We’re used to doing math with numbers (adding, subtracting, multiplying, dividing) but did you know that you can also do math with no numbers at all? There is an area of mathematics called topology, which is the study of shapes and how they can be bent or stretched. Topology can deal with either flat (2 dimensional) or solid (3 dimensional) objects. In the activities below, we’ll be working with 2 dimensional objects—strips of paper.

Activity 1 – Loop

In this activity, we’re going to take a paper strip and make it into a loop.

1. Take a strip of paper and use a glue stick or tape to attach the ends together to form a loop.

This loop is the simplest kind of knot (sometimes called a null knot or an unknot).

Draw a line around the middle of the outside of the loop. Notice that there is no line on the inside of the loop.

2. Now we’re going to cut our loop in half along the line. Before doing that, think about what you’re going to end up with when you cut the loop in half.

We end up with two loops. Each loop is half as wide as the original loop, but each loop looks the same, right? That’s probably just what you expected. But sometimes math can surprise us…

Activity 2 – Loop With a Twist

In this activity, we’re going to take a paper strip and make it into a loop, but we’re going to do it a little differently this time.

1. Take a strip of paper and twist it one time before using a glue stick or tape to attach the ends together.

2. Compare this loop to the first loop we made. What is different?

This loop is called a Mobius strip, named after the German mathematician August Ferdinand Mobius who discovered it. Even though all we did was twist the strip one time before making the loop, everything changed!

3. Use a pencil or marker to draw a line right down the middle of the strip. Keep going until your line connects back to the beginning. Notice that the line is on the “inside” and “outside” of the Mobius strip. That means that the Mobius strip only has one side!

4. Now we’re going to cut our Mobius strip in half like we did before with the simple loop (along the line you just drew). Before doing that, think about what you’re going to end up with when you cut the Mobius strip in half.

Were you surprised that you only ended up with one loop, even though you cut it in half?

Activity 3 – Loop With 2 Twists

In this activity, we’re going to take a paper strip and make it into a loop, but we’re going to do it with two twists.

1. Take a strip of paper and twist it twice before using a glue stick or tape to attach the ends together.

2. Now we’re going to cut our twisted loop in half. Before doing that, think about what you’re going to end up with when you cut the loop in half.

Were you surprised that you ended up with two interconnected loops?

Activity 4 – Cutting a Mobius Strip in Thirds

1. Take a strip of paper and make a Mobius strip by twisting it one time before using a glue stick or tape to attach the ends together.

2. Now we’re going to cut our Mobius strip like we did before but instead of cutting it down the middle, cut it one third of the way from one edge.

Were you surprised that you ended up with two interconnected loops instead of a single loop like when we cut it in half? How are the two loops different than the ones in the loop with two twists?

Activity 5 – Combining Two Simple Loops

1. Take a strip of paper and use a glue stick or tape to attach the ends together to form a simple loop.

2. Take a second strip of paper and use a glue stick or tape to attach the ends together to form another simple loop.

3. Use tape to connect the two loops perpendicular to each other.

4. Cut each loop in half.

5. Unfold and see what you got.

Were you surprised that you were able to cut two circles in half to get a square?

Other Activities

Math is full of lots of surprises and topology is no different. If you want to try some other activities on your own, here are some suggestions:

1. What if you made 3 (or 4 or 5 or more) twists in the paper strip before connecting? Does anything different happen when you cut the loops in half? Are there similarities between loops with an even number of twists? Are there similarities between loops with an odd number of twists?

Note: if you want to make more twists in the loop, it is helpful to use a longer strip of paper.

2. Make a simple loop. Take another strip and thread it through the first loop before attaching the ends. This will make a chain of loops. See what happens when you cut each of the loops in half.

3. Link two Mobius strips together. See what happens when you cut each of the Mobius strips in half. How is the result different than when we did this with simple loops.