Timothy Chartier is an Assistant Professor of Mathematics at Davidson College. Tim holds a bachelors degree in Applied Mathematics and a masters degree in Computational Mathematics from Western Michigan University. He received his doctorate in Applied Mathematics from the University of Colorado at Boulder. His research area is numerical analysis. In his time apart from academia, Tim enjoys the performing arts, mountain biking, nature walks and hikes, and spending time with his wife and two children.
This article develops algorithms to produce coastlines and mountains in two dimensions by adapting mathematical ideas related to the construction of such fractals as Koch's curve. A hand's on activity enables a reader to create a coastline with a rubberband, six-sided die, and thumb tacks. Java applications allow for exploration of these algorithms and the influence of their associated parameters. After discussing 2D fractal mountains, this article extends the 2D algorithm to produce 3D mountains. Finally, mathematical issues in random number generation are discussed. More specifically, linear congruential generators are considered and shown to be suitable as a random number generator for the 3D fractal landscape algorithm. The use of fractal landscapes in movies is also discussed.
All elements of this article (expository text, graphics, applets) are combined into a single Java Archive (JAR) file. You will need the Java Runtime Environment (version 1.5 or later) installed on your computer to view the article. Click on the link to install or update the JRE.