How Short Can A Mobius Strip Be?

The Möbius strip is a fascinating mathematical concept that continues to captivate minds with its seemingly impossible and paradoxical nature. One of the intriguing questions that arise from playing with a Möbius strip is just how short can one be? Today, we'll explore that question and learn more about this topic since many teachers have had students create one in class.

Let's start with a bit of background. The Möbius strip, named after the German mathematician August Ferdinand Möbius, the person who discovered it in 1858. It is a non-orientable surface with only one side and one boundary. This odd creation is formed by taking a strip of paper, giving it a half twist, and then connecting the ends. The result is a single-sided, continuous loop that challenges conventional notions of geometry. 

The Möbius strip's most remarkable property is that it has only one edge and one surface. If you trace your finger along the surface, you would find yourself on both sides without ever lifting your finger. This inh built paradoxical nature makes the Möbius strip a favorite subject for mathematical explorations and artistic creations.

The question of how short this creation could be snagged the imagination of Richard Evan Swartz. He explored the topic using a computer program but due to a mistake in the program, he almost missed finding the answer. However, he kept playing with Möbius strips and that lead him to the answer.

1. The usual method of constructing a Möbius strip involves taking a rectangular strip of paper, twisting one end by 180 degrees, and then connecting the ends. The resulting Möbius strip has a length twice that of the original strip. Experience showed that a long thin strip is easier to make than a short, fat one. 

Back in 1977, several mathematicians theorized that a triangular shaped Möbius strip is as small as the strip can get but no one could prove it . They said the ratio between the length and width would be more than about 1.57 times or pi/2. It took another 50 years before someone one was able to come up with proof.

In order to address this question Swartz focused on the properties of a Möbius strip. At every point, there is a direction that a line travels edge to edge with no curvature. It is completely flat. Swartz recognized that there are places where the two lines cross forming 90 degree angles forming an T shaped intersection.

Swartz used these contortions to find a new length to width ration of 1.69. He moved on to other projects but still thought about this. One day, he realized that he'd made a basic error when he cut open a Möbius strip, realizing it was trapezoidal rather than a parallelogram. This lead to the understanding that he'd made a basic error in the computer program he'd been using to explore the topic.

This small change in understanding lead to the discovery that the ratio is the sqrt 3 or about 1.73 length times its width. In addition, the strip is so short, it ends up flattening into an equilateral triangle. Let me know what you think, I'd love to hear. Have a great day.

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