The Mysterious Loops
For this trick a simple loop of paper is cut with scissors along its length, but instead of producing two half-width loops of the same diameter as you would expect, you make a single loop- half as wide and twice as long! Cutting other loops leads to even stranger results, like loops that are interlocked and different sizes. What could be happening?
What you'll need:
How to Prepare:
How to Perform the "Magic" Trick:
You can decide whether or not to show your audience that the loops are twisted, because no one - except someone who understands topology-- would ever suspect that a simple twist of the paper could possibly lead to such a strange outcome. Go ahead and show them how you make the Möbius Strip. They will still believe it must be magic when you cut it and produce anything but two thinner loops!
There really is no "trick" to his one, it's all math- or at least the fascinating branch of mathematics called topology that studies shapes and surfaces. You see, math is more than just numbers and arithmetic. You just usually must wait until college before they show you the really cool stuff, which is a shame, and we can't have that now, can we? Of course to really understand it is quite complicated (probably why they wait until college), but we'll give it a try.
When you formed the loop with a 180° or half-twist, the shape you created is called a Möbius Strip. Unlike the first simple loop you made, a Möbius Strip has only one side (or surface) and one edge. There is no "inside" or "outside" or "left" or "right" edge, even though it still looks like there must be. To prove it, take a marker or crayon and start drawing a line lengthwise without lifting it off of the paper. You will find that your marker eventually comes right back to where it started, yet the line covers both "sides" of the paper! Another way to think of it: if you were a tiny ant crawling along the paper you could explore both "sides" without ever crossing over the edge (and falling off), because there is only one side! In a sense it is easy to understand how this happened: when you twisted the paper halfway you connected the top side of the original strip to the bottom side, thus there is only one side now, but if you really think about what this means it may just blow your mind, as you clearly see when you begin cutting. You can also use your marker to show that there is in fact only one edge the same way, and since there is only one edge, it must be twice as long as the original paper strip (because both of the original edges have been "combined" to form the new edge).
As you cut the Möbius Strip you create a second edge with your scissors- but not two new edges, because remember that this is a Möbius Strip. When you finish the cut the loop now does have two edges and two sides (check it with a marker), but also two full twists. As long as the loop has an odd number of half-twists (1,3,5,etc.) it will be a Möbius Strip, but if it has an even number of half-twists (two half-twists equals a full twist) then it is not a Möbius Strip, it's just a twisted loop. So when you cut a twisted, non-Möbius loop in half, as you did with the second loop that had the full twist, you will finish with two separate loops, but the twists end up locking the loops together. The fourth example, where you cut around a Möbius Strip closer to one "edge" is a bit more difficult to explain, but try to picture in your mind that one side of your cut is producing one new edge on the original loop and the other side of the cut produces two edges of a new loop. Thus the original loop is now twice the original diameter (and no longer a Möbius Strip) but the new loop (which is a Möbius Strip) is the same diameter, and the original twist again locks them together. To try to explain more of what's happening with this strange topological shape (the simplest of what are called manifolds) is well beyond the scope of this simple "magic" trick (as well as my understanding!), but it is fascinating to experiment, so be sure to try cutting different ways in different loops to see what happens. Some of the videos in the links below may help you understand what is happening.
This one has nothing to do with Möbius Strips, but it's fun: tape two simple, non-Möbius Strip, loops (rotated 90° relative to each other) together, like you might do to make a chain. The loops should actually be fastened to each other however, not just looped or locked together (see the video in the links below). Now cut through through the center of one of the loops lengthwise, just as you did in the original demonstration. This will actually cut open the second loop as well. Once you finish and unfold the cut loop you should have a straight strip of paper with a thinner loop attached to each end. Now cut the straight strip lengthwise down the center, also cutting open each of the loops on the ends. When you unfold it you will have a perfect square frame! Do exactly the same thing starting with two attached Möbius Strips instead, and the final result will be two locked hearts! [Note that you will need to tape the two Möbius Strips together so that their half-twists are in the opposite directions, i.e. one clockwise and the other counterclockwise. Watch the video carefully. What happens if you twist both Möbius Strips in the same direction instead?] All this cutting is amazing enough to fill an entire magic show by itself!
Fun Fact: Did you know that the "twisted arrows" design commonly used to symbolize recycling is actually a Möbius Strip? Look closely- if the three sections are connected they form a loop with one half-twist.
Links to more information and activities:
Wikipedia on topology and Möbius Strips:
Mobius Strip as a magic trick:
Classroom lesson video:
Mathematical model and a challenge (for older students):
Why do we get so many twists after cutting? (explained in a video):
Videos showing the connected Möbius Strips tricks to make a giant square frame and connected hearts:
Another cool mathematical paper trick- climb through a piece of paper!:
Two more topological tricks: