Fun with Geometry
Rahman Ur Izam, Chennai
In mathematics, a Möbius strip, Möbius band, or Möbius loop is called a non-orientable surface. What does this mean? Read on to find out.

Activity 1.
Cut out a thin strip of newspaper and join the ends with a staple or cellotape. Now, with a pen or pencil, start at one end of the joint, and draw a line along the length of the paper. Keep going till you come back to the starting point. Observation: You will see that one side of the paper strip has a line going through it, and the other side is clean. This means (obviously) that there are two sides of the paper strip, which we could have actually found by just looking at it! So what is special? We will find out in Activity 2.

Activity 2.
Cut out another strip of newspaper. Now, twist it once before you join the ends. That is, turn over the edge you have in one hand, and join it to the other. This is called a half-twist. It should look like the twisted strip shown in the figure. Do the same "experiment" as before. Starting at one end, draw a line along the paper and keep going till you reach the starting point.
Observation: What do you see? You will find that there is no side of the paper strip which is not marked by your pen. If you think this through, you will realise that this object that you have just made has only one side! It is called a Mobius strip.
So, now, what do we mean by "non-orientable"? It is a surface, where we cannot distinguish clockwise from anti-clockwise turns. This is because it has only one boundary curve, which is the equivalent of the single pencil line along the strip, covering "both" sides.

Activity 3.
You can check this by starting at the join, colouring one edge using a felt pen, and continuing till you reach back to the starting point. You will find that there are no un-coloured edges left, since there is only one boundary.

Applications:
There are many applications of Möbius strips. One of the most important ones was the realisation that if you have a mechanical belt in the shape of a Mobius strip, it will last twice as long as an ordinary belt where only one side is always exposed to the effects of friction. So, for instance, dual-track roller coasters whose carriages alternate between the two tracks, use Mobius strip belts. Möbius strips also appear in molecules and devices with novel electrical and electromechanical properties.
Interestingly, the Universal recycling symbol (see picture, the outline version with green fill), orginally designed in 1970, is a Möbius loop. So is the older Google dirve logo. However, what fascinates mathematicians are the very curious properties of the Mobius strip.

BOX Do you know?
Heinrich Franz Friedrich Tietze showed in 1910 that the Möbius strip can be subdivided into six regions that all touch each other – three along the boundary of the strip and three along its center line. There is a famous "4-colour" theorem in mathematics which says that objects on any flat surface (like a map) only need a maximum of four colours for uniquely colouring it. For example, you can colour a map of Indian states using four colours so that no two adjacent states (with common boundaries) are touching each other. It appears that you will need 6 colours to achieve this with a Mobius strip.
END OF BOX

Properties of the Mobius Strip
Now that you know how to make a Mobius strip, let's have some more fun with them.

Activity 4.
Cut a Mobius strip along the centre-line with a pair of scissors. Don't you expect to get two Mobius strips? What do you get? A single strip, which has 4 half-twists in it! Again, do the experiment of drawing a line along the length of this cut strip. What do you get? Is it also a Mobius strip with only one side?

Activity 5.
Cut this double twisted strip again along the centre-line. This time you will get two strips, but they will be interlinked and you will not be able to separate them without cutting the paper! Each strip is double-twisted (check). Is each strip a Mobius strip? Find out!

Activity 6.
Instead of cutting a Mobius strip long the centre-line, cut it one-third of the way from the edge. Again, keep going until you reach back to the original point. What do you get? You will now get two strips! Again, these are interlinked and cannot be separated without cutting the paper. Amazing, what just cutting it off-centre will do! Are these both Mobius strips? Do the Mobius strip test of drawing a line along the strip. You will find that one of the strips is indeed a Mobius strip while the other has two half-twists.

Activity 7.
Instead of cutting the strip one-third of the way from the edge, try different widths: cut it at 1/4, 1/5, etc., and see what you get. Do write in to JM with your answers and insights, or even pictures!

	Sources: Images from Wikipedia