ε = 20
4
Fractal with Grid Overlay
Current Box Size:
ε = 20
Boxes Containing Fractal:
N(ε) = 0
Dimension Estimate:
D = —
Log-Log Plot: log(N(ε)) vs log(1/ε)
Slope (Dimension D):
—
R²:
—
Theoretical D:
—
Box Counting Data
| ε (Box Size) | N(ε) (Count) | log(1/ε) | log(N(ε)) | log(N)/log(1/ε) |
|---|
Box Counting Algorithm
1
Choose Box Size
Select a box size ε to create a grid overlay
2
Overlay Grid
Cover the fractal with a grid of ε×ε boxes
3
Count Boxes
Count N(ε): boxes that contain any part of the fractal
4
Record Point
Plot (log(1/ε), log N(ε)) on log-log graph
5
Repeat
Repeat for different ε values
6
Fit Line
Linear regression slope = fractal dimension D
Mathematical Foundation
The fractal dimension D is calculated as the limit of the ratio of logarithms as ε approaches zero:
- ε (epsilon): Box size
- N(ε): Number of boxes containing fractal parts
- D: Fractal dimension (slope in log-log plot)
Practice Problems
Question 1: Dimension Prediction
Before counting, predict the fractal dimension of a Sierpinski triangle. Hint: Each iteration divides into 3 copies scaled by 1/2.
Question 2: Box Counting Practice
For a Koch curve with ε = 1/3 of the total length, how many boxes are needed? What about ε = 1/9?