{"id":472,"date":"2007-07-17T13:17:49","date_gmt":"2007-07-17T13:17:49","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2007\/07\/17\/fractal-curves-and-coastlines\/"},"modified":"2007-07-17T13:17:49","modified_gmt":"2007-07-17T13:17:49","slug":"fractal-curves-and-coastlines","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2007\/07\/17\/fractal-curves-and-coastlines\/","title":{"rendered":"Fractal Curves and Coastlines"},"content":{"rendered":"

\"Von_Koch_curve.gif\"<\/p>\n

I just finally got my copy of Mandelbrot’s book on fractals. In his discussion of curve fractals (that is, fractals formed from an unbroken line, isomorphic to the interval (0,1)), he describes them in terms of shorelines rather than borders. I’ve got to admit
\nthat his metaphor is better than mine, and I’ll adopt it for this post.<\/p>\n

In my last post<\/a>, I discussed the idea of how a border (or, better, a shoreline) has
\na kind of fractal structure. It’s jagged, and the jags themselves have jagged edges, and *those* jags have jagged edges, and so on. Today, I’m going to show a bit of how to
\ngenerate curve fractals with that kind of structure.<\/p>\n


\n\"koch.jpg\"<\/p>\n

The canonical example of a fractal curve is the Koch curve. To generate the canonical Koch curve,
\nyou take a line segment, divide it into three sections, and replace the middle with
\nthe upper two edges of an equilateral triangle. Then you repeat that process for each of
\nthe straight edges in the resulting figure. Over to the side are illustrations of the Koch
\ncurve after 1, 2, 3, and 4 iterations.<\/p>\n

This kind of curve obviously isn’t ideal as a model of something like a coastline. It’s perfectly regular, which natural structures like coastlines aren’t. But it’s a starting point; we can eventually work irregularity and randomness into these kinds of curves. But
\nfor now, we’ll stick with the simple ones.<\/p>\n

The Koch curve can be described in terms of a simple grammar, which guides
\na recursive rewrite system. (In fact, that’s how I implemented the code to generate
\nthe diagrams for this post: I’ve got a simple rewrite system, and I’m feeding the
\nresults into a turtle drawing program which traces out the curve.)<\/p>\n

In a rewrite system for fractal curves, you start with a grammar symbol called
\nthe initiator<\/em>, and then specify a rule system called the generator<\/em>. The generator is a set of replacement rules. Each iteration of the algorithm, you scan through the current state of the curve (starting with the initiator on step 0), and
\neach time that you encounter a symbol which is the left hand side of a rule in the generator, you replace that symbol with the right hand side of the corresponding rule.<\/p>\n

For example, the Koch curve is specified by:<\/p>\n