Tag Archives: math behind the mobius strip

Every now and then I get a little bit obsessed thinking about things that I know exist but are very strange in such a way that I halfway don’t want to believe they can exist… like the Mobius Strip! The definition for the Mobius Strip on Dictionary.com is “a continuous, one-sided surface formed by twisting one end of a rectangular strip through 180 degrees about the longitudinal axis of the strip and attaching this end to the other”. — OK, so that is a bit difficult to follow but how about “a surface with one side and one edge”?

So now you’re saying “Wait — how did that rectangle become something with only one edge? I am quite sure that rectangles have FOUR edges!” and you would be right – but the mystery of the Mobius Strip lies in that little half twist of the rectangle – and once you have your Mobius Strip in your hand you can see how wonderful they are! I am going to give you some instructions on how to make your own Mobius Strip that I found on this website– I think they are pretty simple to follow but I’ll include some other cool Mobius related links below!

So – let’s make one:

To construct a Mobius Strip requires only a piece of paper, scissors, and some tape.

Simply cut the paper into a single, fairly long strip. Now, holding each end of this strip, give it a half twist (be sure not to give it a whole twist – just flip one of the ends around so that the sides are facing the opposite direction). Now, all that is left is to do is to attach the two ends together into a loop with the tape.

Here are some photos to help you with the first steps:

So as you make the simple Mobius Strip remember you just want to add one half twist to expose the other side of your rectangle. I added letters on the ends to show you how it should be taped together.

So now what? Do you have a single sided surface with only one edge? Let’s check!

Start with a marker and draw a line down the middle of the loop – I started between the letters on the seam so I would be able to easily find my starting point. As you draw your line you will turn the loop around and around and realize that you come back to exactly the same place — ONE SIDE! If you follow the edge around with your fingertip you can also discover how this has only one edge… amazing isn’t it!?

For those of you who like to know the real life application for things that seem incredible and possibly otherwise worthless this is useful for typewriter ribbons and conveyor belts to help keep the wear evenly distributed rather than a regular looped belt without a twist that would have to be flipped periodically to allow for consistent wear.

Ok, next let’s see what happens when you cut along your line in the middle of the Mobius Strip. I’m not going to tell you what happens next but I promise it is pretty cool and you should try it. Then – see what happens if you were to cut half way between your midline and the edge, what happens if you take the result of your first cut and try to cut it in half again, what happens if you put two half twists in your first Mobius Strip and then cut it!? There are lots of fun things to try… what can you come up with?

If you like the Mobius Strip – you’ll love the Klein Bottle! if you know someone who’s particularly interested in topology and mysterious math you can actually buy a Klein bottle and there’s still time to get it before the holidays!