Making the mathematical wonder known as a Mobius strip is surprisingly easy, though understanding it on a mathematical level adds a whole new complexity.
The Mobius strip was first developed and defined by the German Mathematician August Ferdindand Mobius in 1858. In essence, this strip is an easily contstructed topological (which refers to the study of geometric space) paradox:
A surface with only one side and one edge.
Imagine a piece of paper. Clearly, this surface possesses two sides and four edges, creating a perfectly standard shape, easily defined in a topological sense. Now, imagine the same shape, but with three of the edges taken away, and with only one side... does this seem impossible?
It's actually quite simple
Making a Mobius Strip
To construct a Mobius strip requires only a piece of paper, scissors, and some tape or glue.
Simply cut the paper into a single, fairly long strip. Now, holding each end of this strip, give it a half twist (be sure not to give it a whole twist - just flip one of the ends around so that the sides are facing the opposite direction). Now, all that is left is to do is to attach the two ends together into a loop with the tape or glue.
What should be left is a half-twisted loop of paper, which on the surface doesn't seem very interesting.
It is easy to prove, however, that this shape possesses only one side and one edge. Try to draw a line down the middle of the surface of the strip, beginning at one location and continuing on forever. Eventually, this line will have covered the entire surface of the shape and end up right back where it began. Therefore, there is only one surface! The same test also works by running a finger along the edge of the paper.
This in itself is mathematically interesting, but it gets better.
Now that a line is drawn along the center of the strip, what happens when the strip is cut in half along this line? Wouldn't the expected effect be that two identical strips are produced?
Against expectations, the result is not two Mobius strips, but one larger one.
What happens when this larger strip is cut in half? The reader is encouraged to attempt this, knowing only that the resulting figure is very difficult to predict.
The Math of a Mobius Strip
In topology, the mobius strip possesses many interesting characteristics.
For one thing, the aforementioned experiment where the strip is cut twice in half has a very precise analogue in the untying of a trefoil knot in knot theory.
In topology, a mobius strip is considered to be a non-orientable surface which has its three-dimensional analogue in the "impossible" figure known as a klein bottle.
Mobius strips possess the feature of being chiral, which means that it is subject to "handedness." In other words, there are right handed mobius strips and left handed mobius strips, depending on the direction of the twist. This feature of topology has many applications in both mathematics and in physical science (subatomic particles often obey chiral laws).
Mathematicians have been able develope the exact mathematical formulas necessary for graphing the characteristics of a mobius strip using relatively simple three-dimensional mathematics (this formulation can be found in detail at Wolfram's Mathworld Mobius Strip Page).
In electronics, the idea of a mobius strip has been translated into a device known as a mobius resistor, which uses a 180 degree twist in order to cancel out its own self-inductance, which means that it resists electrical flow without causing magnetic interference.
To conclude, there are two definite benefits to knowing about mobius strips: There is the obvious benefit of having a simple form of entertainment, and the slightly more difficult benefit of having a true, deep mathematical problem with many applications throughout several sciences.
References:
"Mobius Strip." Cut the Knot.org http://www.cut-the-knot.org/do_you_know/moebius.shtml
"The Colossal Book of Mathematics." Martin Gardner. W.W. Norton & Co. 2001.
"Mobius Strip." Wolfram's Mathworld.
