Mobius Strip
A mobius strip has only one side. It sounds impossible, but it does. To understand it, you have to make one. Make a mobius strip by twisting a strip of paper 180 degrees and tape the ends together. Color each side a different color before taping. The fact that it only has one side will be more obvious. Try tracing a finger all the way around the strip.
The Mobius Strip is named after August Ferdinand Mobius, an astronomer and mathematician. Mobius was born in 1790. He discovered the mobius strip in September of 1858. He wrote an article on it in 1865. Though Mobius got the credit, the first person to both discover the strange three dimensional figure and publish his finding was Johan Benedict Listing. He found it in July 1858 and published his findings in 1861. Even so, knowledge of the strip may date back to ancient times where mention of something similar is made in an Alexandrian manuscript.
If a mobius strip is cut lengthwise (all the way around) it will end up with a loop that is twice as long as your original mobius strip. It will have two 180 degree loops in it. Cut it another time, there will be two of the same kind of loops interlinked. Cut it another time on each side, there will be four of those loops interlocked.
Similar to the Mobius Strip is the Klein Bottle. It was invented by Felix Klein, who lived from 1849 to 1925. It is a threedimensional shape that is onesided like a mobius strip. It can only exist as a computer generated model. Cut it in half lengthwise, it turns into two mobius strips.
Mobius strips have been put to use by engineers. Some conveyor belts are made with a half twist so that the wear and tear on the belt is equal on both sides  the belt only wears out half as fast. Continuous loop recording tapes were made in this way  to double the recording surface using the same amount of tape.^{1}
The Afghan Bands is a magic trick that uses mobius strips. To perform the trick:
Before the show, cut three strips of paper. For one of them, just tape the ends together. For the next one, make a mobius strip. For the last one, twist it 180 degrees in one place, and do it again in another place. During the show, cut them in half lengthwise. The first one will turn into two loops, as expected. The second one, as noted above, will turn into a loop that is twice as long and has two 180 degree twists in it. The third one will turn into two loops that are linked together. The trick is to not let the audience know about the twists.
Mathematics in the Universe
1. BCC on the Mobius Strip
