Mobius Strip

If you are familiar with the concept of infinity, you would recognize its symbol that looks like an eight on its side. That symbol is called a lemniscate, which is Greek for “ribbon”. The lemniscate is directly related to a real-world mathematical shape: the Mobius Strip!

The Mobius strip is particularly fascinating because it is a shape that has only one side and one edge! Let’s compare it to a simple piece of paper: if you draw a line on one side of the paper, there’s nothing on the other side. There are multiple edges to this paper (or just one, if it’s a circle). For a Mobius strip, you can only draw on one side, because there is no other side. If you try to trace your finger along the edge of the strip, you’ll be able to go around and around and around, but it might look like you’re tracing the “other” side, but you never lifted your finger!

This means that a Mobius strip is “non-orientable.” Most objects and surfaces are orientable, and it can be tested by attempting to paint the sides different colors. With a cube, you can have up to six different colors, but with a Mobius strip there’s only one. There’s a famous painting, titled “Mobius Strip II” by M.C. Escher where ants are pictured walking along a Mobius strip. Just like what we’ve described above, the ants (or anything really!) could walk along the strip indefinitely!

The Mobius strip isn’t just some random shape made up for fun; it can actually be applied to real life! ​

Conveyor belts have used the Mobius strip design, which made it last twice as long. That’s because the entire surface of the strip was used, not just one side got worn down like it would in a normal ring. Manufacturers have stopped using the Mobius strip system because modern-day technology allows multiple layers to support the belt. Typewriters also used the strip’s design in their ribbons. With a ribbon twice as wide as the print head, both halves of the ribbon can be used evenly. This saved a lot of money back when typewriters were popular!

History of the Mobius Strip

In 1858, August Ferdinand Mӧbius invented this shape and named it after himself. John Benedict Listings then independently discovered it and was published following his findings. The two German mathematicians never collaborated, but both are recognized as the strip’s founders. August Mӧbius also singularly discovered the Mobius Ladder, which is an alteration of a prism graph with a twist in it. Another similar mathematical shape is the Klein Bottle, shown below the ladder.

Mobius Experiments!
You can make your very own Mobius strip at home with a piece of paper, some scissors, and tape.

• Cut out a strip of paper about 1-2 inches wide
  
• Twist one end halfway
  
• Tape it together
  

Now you can try out the different examples from before: try to draw a line on “both” sides, or paint them two different colors. If you cut it lengthwise down the middle, what happens? There are many more videos out there that describe other interesting things you can do with a Mobius strip, like this one!


References:

Weisstein, Eric W. "Möbius Strip." MathWorld--A Wolfram Web Resource. Accessed March 25, 2017. http://mathworld.wolfram.com/MoebiusStrip.html
Teplitskiy, Abraham. “Student Corner: Mobius Strip”. The Triz Journal. January 1, 2007. https://triz-journal.com/student-corner-marvel-of-the-mobius-strip. Accessed March 26, 2017.


Image Credits:

Benbennick, David. “Mobius Strip”. Released into the public domain. Uploaded on March 26, 2017 from wikimedia.org
Kapp, J. Lehman. “Endless Belt”. Released into the public domain. Uploaded on March 26, 2017 from US Patent #3991631
Eppstein, David. “Mobius Ladder”. Released into the public domain. Uploaded on March 26, 2017 from wikimedia.org
“Structure of a Klein Bottle”. Released into the public domain. Uploaded on March 26, 2017 from wikimedia.org