Fractal Dimension of Bread
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Abstract
The purpose of this lab is to determine the fractal dimension of white and wheat bread.
Introduction
Fractals are complex geometric figures made up of small scale and large scale structures that resemble one another. There are two types of fractals: geometric or regular and random. A regular fractal consists of large and small structures that look like exact duplicates of each other. In random fractals, there are large and small structures, but they do not look alike. Instead, the structures are mathematically related. Many random fractals are irregular patterns, which can be found in nature.
In mathematics, objects are classified by dimension. For instance, a line is classified as 1-dimensional, a plane is classified as 2-dimensional, and a cube is classified as 3-dimensional. However, fractals do not have a whole number dimension. Their dimensions are in between two integers. Bread, for example, is a cube with many spaces inside of it. So, when broken down, it is really just a combination of infinitesimally small planes. Hence, since it is not really a cube, and it is not really a plane, its fractal dimension must be in between two and three.
Procedures
Take a whole slice of white bread and make it into a tight ball, as spherical as possible. Next, find its mass using an electronic scale. Then using a caliper, measure the diameter of the ball. Take another slice of bread and divide it in half, and repeat the process of balling, weighing, and measuring the diameter. Continue this process using slices that are sized in quarters, eighths, sixteenths, thirty-seconds, sixty-fourths, one hundred twenty-eighths, and two hundred fifty-sixths. Do a second trial with these procedures to minimize variability. Then repeat the entire process, using the two trials with wheat bread.
Using the white bread data, graph the natural log of the diameter (x-axis) versus the natural log of the mass (y-axis). Find the slope of the linear regression line. This value is the fractal dimension of white bread.
This whole procedure was done twice. It was first done with Wonder Bread -- White Bread, and once again with Wonder Bread -- Whole Wheat Bread.
Analysis
White Bread
To see a larger image, click here
| Mass (grams) | Diameter (cm) | ln (mass) | ln (diameter) |
|---|---|---|---|
| 21.8 | 3.1 | 3.082 | 1.131 |
| 22.9 | 3.4 | 3.131 | 1.224 |
| 12.5 | 2.6 | 2.526 | 0.9555 |
| 11.8 | 2.5 | 2.468 | 0.9163 |
| 4.6 | 1.9 | 1.526 | 0.6419 |
| 5.0 | 2.1 | 1.609 | 0.7419 |
| 3.0 | 1.65 | 1.099 | 0.5008 |
| 3.1 | 1.6 | 1.131 | 0.4700 |
| 1.3 | 1.2 | 0.2624 | 0.1823 |
| 1.9 | 1.35 | 0.6419 | 0.3001 |
| 0.6 | 0.9 | -0.5108 | -0.1054 |
| 0.6 | 0.9 | -0.5108 | -0.1054 |
| 0.2 | 0.7 | -1.609 | -0.3567 |
| 0.3 | 0.7 | -1.204 | -0.3567 |
| 0.2 | 0.6 | -1.609 | -0.5108 |
| 0.2 | 0.6 | -1.609 | -0.5108 |
| 0.1 | 0.4 | -2.303 | -0.9163 |
| 0.1 | 0.4 | -2.303 | -0.9163 |
Whole Wheat
To see a larger image, click here.
| Mass (grams) | Diameter (cm) | ln (mass) | ln (diameter) |
|---|---|---|---|
| 28.5 | 3.7 | 3.350 | 1.308 |
| 25 | 3.6 | 3.219 | 1.281 |
| 12.4 | 2.8 | 2.518 | 1.030 |
| 12.2 | 2.6 | 2.501 | 0.956 |
| 5.2 | 1.9 | 1.649 | 0.642 |
| 6.7 | 2.1 | 1.902 | 0.742 |
| 3.1 | 1.7 | 1.131 | 0.531 |
| 2.4 | 1.6 | 0.876 | 0.470 |
| 1.8 | 1.4 | 0.588 | 0.336 |
| 1.4 | 1.1 | 0.336 | 0.095 |
| 0.7 | 0.9 | -0.357 | -0.105 |
| 0.6 | 0.8 | -0.511 | -0.223 |
| 0.3 | 0.7 | -1.204 | -0.357 |
| 0.3 | 0.7 | -1.204 | -0.357 |
| 0.2 | 0.6 | -1.609 | -0.511 |
| 0.1 | 0.5 | -2.303 | -0.693 |
| 0.1 | 0.4 | -2.303 | -0.916 |
| 0.1 | 0.5 | -2.303 | -0.693 |
Conclusion
The fractal dimension of white bread is 2.69, and the fractal dimension of wheat bread is also 2.69. We believe these are reasonable estimations because the fractal dimension of bread should be between 2 and 3.
Sources of Error
- One source of error that occurred during this experiment was the expansion of the bread balls after tightly packing them. The diameter should be measured soon after they stop expanding.
- Another source of error that occurred during this experiment was the inability to create a sphere. The diameter measured was therefore subject to variability depending on the orientation of the calipers.
- Another source of error is the use of crust while balling up the bread. Since the crust's structure is different from that of the bread, and since it balls up differently, the fractal dimension was really measured for the crust and bread, rather than the bread itself.
Samuel Bernard, Samantha Dong, Sabrina Khan, Anicia Ndabahaliye -- 2002
Chaos Project