Fractal Dimension
Learning Outcomes
- Define and identify self-similarity in geometric shapes, plants, and geological formations
- Generate a fractal shape given an initiator and a generator
- Scale a geometric object by a specific scaling factor using the scaling dimension relation
- Determine the fractal dimension of a fractal object




Scaling-Dimension Relation
To scale a D-dimensional shape by a scaling factor S, the number of copies C of the original shape needed will be given by:[latex]\text{Copies}=\text{Scale}^{\text{Dimension}}[/latex], or [latex]C=S^{D}[/latex]
Example
Use the scaling-dimension relation to determine the dimension of the Sierpinski gasket. Suppose we define the original gasket to have side length 1. The larger gasket shown is twice as wide and twice as tall, so has been scaled by a factor of 2.
[latex]3={{2}^{D}}[/latex]
Use the exponent property of logs.[latex]\log(3)=\log\left({{2}^{D}}\right)[/latex]
Divide by log(2).[latex]\log(3)=D\log\left(2\right)[/latex]
The dimension of the gasket is about 1.585.[latex]D=\frac{\log\left(3\right)}{\log(2)}\approx1.585[/latex]
Scaling-Dimension Relation, to find Dimension
To find the dimension D of a fractal, determine the scaling factor S and the number of copies C of the original shape needed, then use the formula[latex]D=\frac{\log\left(C\right)}{\log(S)}[/latex]
Try It
Determine the fractal dimension of the fractal produced using the initiator and generator.
Answer:
Scaling the fractal by a factor of 3 requires 5 copies of the original.
[latex]D=\frac{\text{log}\left(5\right)}{\text{log}\left(3\right)}\approx1.465[/latex]
Licenses & Attributions
CC licensed content, Original
- Question ID 131177. Authored by: Lumen Learning. License: CC BY: Attribution.
- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
CC licensed content, Shared previously
- Fractal dimension . Authored by: OCLPhase2. License: CC BY: Attribution.