How can you explain the apparent paradox of the double Möbius strips?
The Möbius loop is a topological surface first discovered by August Ferdinand Möbius in 1858. Möbius was a mathematician and professor of astronomy whose work in topology revolutionized the field of non-Euclidean geometry. A Möbius loop can be constructed by connecting two ends of a strip of paper after giving one end a half twist. This results in a baffling surface which has only one side and one edge. The Möbius loop has been immortalized by artists like M.C. Escher, who used it in his print Moebius Strip II, which depicts ants marching in an endless line around a Möbius loop. It also has practical applications in the industrial world, where the large belts in some machinery have been designed with a half twist so that both sides get equal wear. This puzzle presents a fascinating variation of the Möbius loop in which two apparently disconnected loops turn out to be joined together. Students will be challenged to explain this phenomenon as they explore topology using the Möbius loop.
1. This puzzle works best if you construct a model in front of the class, move the pencil between the two loops to show that they are not connected, and then try to pull them apart, showing that they are, in fact, connected.
2. When moving the pencil between the two loops, you will find that after one rotation the pencil will be facing the opposite direction than it was when you started. It is necessary to make two complete rotations to return the pencil to its original orientation. This realization is an important part of explaining the puzzle, and students should be allowed to make the discovery for themselves without having it pointed out to them.
3. Once you have demonstrated the puzzle for the class, give students the necessary materials and have them construct their own version of the puzzle. It is better if the paper students are using is plain so that it is more of a challenge to distinguish between front and back.
Cut two identical strips of paper that are about 11 inches long and one inch wide.
Place one strip on top of the other, holding at the end between your thumb and first finger.
Give the strips a half twist and bring the end together.
|Tape the ends, together -top to top and bottom to bottom. You should now have two Möbius loops nested right next to each other.||Take a pencil and place it between the two loops.||Move the pencil around the loop one time until it returns to the place you began.|
Answer these questions after you have made your loops and followed the directions of the first worksheet.
1. What direction is the tip of the pencil facing now?
2. Is this the same or different than the direction it was facing when you began?
3. Move the pencil around the loop one more time. Now what direction is it facing?
4. Pull the two loops apart. What happens?
5. How can you explain this?
Coloring each side of each strip of paper a different color before the band is assembled can help students see which strips are being attached to each other.
Click the arrow below to view the solution.
In order to understand what is happening with the two strips of paper, it is important to examine how they are taped together. Before the two strips are twisted and taped together, they are placed one on top of the other. Each strip in the beginning configuration has a left and right end. (It might help to label these ends beforehand: TL, TR, BL, & BR..) When the strips are given a half twist and their ends joined, the right end of the top strip ends up being taped to the left end of the bottom strip. Likewise, the left end of the top strip ends up being taped to the right end of the bottom strip. In this way the finished product appears to be two separate Möbius loops nested within each other when in actuality they form one large loop with two half twists. This means that when the pencil is inserted between the two pieces of paper, it must travel twice around the loop to return to its original orientation.