Loopy Geometry


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Experiment Category: 


Discover geometry by creating shapes from loops of paper.

What You Need: 

  • A few sheet of sturdy paper (construction paper works well)
  • A pair of scissors
  • Tape

To Do and Observe: 

1. Cut a few equal length strips of paper 1/2 inches in width and at least 11 inches long. Tape the ends securely together to create paper loops. If you take one loop and slice it into lengthwise you get two thinner and equal loops from the original, right?

2. Now this time take two new loops and tape them together, with one turned 90 degrees to the other. Kind of like a figure 8 with a right angle twist... Now, take hold of one loop and slice it again right down the middle. I know, I know it cuts through the second loop but do it any way. What shape do you get?

3. "So what?" you might first say, but wait! Cut the strip in two that used to be a loop before you sliced the first loop. Now what do you get? Hmmm. Is that what you thought it would be?

4. You are just starting! Cut three more equal lengths of half-inch paper and make three new equal loops. This time, link all three in a chain securely taping both the inside and outside of each loop together. Be sure to securely tape all the edges down when you link loops. (If you don't, the loops and strips begin to disintegrate when you slice them length-wise.)

5. Now once again slice the loops right down the middle. It doesn't matter which one you start with, just remember that all of the loops must be halved right down the middle, whether they have been severed into strips by a previous cut or not!

6.Can you imagine what 4 loops would give you when you slice them? It is possible to lay the resultant paper out in a 2 dimensional structure with no folds or bends - but it's a bit of a puzzle! Try it!

What's Going On: 

The morphing loops give you a totally new structure when you cut them. To figure out what it will become, there are some features you should notice. How long are the strips of paper before you tape them? Were the two strips the same length to begin with? Will they be the same length afterwards? What about the angle in which you tape the two loops together? In geometry, an angle that is "exactly sideways" like this is called 90 degrees. Can you think of some shapes that have only 90 degree corners to them? What are they?

Parent/Teacher Tips: 

Changing the angle that you link loops together will radically change the shape of things. Instead of 90 degrees, try orienting loops at 60 degree angles from each other.