The Amazing Mobius Strip
Created | Updated Jan 28, 2002
The Mobius strip is an amazing topological device with many exciting and sometimes frightening properties.
Making your first Mobius
To construct your very own Mobius strip simply take the product of two intervals (or a rectangular strip of paper if you don't have any intervals lying around), and identify the bottom edge with the top edge in reverse, so that each of the bottom corners is identified with the OPPOSITE top corner (that is twist it once in the middle and glue the ends together).Look! You just made your very own Mobius strip - what fun you'll have! You'll find that no matter how you twist or bend a Mobius strip (short of tearing it in two and gluing it back together again) you'll never be able to get rid of that twist and turn it into just a normal loop (if you want to sound really clever you can say that the Mobius strip is not topologically equivalent to a cylinder).
The mystery of the Mobius
But what's so special about this strip? Well, try colouring one side of your strip red and the other side green. You'll find you just end up colouring the whole thing a sort of swampy brown. Why is that? Because the most amazing thing about the Mobius strip is that it only has one side. Follow it round with your finger if you like, and you'll find you end up on the opposite side from where you started without going over the edge. This is where the sybol for infinity comes from - the sideways figure 8 you see meaning infinity is actually supposed to represent a Mobius strip, since the Mobius strip goes on forever!More fun with your Mobius
So what else can you do with your Mobius strip? Well, since they're so much fun, why don't we try to turn your Mobius strip into TWO Mobius strips? Get a pair of scissors and cut it along the length of the strip. Careful not to cut in two - then you'll just end up with a boring strip of paper, not an exciting topological space at all. When you get back to the start of your cut after cutting all the way around the strip you'll be able to take your two Mobius strips apart and give one to a friend. Only kidding! The Mobius strip is much more tricky than that! You'll find that you haven't cut your Mobius strip in two at all - you've still just got one long loop with a couple of extra twists thrown in for good measure!Still not satisfied? Then try cutting it in two again! Do it the same way - all the way around the strip, and this time you really DO get two strips. But the Mobius stip still has another trick up its topological sleeve. The two strips you now have are LINKED TOGETHER! And cut them in two and what have you got? Pretty much just a load of swampy brown confetti, so I guess its time to make a new Mobius strip and start the fun all over again. (don't forget to recycle that old one!)
The Final Twist
And as if all that wasn't enough there's one more piece of fun you can have with Mobius strips. If you just take a normal cylinder and glue the two ends together you get a torus (or donut), but if you glue the two edges of a Mobius strip together you'll produce the even more exciting and mysterious KLEIN BOTTLE. However do not become frustrated if you cannot construct this exciting vessel, since a true Klein Bottle can only exist in 4 dimensions - how many other sorts of bottles can you say that about?
