Geometry Forum - Problem of the Week
Solutions - The Mobius Band, Oct. 10-14, 1994
Annie says:We try to offer problems of varying difficulty and types, hoping that students will be challenged and interested at different levels. This mobius problem was a difficult one whose full solution we did not expect to receive but we were interested to see the explorations of those students who attempted it. The Fairfield group was very accurate in their estimate of the shortest length, although we have no information about how they arrived at their answer. The submissions also contain some other tentative explorations of what is happening as the band is cut in various ways.
At the end of the submissions presented below we include the answer of Dan Asimov, who sent us the problem.
Samantha Brenner, Julie Black, Mike Henry, Justin Davies
1. The shortest length of a rectangle is about 1.75 inches. If 1 inch is the length, however, the other dimension is as small as you can cut it.2. After cutting down the center of the Mobius strip, it becomes one large Mobius strip with 2 twists. If you cut the already cut Mobius strip in half again, it will form two intertwining Mobius strips approximately equal.
If you cut another Mobius strip 1/3 of the way in, it will form a long skinny strip adjoining with a wider, shorter, one twist strip. The smaller the fraction of the width that you cut will make for a skinnier long piece and a shorter, fatter short piece.
Michele Gibney
A mobius strip is a single closed curve. Also in the making of a mobius strip, you take a long thin rectangular piece of paper, give one of the narrow ends a 180 degree twist and glue or tape the ends together. If the paper had not been twisted before joining the ends, it would have made a cylinder with two edges. So, the mobius strip is closely related to the figure known as a circle. The cylinder would equal l=(pi) d, therefore the length of the mobius strip is going to equal 3.14 W. The width of our mobius strip is 1, so 3.14 ( 1) = 3.14 THIS IS THE SMALLEST MOBIUS STRIP.Joe Fuston
I made a mobius and pulled one end tighter until it could not move anymore. 3 and 1/4 widths was the shortest I got.When I cut the mobius lengthwise the size doubled. I believe half twist prevented it from separating. When I cut the mobius again, it created 2 linked loops. When you cut the first mobius it has a half twist so when you trace a line on it, it shows on both sides, therefore it stays together. When you cut it again, it's basically a mobius with a full twist therefore you only can trace a line on one side which means it will separate but they were linked.
Dan Asimov
And now the source of this problem, Dan Asimov, offers his thoughts:First the easier part:
What happens when you cut a Mobius band all around its length ?
This is so much fun that you *must* try this yourself, if you haven't already. But the result is...you get just one band of paper, not two, amazingly enough. This band turns out to be what you get if you start with a rectangle (twice as long and half as wide as the original one) and join the ends together after giving one of the ends a full 360-degree twist. This long, skinny, twisted band (call it LST) is topologically equivalent to a cylinder, and not to a Mobius band. One way to see this intuitively is to check that LST is two-sided, like a typical surface, and not one-sided like the Mobius band. (You may wonder if this means that LST can be moved around in space until it looks just like an ordinary cylinder. In fact it cannot, because even though it is topologically equivalent to a cylinder, it is embedded in space differently from the ordinary cylinder. This is similar to the fact that a simple closed curve in space that is knotted cannot be moved around in space to look like an unknotted circle, even though they are topologically equivalent.)
Can you guess what happens to LST if once again, you cut it all
around its length? (Try it and see.)
What is the longest length of any rectangle -- of width 1 -- that cannot be made into a Mobius band (without any stretching or tearing)?
The answer to this question is amazing. It turns out that ANY rectangle, no matter how short, can be made into a Mobius band. So the answer is 0. BUT in order to do this, it is necessary to use certain kinds of crinkling that result in a Mobius band that is not very smooth. HOWEVER, when I posed the question, I had in mind a smooth Mobius band. So -- the question I really meant was, What is the longest length of rectangle having width 1 that cannot be made into a smooth Mobius band (without any stretching or tearing)?
The answer to this question is length = sqrt(3) = 1.7320508075688772...
Here's the limiting case: Consider 3 equilateral triangles of altitude = 1. Join them in a row of three triangles. Call them ABC, BCD, and BDE. (This is easy to make with paper, scissors, and tape.) Now fold along the lines BC and BD. (Yes! You are right -- folding is not smooth. We will weasel out of this later.) Now the edges AC and DE are in the same place, so tape t hem together. Voilà -- you have created a Mobius band.
This thing is so folded up that it's tricky to see that it really is a Mobius band, but it really is. Now note that if we call the midpoint of the edge AB by the name F, then before folding BC and BD, we could have lopped the right triangle AFC off one end of the strip of triangles, flipped it over, and taped in onto the other end (in other words, so that AC gets taped to DE). This creates a rectangular strip of dimensions 1 x sqrt(3) which can be folded and taped to become the same Mobius band as described above.
BUT, the above two paragraphs exaggerated a bit, since this Mobius band cannot be created smoothly unless the length is just a bit -- no matter how small -- longer than sqrt(3).
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