The Physics Factbook™
Edited by Glenn Elert  Written by his students
An educational, Fair Use website
topic index  author index  special index
The purpose of this lab is to determine the fractal dimension of broccoli.
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:
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 selfsimilarity by chopping broccoli.

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.

The fractal dimension of a broccoli is 2.66.
Adam Kapelner, Vitaliy Schupack, Max Golomshtok, Johnny Alicea  2002
Chaos Project pages in The Physics Factbook™
Another quality webpage by Glenn Elert 
home  contact bent  chaos  eworld  facts  physics 
