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	Topology

See the Science Chapter for more instructions, hints, tips, and ideas.

For the paper, you can use calculator paper, streamers, or cut your own.

Topology is the study of surfaces.

Let's start with a simple shape - a ring. Make a ring about 12" in diameter with wire. Place it on the table. Now place your finger on the table outside the ring. To put your finger inside the ring, you must cross the ring exactly once. Then to get your finger out of the ring, you must cross the ring exactly once.

Now take the wire and make a long loop out of it. The above still applies - one cross of the wire to get inside, another to get out. Next, take the long wire loop and wrap it into a spiral. Now pick a place inside the spiral and place your finger in there. Is your finger inside or outside the loop? To find out just count the number of times you must cross the wire to get to the outside of the spiral. If it is an even number, then you finger was outside the loop. If it is an odd number, your finger was inside the loop. Let's test this. Place your finger inside the spiral so there are an odd number of wires to get to the outside. Remember that this means that your finger is inside the loop. Now untwist the loop and your finger should be inside the loop!

This technique can be applied to solve any maze problem. Simply start at the beginning and follow a wall, it can be either the right or left, but you must follow it ALL the time. Don't pay attention to the corridors, just follow the wall. You will eventually get to the destination without taking a wrong turn!

Want to try it out? Build a maze and try it out!

Now let's try something else with a ring. Take a strip of paper and straighten it out. How many edges does it have? Four (right, left and two ends). Now tape the ends together. How many edges does it have? You got rid of two by taping them together, so it just has two. How many surfaces? To figure this out place a pencil near the joint and hold it against a table. Keep the pencil on the paper all the time and move the paper beneath it. Trace a line until you can connect with the beginning of the line. This covers one side. Therefore, you have discovered that the surface has two sides. Now cut the loop in the middle of the paper all around the ring. What do you get? Two rings.

Now let's try something just a little different. Take another strip of paper. Straighten it out but before you tape the two ends together, give one end a half twist. This is a very famous shape called a Mobius strip. Now how many edges does it have? Two. How many surfaces? Let's do the same as above with the pencil. This time you will notice that you go over the joint, but the pencil mark isn't visible, it's on the other side. Keep drawing the line on the paper until they meet. This proves that the surface has only one side. Now cut the loop in the middle of the paper all around the ring. What do you get? One ring with one half twist! What would happen if instead of cutting the paper ring in half, you cut it in thirds?

Okay, let's do the same thing only this time put a full twist in the paper strip before you tape the two ends together. It still has two edges. Use the pencil technique to count the number of side. It's two. Now cut the ring in the middle of the paper all around the ring. What do you get? Two loops that are linked together.

Finally, let's put one and a half twists in the paper strip before you tape the two ends together. It still has two edges. Any guess as to how many sides it has? That's right only one. Well, what's going to happen when you cut it in half?

Here's a topology trick. Take a piece of rope and hold one end in each hand. Can you tie a knot in the rope without letting go of the rope? The answer is no. However, if you tie a knot in your arms first, then pick up the rope and untie you arms, the knot will be transferred to the rope. Go ahead and try it. Just fold your arms. One hand needs to be behind your arms the other in front. Now grab the rope and un-fold your arms.

A mobius strip makes a great neckerchief slide.

• Cut the strip of leather using a sharp leather scissors or a knife.
• Form the strip into a circle. Take one end and turn it over (giving it a half twist) so the hair side of one end is joined with the flesh side of the other end.
• Note: You may need someone else to help you with this. Put a little E-6000™ glue on the ends, hold them together and clamp in place with a binder clip. Don't use too much glue or you will glue the clip to the leather.
• When dry remove the clip and add more glue.

To wear as a neckerchief slide, place the loop on the neckerchief. Bend it into a figure 8 so the pieces cross in front of the neckerchief. Pull the tails of the neckerchief out through the bottom of the figure 8.