Fractal Dimension of Broccoli

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Abstract

The purpose of this lab is to determine the fractal dimension of broccoli.

Introduction

A fractal is an object that has a fractional spatial dimension. The dimension of a point is 0, a line is 1, a plane is 2 and a cube is 3. The dimension of a fractal that can be drawn from a line is between 1 and 2 and has infinite points and finite length. The dimension of a fractal created from a shape on a plane has a fractal dimension between 2 and 3 and has infinite length and finite area.

Broccoli is a fractal because it branches off into smaller and smaller pieces, which are similar in shape to the original. The fractional dimension can be calculated by:

equation

where N is the number of closed balls of diameter S needed to cover the object.

In this study, the fractal dimension of a broccoli is determined by calculating the slope of the graph of ln(N)/ln(S). Broccoli is especially interesting for its fractal properties. You can explore the concept of self-similarity by chopping broccoli.

Procedure

  1. Buy three bunches of broccoli.
  2. Eat two.
  3. Measure and record the size of the remaining bunch, using either a micrometer (for smaller pieces) or a ruler (for larger pieces)
  4. Break up the broccoli into smaller pieces and repeat step 3, three times in order to measure 3 levels of the fractal.
  5. Eat the remainder of broccoli

Analysis

Size Range (cm) Number of pieces
0.25–1 117
1–2 80
2–4 9
4–8 2
8–16 3

The Fractal dimension of broccoli is the magnitude of the slope of the graph of ln(number) vs. ln(size). According to the line of best fir, the slope is 1.5. But wait, that doesn't make sense since broccoli exists in three dimensions, so the fractal dimension must be somewhere near 3. Something must be wrong. What should we do?

Let's get rid of the first and last data points. They are outliers that don't fit with the remaining points. Aha! Now the slope is 2.66, which is closer to 3 and thus makes more sense.

Size Range (cm) Number of pieces
1–2 80
2–4 9
4–8 2

Conclusion

The fractal dimension of a broccoli is 2.66.

Sources of Error

  1. It was difficult to measure the diameter of irregular pieces. We were forced to average the major and minor axes.
  2. It was very difficult to determine where to split the broccoli as the size became smaller.
  3. It was ever more difficult to accurately split it, some of the buds shed off when it was being handled.

Adam Kapelner, Vitaliy Schupack, Max Golomshtok, Johnny Alicea -- 2002



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