## The Dragon Curve Explained

## Try It Yourself

Are you bored, or dying to do something tedious? Do you need a geometry project that will keep your students busy all afternoon? Try drawing a giant dragon curve. No computer necessary!

**Supplies required:**

*sharp*colored pencils,- sturdy eraser,
- ruler (optional if you use graph paper),
- several pieces of plain or graph paper,
- clear tape, for sticking pages together to make an even bigger dragon,
- a light table (or tape your paper to the window for tracing),
- and a large dose of OCD-quality patience.

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I often find it fun to make a dragon curve, by taking a long strip of paper (like rolls for a cash register) and folding it in half repeatedly, then unfolding it and setting it so that each fold is a right angle.

There’s a lot of interesting math here, like why it can touch itself but never cross itself, so that it is actually possible to make it out of a sheet of paper like that! The right angles never steer you in an impossible direction “through” a previously placed bit of the strip of paper.

I like the paper-folding approach, too, except I find it easy to forget which way I’m going and flip my paper over, so the folds go different ways. I did draw the curve ages ago by repeated use of copy, paste, and group in Microsoft Word — which is about the limit of my techiness.

Is there a proof that the curve never crosses itself that can be understood by someone who doesn’t remember much beyond high school math?

I have kids tape one end of the paper to the table, and always fold toward the tape, before finally removing the tape and unfolding it.

The proof is understandable indeed, but hard to discover. Maybe the most straightforward path starts out by putting the curve on the coordinate plane, with turns each one unit, so that you can categorize points into even and odd based on whether the sum of the coordinates (or the total distance traveled to get to that point) is odd or even. Then take a look at whether you’re moving up or down, or right or left, when you arrive at that point, and look at whether you have a valley or mountain fold and how that affects which way you turn.

I did this by getting a sheet of paper with a big grid on it, at least 1cm per square, so I could easily label each vertex with all the relevant information and see what patterns started emerging. Once I made the diagram and saw the patterns, it wasn’t too huge of a leap to figure out why the folding pattern made the patterns turn out that way, and why the patterns guaranteed no self-intersection.

Thanks for the kind mention though I don’t know about “dying to do something tedious” ;) As you can see I did a number of posts on the “Fractals You Can Draw” topic and they should all be getting an update sometime soon (They’re some of the most popular posts on my blog!). In addition, I’m compiling those posts as a booklet as a companion piece to my forthcoming fractal book (Fractals – A Programmer’s Approach) soon to be available on Amazon and Bundle Dragon (okay shameless plug over). Thanks again for linking to the posts.

The “tedious” comment was yet another example of why I should stick to being the straight man. Trying to be wry just doesn’t work online. To tell the truth, even in real life it can be awfully trying…

Readers:Check out Ben’s fractal posts here.Don’t worry about it, wry is how I took it ;)

Very cool. Thanks for posting this on Math Monday Blog Hop!!