The Möbius strip is a curious object.The strip was discovered simultaneously and independently by German mathematicians August Ferdinand Möbius and Johann Benedict Listing. Möbius took the concept farther than Listing by exploring the properties of this strip. Let us discuss some of these properties.
If you happen to have a ribbon or something like it close by, lay it out flat. Twist one end 180° and connect it to the other end. Now look at the object created. The Möbius strip is two dimensional. Notice it only has a length and width. That means that it is a two dimensional object that exists in three dimensional space! Not only that, if you attempt to press your strip down on a table, it starts to cross itself and lose its properties. So the strip cannot exist in 2D space.
Imagine, or try, to color one side of this strip. You may have realized that if you tried to color just one side, you actually color the whole thing! This means that this surface has only one side. That is, the Möbius strip is non-orientable. To see this further, fix the strip still in space. Now take a point and give it a direction “outwards.” If you move this point around the strip, by the time you return to its original position, the direction will be reversed! See the applet from Math Insights for an interactive visualization. Not only is this strip non-orientable, it is in some sense a basis for determining non-orientability of other surfaces as noted in the following theorem.
Theorem: A surface is non-orientable if and only if it contains a subspace that is homeomorphic to a Möbius strip.
On this last note, we leave a short exercise. Draw a line in the center of your Möbius band. Cut along that line and explore what happens.
Topology is full of fun and interesting shapes and surfaces. In the next post, we will look at some other topological spaces you can make with a simple rectangle! We will be exploring the strange and the unusual in this exciting series!