The Möbius strip: a shape with one side
Tonight's idea
Take a strip of paper and tape the ends together. You get a plain loop, like a bracelet. It has two sides, an inside and an outside, and two edges, top and bottom. Nothing surprising.
Now take another strip, and before taping, give one end a half twist (turn it over once). Tape. This is a Möbius strip, named after August Möbius who described it in 1858.
Put a pen in the middle of the strip and draw a line along it, never lifting the pen. Keep going. You will arrive back where you started, having drawn on what looked like both sides. There is only one side. Run a finger along the edge and you will find there is only one edge too.
This belongs to topology, the study of what survives when you stretch, bend and squash a shape without cutting or gluing. A topologist cannot tell a coffee cup from a doughnut, because each can be squashed into the other: one hole each. Number of sides, number of edges, number of holes: these survive stretching. Lengths and angles do not.
The experiments below are the point of tonight. For each one, write your prediction before you cut. Being wrong is where the learning is.
Watch one: the first cut
Take the plain two-sided loop and cut it lengthwise down the middle. You get two thinner loops. Obviously.
Now take the Möbius strip and cut it lengthwise down the middle, following the line you drew. Before cutting, most people predict two loops. What you actually get is one long loop, twice as long, with two full twists in it. Check: is it still one-sided? Draw a centre line and see whether you cover both faces.
Why one piece? The cut line is a single closed loop in the middle of a single-sided surface. Removing it leaves one connected strip of paper. The "other side" you would have been cutting off is the same side.
Your turn: predict, then cut
- Edges and sides. Make a fresh Möbius strip. Count its sides by drawing, then count its edges by following with a finger. Now make a loop with two half twists (a full turn). How many sides and edges does that one have?
Prediction:
Result:
- Cut the Möbius strip down the middle. (Fresh strip.) How many pieces? How many twists in the result?
Prediction:
Result:
- Cut the long result from question 2 down the middle again. How many pieces this time? Are they linked?
Prediction:
Result:
- Cut a fresh Möbius strip one third of the way in from the edge, and keep cutting at one third until you come back to the start. How many pieces? Same size?
Prediction:
Result:
- Cut the two-half-twist loop down the middle. How many pieces, and are they linked?
Prediction:
Result:
- Explain in a sentence why you can be sure a plain loop will always give two pieces when cut down the middle, but a Möbius strip cannot.
Stretch: three half twists
Make a loop with three half twists. Is it one-sided or two-sided? (Rule to find: odd twists are one-sided, even twists two-sided. Check it.) Now cut it down the middle. You get one loop, but it is tied in a knot. Which knot? Count the crossings when you lay it flat as neatly as you can.
Harder: explain why the Möbius strip has one edge by tracking what the half twist does to the two edges of the original flat strip. Where does the top edge arrive after one trip around?
Answers
- Möbius strip: one side, one edge. Two half twists: two sides, two edges, like a plain loop.
- One piece, twice as long, with four half twists (two full twists). It is two-sided.
- Two pieces, each the length of the long loop, and they are linked together like chain links.
- Two pieces: one short Möbius strip (the same length as the original, one third as wide) and one long two-sided loop twice the original length. They are linked.
- Two pieces of equal length, linked together, each with two half twists.
- A plain loop has two edges and the middle cut separates the strip into an "upper" piece and a "lower" piece, each keeping one of the original edges. A Möbius strip has only one edge, so there is nothing to separate: the paper on one side of the cut is connected to the paper on the other side by going round.
Stretch. Three half twists is odd, so one-sided (draw the centre line to check). Cutting it down the middle gives a single loop with a trefoil knot in it: three crossings when laid flat, the simplest knot there is. For the one-edge argument: the original flat strip has a top edge and a bottom edge. The half twist joins the top edge at one end to the bottom edge at the other. Follow the top edge around: after one lap it has become the bottom edge, and only after a second lap does it return to the start. So the two edges are joined into one closed edge of double length.