What is a Möbius Strip?
We all know that one of the areas that mathematicians are interested in is geometry. In this post I will give an introduction to a simple geometric object that mathematicians like to work with called the Möbius strip which is named after a mathematician of the same name.
The construction of the Möbius strip is quite simple. First we start with a rectangle as below.
Notice that the left and right edges of our rectangle have arrows on them that are pointing in the opposite directions. This signifies that we what to identify these two edges in a manner so that the arrows will line up and point in the same direction. You can think of this process as gluing the two edges together so that the arrows line up.
In order to do this pretend that the rectangle is made up of some malleable substance such as rubber. We can then take the left edge of the rectangle and give it a half twist so that the direction of the arrow on the left side is now pointing in the same direction as the arrow on the right edge. We then take this left edge and pull it over to the right edge and identify them. Intuitively think of the identification as gluing or taping the two edges together. We now have a curved surface, known as the Möbius strip which looks like the following (make sure you can visualize this):
If you have trouble visualizing this it can be a useful exercise to cut out a rectangle from a piece of paper and glue together two of the edges with a half twist. You will then have your own real life model of a Möbius strip which you can use to help you visualize what is said below. Note that if we glue together two edges of a rectangle without a half twist then we would just get a cylinder.
We have constructed a Möbius strip from a rectangle by gluing together two of the edges with a half twist. There are several reasons why mathematicians are interested in the Möbius strip. The first reason is that it is a very simple geometric object. Mathematicians like to work with simple objects because it helps them to build their intuition when working with more complicated objects that might be impossible to visualize.
The Möbius strip also has several interesting properties. The first property is that the boundary of the Möbius strip is a circle. The boundary is just the black edge show in the picture above. If we pick a point on the boundary and then traverse the boundary until we arrive back at the point we started then it is clear we are tracing out a circle. The circle is twisted but it is still a circle.
The other important property of the Möbius strip is that it is nonorientable. This just means that we can't put an orientation on the Möbius strip in a consistent manner. To see this suppose you are standing at some point on the Möbius strip and that you walk around the Möbius strip until you arrive back at the point you started. You will find that you are upside down. Try to visualize this! This means that it is impossible to have a consistent notion of clockwise or counterclockwise on the Möbius strip.
In this post we have seen that a Möbius strip is constructed by gluing together two edges of a rectangle with a half twist. We have studied some of the interesting properties of the Möbius strip including the fact that it is a nonorientable surface. Not only is the Möbius strip nonorientable but we can use it to create other more complicated nonorientable surfaces by taking a sphere and cutting out disks from the sphere and then gluing a Möbius strip in the disks place. We can do this because the boundary of a disk is a circle and the boundary of a Möbius strip is also a circle as mentioned above. The Möbius strip is usually encountered in a first course on topology at the undergraduate level.
References:
https://en.wikipedia.org/wiki/M%C3%B6bius_strip
http://mathworld.wolfram.com/MoebiusStrip.html
Just a remark, the Möbius band, as you presented, is not a geometrical construction is a topological object.
The terms geometry and topology can be used interchangeably especially when talking to a broader audience since the term topology is not well known to nonmathematicians. In a broader sense I would say that topology is an subset of the general subject of geometry. But yes a geometric structure would be considered something with a metric or something with a group of isometries acting on it whereas a topological space does not require this.
I understand that you are addressing to a general audience (no mathematicians), the general public do not well know the difference between geometry and topology. However in mathematics both disciplines are well differentiated, even though they have no empty intersection.
Yes, this is why I mentioned at the end that students would learn about the Möbius strip in the first topology course.
Nice!! I can use this for my sons next show and tell. I'll make a paper model and glue some figures on both a flat rectangular strip as well as on the mobius model to display the differences in orientation.
Good idea!
Thanks for the fun read! Ever since watching Sagan's Cosmos, I've always wanted to experience this like a flatlander :) Would you be interested in doing one on a Klein Bottle?
Yes, I will probably do the Klein Bottle as well and I have ideas for a couple others. Thanks!
...Geometry... use it everyday... I hear hypotenuse about 15 times a day. Now what about the Morpheus Strip?
If they don't believe in the Morpheus Strip, that's okay! His beliefs do not require them to. :)
This is Greatness!!!
It's a totally different construct
They always fall the first time
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