To accompany ‘A twisty mystery about Möbius strips has been solved at last‘
1. With scissors, cut three long strips from a sheet of paper. Make sure each strip has the same width and the same length. Using a ruler, measure the length and the width of each strip.
2. Take one strip and make a loop by taping one end of the paper strip to the other. Take the second strip and twist the paper halfway around once before taping the two ends. If you need a visual, check out this video. The normal loop has two surfaces, an inside and outside. How many surfaces does the half-twist loop have? You can figure this out by running your finger along the loop until you arrive back at the point you started.
3. Take the third strip and create another loop with a twist. Before taping, try to make the loop as small as you possibly can. What happens to the loop if you make it too small?
1. How does the number of surfaces differ between a strip of paper curled into a loop vs. a Möbius strip?
2. What problem occurs if you attempt to make a Möbius strip from too short of a strip of paper?
3. Explain the meaning of the symbol √3.
4. In 1977, what did mathematicians hypothesize regarding the limitations of the Möbius strip?
5. In 1977, mathematicians showed that a Möbius strip’s ratio of length and width must exceed a particular value. What is that value?
6. What assumption did Schwartz make regarding the shape of a sliced open and flattened Möbius strip? What did he discover the shape to be?
7. What did Schwartz prove after correcting his previous mistake?
1. What does this story tell us about the value of trying multiple approaches in addressing challenging problems?
2. What was the proportion of your Möbius strip’s length to width? Did it fall within the proportion proved by Schwartz? Explain.
3. Come up with a potential application for a Möbius strip and describe it in one sentence. Feel free to be creative!