Fractal dimension
The dimension of a self-similar figure, counting boxes at the levels of a construction, and Moran's equation for copies of different sizes — the Koch snowflake, the Sierpiński carpet and triangle.
How to solve it
A segment halved fits into itself $2$ times, a square $4 = 2^2$ times, a cube $8 = 2^3$ times. The exponent is the dimension. For fractals it comes out fractional: the Sierpiński triangle is $3$ copies at scale $\frac{1}{2}$, and $2^D = 3$ gives $D \approx 1.58$.
Step by step
- A self-similar figure of $N$ copies shrunk $k$ times: $k^D = N$, that is $D = \frac{\ln N}{\ln k}$.
- Box counting: at each level of the construction the number of boxes multiplies by $N$ and their size divides by $k$.
- Copies of different sizes: solve Moran's equation $\sum r_i^D = 1$, where $r_i$ are the scale factors. A substitution $u = r^D$ often helps.
- A check: the dimension should lie between that of the “skeleton” and of the space, for example between $1$ and $2$ for figures in the plane.
Common mistakes
- Dividing $N$ by $k$ instead of their logarithms: $D = \frac{\ln N}{\ln k}$, not $\frac{N}{k}$.
- Confusing the shrink factor $k$ with the scale $\frac{1}{k}$ and getting a negative dimension.
- Forgetting in box counting that the first level already has $N$ boxes, not one.
- Averaging the scale factors for copies of different sizes instead of solving Moran's equation.