Monday, May 24, 2010
Martin Gardner, 1914-2010: A Personal Remembrance
A dragon curve is a space-filling, self-similar curve that belongs to the family of shapes known as fractals, although like many other fractals it was discovered and analyzed before the term "fractal" was coined by Benoit Mandelbrot in 1975. It was discovered through a simple, recursive paper-folding procedure: take a strip of paper, fold it in half lengthwise, then unfold it so that the crease forms a right angle. The path that the paper makes when viewed on edge is a dragon curve of order 1. Now re-fold the paper, then fold it in half again. Open it up so that creases again make right angles. When viewed on edge it makes a dragon curve of order 2. Repeat: fold it in half, in half again, and in half yet again. Unfold it, and you have a dragon curve of order 3. Each curve of order n has the property that it never crosses itself, and is space-filling in the sense that multiple copies can tile together to fill the unit lattice. As the order increases, the curve takes on a vaguely dragonish shape: head, tail and two legs, hence the name. A dragon curve of order n is made up of two curves of order n-1 placed head to tail, which makes sense when one reflects on the paper-folding procedure that generates the curve.
The curve appealed to me, and still does, because (1) it's simple to understand and to draw on graph paper, (2) there are multiple ways to define it, hence multiple ways to generalize it, and (3) the plane-tiling aspects (if you consider each unit segment as the diagonal of a square and replace segments with squares) create visually appealing patterns. The paper-folding procedure originally used to define the curve can be replaced by sequences of 0s and 1s representing right and left turns in a turtle-geometric tracing of the curve. These binary sequences can then be generated different ways mathematically, and their properties investigated.
I used dragon curves as the topic of an independent study project in 1971 while a freshman at New College in Florida. This was well before Google and the internet, and my library research skills were limited to card catalogs, so to find out more about dragon curves I wrote directly to Martin Gardner. I still have his postcard reply: