MOBIUS MYSTERIES - CHARMING TOPOLOGY

When you take and connect the ends of a strip of paper, you make a simple loop with two sides, two edges, an inside and an outside. If you draw a line around the outside of the loop, the line joins with itself and remains on the outside. If you draw a line around the inside of the loop, the line joins with itself and remains on the inside. The line drawn on the inside never meets the line drawn on the outside and vice versa. But you can make a different kind, a Mobius loop (it's also called a Mobius strip, named after the 19th century German mathematician August Mobius), by taking a strip of paper and giving one end a half-twist before you join it to the other end. This way, the top of the strip becomes the bottom, and the bottom of the strip becomes the top. And you'll have a strange loop with only one side  and one edge. 

To prove it, draw a line around the loop. Eventually, your line goes completely around and meets itself.  
Imagine two strips of paper sandwiched together and twisted into a Mobius loop. Now imagine a bug crawling between the strips of paper. He could crawl around the loop forever with a "ceiling" above him and a "floor" below without knowing that the ceiling and floor were actually the same surface. If he drew a big X on the floor and continued crawling, he would eventually reach his X again—on the ceiling. These are some of the reasons Mobius loops fascinate topologists. Just as we've transformed other shapes, we can now change a Mobius loop into something quite different. Cut two strips of paper, but before you connect their ends, draw a dotted line along the lengths of each strip. Connect the ends of one strip so that you have an ordinary loop, and connect the ends of the other strip so that you have a Mobius loop. Use a small amount of glue to connect the ends of your loops, so that they don't come apart when you cut them

What do you think will happen if you cut each loop along the line? 
First, cut along the dotted line on the ordinary loop. You can see that the cut divides the loop into two loops, as you might expect. Next cut along the line on the Mobius loop. Although you might expect to make two separate loops, you wind up with one long loop with two twists.  

Deluxe Mobius Loop 

Take another strip of paper and draw a line along its length as before. This time, give one end a complete twist before you attach it to the other end. Your new Mobius loop should have a smaller loop inside.

Cut along this line as before. This time you'll wind up with two twisted loops linked together. 

Super Deluxe Mobius Loop 

Finally, make a Mobius loop where you give one end one and a half twists before joining it to the other end. Cut this strip along the line you drew, as before. You'll wind up with something completely different this time: a single loop linked together three times. 

Try looping strips together in different ways and cutting them apart. See what strange topological results you can discover.


 No wonder many mathematicians choose to become topologists. 

It allows them to play with Mobius loops!