More Experimental Topology

I was thinking that there might be a way to get a hexagon along the lines of the methods of the post Math with scissors. I told my kids about the idea and they were excited to do some math experiments.

I started with three strips of paper.

Taped them together like this:

Then I wanted to tape the three ends up like this,

but I knew there had to be some twisting of the strands involved. In the end we are going to cut each of the strands down the middle.

So we experimented. I knew that at least two of the strands had to have some odd number of half twists in them to have any chance of getting a hexagon (can you see why?). It took us several tries, but my son and I both came up with our own solutions for how to get a hexagon:

Try it and see if you can get it. Here are our solutions:
My son’s solution: Put a single half twist in each of the three strands, twisting two in one direction and the other one in the opposite direction.
My solution: Put a half twist in each of two of the strands, twisting them in opposite directions. Leave the third strand untwisted.

Of course, the experts will want to conjecture and prove necessary and sufficient conditions to get a nice flat hexagon. I also have an idea for making a five-(or more)-pointed-star along similar lines. I’ll let you know what I come up with.

Math with scissors

The other day I visited my son’s class and did the following math demonstration.

Take a strip of paper and draw a line down the middle. Our line is red.

Tape one end to the other, but put a “half twist” in it, just like shown here. This is a Moebius strip. It only has one side to it. A pretty cool object in and of itself.

Now for some real fun. Take a pair of scissors and cut the Moebius strip in half along the red line–But before you do that try to guess what you will end up with!!! I’m not going to show you. You have to try it yourself. Just make sure that you only cut along the red line!

There is more that we can do. Let’s make a + out of paper with two red lines drawn as shown here.

Now tape two opposite ends to each other with no twists. Like this.

Tape the other two ends to each other, but put in a half twist. What you have here is actually a Klein bottle (whatever that is) with a hole in it.

Now cut along the two red lines. Make sure not to make any other cuts except along the lines.

Keep cutting…

Keep cutting.

In the end what do you get?

(Make a guess first.)

(!)

One more: Make a star shape just like this. Make sure that all of the strips are the same length and that the angles between the strips are the same. Draw three red lines. One down each of the strips.

Here is what it should look like.

Now tape two opposite ends together with no twist.

Take another two ends…

…and tape them, with no twist, just like this.

Now for the tricky part. You have to do this part right to get nice shapes in the end. But, don’t worry too much. If you do it “wrong” you’ll just get something other than what we got. Flip your paper over. Tape the last two opposite ends with a full twist to the left. Just like shown here.

Here is another view. Now cut along the red lines and see what you get. Bailey’s class loved it.