Möbius Strips
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www.worldofescher.com The geometric artist Escher popularized the Möbius strip in his woodcarving "Möbius Strip II." Which side are the ants on? |
In 1858, while working on the geometric theory of polyhedra, Möbius discovered the "Möbius strip." The simple object consisted of a loop of paper with a half-twist. Strange as it may seem, the Möbius strip has only one side. Try tracing your finger around the "outside" of the strip, and you find your finger on the "inside". It has no outside or inside, so there is no way to paint only one side.
This paper model isn't a true Möbius strip because it has a seam. Real Möbius strips have no seam, but it doesn't seem possible to make one like that. Maybe someone could make seamless one-sided Möbius rubber bands J . Möbius wasn't the first one to discover this strange surface. By either publication date or date of first discovery, Listing found it first.
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There are a few tricks you can do with a Möbius strip. Draw a point on the strip. If you trace around the strip, you will touch the point, even if you started on the opposite side.
Now for something more unexpected. If you draw a line around the center of the strip and cut the Möbius strip in half on that line, you'll end up with a long two-sided strip.
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A mathematician confided
That a Möbius band is one-sided,
And you'll get quite a laugh,
If you cut one in half,
For it stays in one piece when divided.
http://isaac.exploratorium.edu/~pauld/activities/mobius/mobiusdissection.html
Also, if you cut the Möbius apart exactly 1/4th of the distance from the edge (this will take twice as long, because you'll have to cut along the "outside" and the "inside"), you'll create two linked strips - except the longer loop is normal and the smaller one is a Möbius strip.
Here's a cool formula for Möbius strips - if you cut along a line 1/n from one side of a Möbius band of width W, there will be a strip left in the middle with a width of W - W/n-W/n. Of course, if n is 2, then the width of the strip in the middle is zero. That's why when you cut a Möbius strip in half as described earlier, there is no strip in the middle.
You can make ultra- Möbius strips with 3 or more half twists. There's another formula that has to do with the number of twists. If you bisect a strip with an even number, n, of half twists you get two loops each with n half-twists. So a loop with 2 half-twists splits into two loops each with 2 half-twists. If n is odd, you get one loop with 2n + 2 half-twists. So a Möbius strip with 1 half-twist becomes a loop with 2+2 = 4 half-twists.
