Try this: Dragon curves

dragon4.1You will need

  • Several long strips of paper

What to do

  • Take a strip of paper and fold it in half end-to-end and make a crease. Open it out and then re-fold to make the crease into a right angle on the inside.
  • Take a new strip of paper. Fold it in half and then fold it in half again, putting the left end over the right end each time. Open it up, and then adjust each crease to be a right angle.
  • Take a third strip of paper. Fold it in half three times being careful to put the left end over the right each time. Lie it on its side and adjust the creases to be right angles, as before. Follow this pattern to make a 4- and a 5-fold shape.
  • Make two 4-fold shapes and see if you can make the 5-fold shape out of them. Can you see a pattern in these shapes?
  • See if you can make a 6-fold shape by arranging 5-fold shapes.

What’s happening?

The shapes you’ve been making are called dragon curves. You can make a bigger dragon curve by starting with a larger piece of paper and folding it in half more times. However after about five folds, it becomes difficult to fold the paper in half.

You can also combine two smaller dragon curves to make a larger one. To do this, you’ll need to put the correct ends together so they meet at a right angle. There are a few different ways you could do this, but only one of them is correct – to check, compare it to a single folded curve.

Although we’re calling all these shapes “dragon curves”, there’s actually only one shape that’s truly a dragon curve. The true dragon curve is the shape you get from an infinite number of folds.

Working mathematically

Because you’re folding in half every time, the distance between creases is always the same on a strip. The angle of the fold is also the same. The only difference is whether it bends left or right. So we can describe a dragon curve by listing the directions of the folds.

The first dragon curve has only one crease – we can assume it’s a right. When you fold this in half again, you add two creases – a right (R) before the original crease, and a left (L) after it. So your paper has three creases – R, R, L. A third fold creates four new creases. The sequence becomes R, R, L, R, R, L, L.

There are many different ways of making the next list of creases from the previous one. Here are two of them:

Write the previous sequence with gaps in between the letters. Then put R in front of this, L in the first gap, R in the second gap, L in the third gap. Keep adding R, L, R, L in the gaps until you get to the end. You should end up putting an L after the last letter. So the third fold would be R, R, L, R, R, L, L.

Write the whole of the previous sequence. Then write R. Now, write the previous sequence, only back to front (so write the last letter first) and with R and L switched. So the third fold would be R, R, L, R, R, L, L. (Note that R, R, L back to front and with the letters switched is R, L, L.)

These two completely different techniques give exactly the same answer, as long as you start with a dragon curve!

dragon1.1

Fold a strip in half, then in half again. Then unfold so each crease is a right angle.

dragon2.1

A 4-fold dragon curve.

dragon3.1

You can combine two 4-fold curves to make a 5-fold one.

dragon4.1

You might need to use sticky tape to hold bigger dragon curves in shape.

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