\approx 0.6309 $$ For $$ R = 9 $$, $$ N = 4 $$: $$ D = \frac{\log(4)}{\log(9)} = \frac{2\log(2)}{2\log(3)} = \frac{\log(2)}{\log(3)} \approx 0.6309 $$ ### 5. Fractal Dimension of the Cantor Set The fractal dimension remains constant regardless of the scaling factor $$ R $$, indicating that the Cantor set has a fractal dimension $$ D \approx 0.6309 $$. ### Why is the Fractal Dimension Less Than 1? The fractal dimension is less than 1 because the Cantor set is not a continuous line. It is a set of points that becomes increasingly sparse as more iterations are applied. The dimension being less than 1 reflects the fact that the Cantor set occupies less "space" than a one-dimensional line but more than a zero-dimensional point set. --- Would you like further details or have any questions? Here are five related questions you might find interesting: 1. How is the Cantor set an example of a self-similar fractal? 2. What would happen if you continued the process of creating the Cantor set infinitely? 3. Can the concept of fractal dimensions be applied to other fractals like the Sierpinski triangle? 4. What is the Hausdorff dimension, and how does it relate to the fractal dimension? 5. How does the Cantor set relate to concepts in topology and measure theory? **Tip:** When calculating the fractal dimension, logarithms provide a key insight into how self-similar structures scale. retry