will need to continue hopping another circuit in order to get back to its original position. 

It takes two circuits to circumnavigate the entire Möbius strip. This concept is a valuable tool in manufacturing because it allows for conservation of resources. Both sides of a conveyor belt can be used with this concept because a Möbius conveyor belt wears evenly on both sides.

If you combine two mirror images of a Möbius strip, you will get a Klein bottle which is a unique vessel which has only one surface. When water is poured into it, the water goes out the same hole.  The Klein bottled was first described by Christian Felix Klein, a German mathematician in 1882.

 Klein bottle image from Wikipedia.

I have recreated this Klein bottle using paper. 

One end of the U-shaped tube flows into the bottle and the other tube flows through the bottle and comes out the bottom.

  Another view with lid retracted.

When my paper model was cut in half, two Möbius strips were produced.

This cross section view of the Klein bottle is similar to mine above.

If you would like to recreate my Klein Bottle, I am including a PDF file.  Print out the file, cut out the pieces and glue them together.

Here is the PDF for my paper Klein bottle.  I used 65 lb. cardstock.
https://drive.google.com/file/d/1Do29_dDjQREzOe8vibK13nDx0edUj5tE/view?usp=sharing

Here is the .Studio file if you have a Silhouette brand cutting machine.
https://drive.google.com/file/d/1XJCl-3k9Ot8gVfuqnOPemcTlDfQNjy-w/view?usp=sharing

Möbius strips can be explored in many unusual ways. Check out this tasty article which transforms a bagel into a Möbius strip http://www.georgehart.com/bagel/bagel.html

Here is another magic trick with a Möbius strip and a loop.

Cut two 2 inch x 11 inch strips of paper.  Tape one into a loop and the other into a Möbius strip.

Tape the two pieces of paper at a right angle to one another as shown above.

Cut the loop down the center.  The result will be the above figure when untangled. Next cut the white strip (see above, the right is a loop too, sorry for the bad picture) down the middle from the one loop to the other loop.

The result is a square.

Möbius strips are magical!