A fractal Christmas

College Park, MD (December 1, 2003) -- Take a close look at a Christmas tree this holiday season - it’s a terrific, natural example of a fractal.

In the case of the fir tree standing in a Christmas tree lot, each branch has structure that, when studied up close, resembles a whole tree, and on each branch there are a smaller, tree-like branches. Even the needles on the tiniest branch are arranged in patterns reminiscent of the structure of the entire fir tree.

The tree's structure can be described by a mathematical equation. The equation and the tree are fractals, patterns that repeat at various scales. Fractals also describe many real-world objects that do not have simple geometric shapes, such as clouds, mountains, turbulence, and coastlines.

Fractals will look the same regardless of the scale shown. For instance, an ice crystal viewed at full size and a detail of one small branch of the ice crystal have a similar structure -- it's often difficult to tell which is the magnified image.

Mathematicians and scientists have identified numerous fractal structures in our daily life. The delicate shape of a snowflake, the patterns formed by highways, streets, and cul-de-sacs, even the intricate relationships among the notes and voices in a Bach Christmas Cantata can be describe by fractal relationships, at least to a certain extent.

Here are some fractals to ponder this holiday season.

• Snowflakes are among the most commonly cited fractals in nature. The Koch snowflake in particular is built with an algorithm that, at its second stage, resembles a Star of David.
  
  Examples and brief discussions of the Koch Snowflake:
  
  http://ecademy.agnesscott.edu/~lriddle/ifs/ksnow/ksnow.htm
  
  http://mathworld.wolfram.com/KochSnowflake.html
  
  
• The ebb and flow of traffic on the way to Grandma’s house appears to be fundamentally fractal in nature, according to recent research (M. Blue and B. W. Bush, Physical Review E, 67, 2003). And viewed from an airplane, the patterns of roadways have fractal structure that is reminiscent of the fractal layout of the veins in a tree leaf. This web page shows the results from a high-school-level experiment that measures a parameter known as the fractal dimension for several leaf types.
  
  
• A crumpled piece of wrapping paper is a fractal
  
  
• Music, including the work of Bach, is often fractal, and the math of fractals has been used to generate music directly.

For further information on fractals and the upcoming holiday season, contact:

H. E. “Gene” Stanley
Boston University
617-353-261

James Riordon
American Institute of Physics
301-209-3084