A piece from the gallery
The dragon curve
Take a strip of paper and fold it in half. Then again, and again. What shape do you think is hiding in those creases?
Fold, fold, fold, then open
Fold a strip of paper in half, bringing one end over onto the other. Fold it in half again the same way, and again. Then unfold it, and open every crease to a right angle, like the corner of a book standing half open.
That's the whole rule. Look at the strip from above and see what the creases make.
Each new fold doubles the creases and adds one more in the middle: 1, 3, 7, 15, 31. After ten folds there are 1,023 creases, and the strip has twisted into the dragon at the top of this page.
Why does it keep its shape? When you unfold the strip, the second half is a copy of the first half, turned around the middle crease. So every dragon is made of two smaller dragons, each made of two smaller ones, all the way down.
A line that never crosses itself
However many times you fold, the strip twists and turns and brushes past itself at the corners, but it never crosses its own path.
And there's more. Take four dragons, start them all at the same point, and turn each one a quarter turn from the last. They fit together like puzzle pieces, with no gaps between them and no overlaps.
Another fractal: like the Koch snowflake, the dragon is a fractal, a shape whose small parts look like the whole. And a turtle could draw it, since every step is just forward, then a turn to the left or the right.
Try it with a strip of paper
This one starts in your hands, so it works well in a classroom. You don't need to know any math first, just how to fold.
- Cut a long, narrow strip of paper, about 2 cm (1 inch) wide. A strip from the long side of a sheet works, and a roll of receipt paper is even better.
- Fold it in half, bringing the right end over onto the left end.
- Fold it in half again the same way, right end over onto left. Do this four times in all, or five if your paper is thin.
- Unfold it and open every crease to a right angle.
- Stand the strip on its edge on a table and look down from above. There's your dragon.
Before you unfold, ask the class to guess how many creases there will be. Then count them together, and come back to the playground to fold far more times than paper allows.
Fold your own
Fold as many times as you like, open the creases wider or narrower, or spin a few dragons together.
10 folds: 1,023 creases and 1,024 straight pieces.
The SVG stays sharp at any size: print it, cut it out, or open it in a drawing app. The PNG is 1200 × 630, the shape link previews use, with your dragon centered on the background you picked.
Things to try
- Start at 1 fold and add one at a time. Can you spot the last dragon inside each new one?
- Press "Watch it unfold" to see the folded-up strip open out into a dragon.
- Choose four dragons. Then move the crease angle away from 90°. Do they still fit together?
- Try opening the creases to 60° or 120° instead. What kind of creature appears then?
Where next?
Here, folding made a fractal. In the Koch snowflake, adding bumps makes one, and the turtle draws whole spirals from just forward and turn.