Fractal Dimension of the Coast of Maine

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Abstract

The purpose of this experiment is to determine the fractal dimension of the Boothbay, Maine coastline.

Introduction

Math is centered around the concept of dimensions. Points, which are of infinitesimal size, have zero dimension. Straight lines, which can be measured, are one dimensional. Flat shapes such as squares are considered to be two dimensional because they can be measured in two directions. Voluminous shapes such as cubes and spheres are three dimensional because they can be measured in three different directions.

A fractal is something that does not have an integral dimension value. An example of something that exists with a fractional dimension is a coastline. Consider the Atlantic shoreline of the United States. When viewed from outer space, the coastline from north to south seems to be about two to four thousand miles long depending on smoothness. However as you get closer and into higher detail, the coastline would appear to be much more intricate and therefore longer, even though the displacement between the north and south borders has not changed. Due to this varying change in detail, the coastline cannot necessarily be classified as one dimensional, nor two dimensional. It is somewhere in between.

Procedure

  1. Using a ruler, measure the length of the Boothbay, Maine coastline in increments of 10 cm. Try to measure along the length of the entire coastline including bays, inlets, and channels.
  2. Repeat the same process measuring the length of the coastline in increments of 8 cm.
  3. Line up the two tips of a compass with the ruler so that they are 5 cm apart. "Walk" the compass points along the coastline so that you measure the length of coastline in increments of 5 cm.
  4. Repeat with increments of 3 cm, 2 cm, and 1 cm.
  5. Have several members in your group count the number of steps the compass "walks" and the ruler makes at each increment of measurement (10 cm, 8 cm, 5 cm, 3 cm, 2 cm, and 1 cm).

Analysis

The fractal dimension of the Boothbay, Maine coastline is 1.27.
Click here to view the source data.

Conclusion

The fractional dimension of the Boothbay, Maine coastline is 1.27.

Sources of Error

  1. The tips of the compass slipped against the map surface while "walking."
  2. The distance between the ends of the compass varied slightly and uncontrollably as it was used to measure the coastline.

Jordan Levine, Leah Oppenheim, Jonathan Hamill, Jason Atkins, Anand Maharaj, Tony Schneider, Keith Singh -- 2002



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