fig. 4: For the Koch curve = 3
3
= 3
A more robust method for calculating dimension of a fractal is the “box-counting” method. As the name implies you count how many squares of differing sizes are required to cover a fractal pattern. By using the Koch curve again, [ fig. 5 ], we can see that if you start by trying to cover the curve with squares 1/3 the width of the fractal 3 squares are needed, shrink the squares by one 1/3 again to 1/9 th of the original shape and twelve boxes are needed. This is because as the size of the squares decreases, the squares have to rotate to the shape of the pattern. As the squares get smaller and smaller they become indistinguishable from the fractal they are trying to cover.
is the width of the th square ) is the number of squares of size
Where and (
that are needed to cover the fractal
r 1
= 1/3, N(r 1
) = 3
r 2
= 1/9, N(r 2
) = 12
fig. 5: The Box-Counting Method
Soon a pattern emerges,
is the fractal dimension according to the box counting method, where
is defined as:
(
)
= lim →0
1
(
)
Where lim is the limit approached by the ratio of logarithms as the ratio r approaches 0. The answer as with the similarity function is log4/log3 = 1.2618... . Mandelbrot showed that d s =d b in his 1975 paper. However, the box counting method can be used on a much wider range of fractals such as the self-affine fractals of financial charts.
Fractals as Mathematical Curiosities
Fractals are most definitely curiosities of pure mathematics. They pop up all over the place, and are an interesting way to look at iterative functions continued to infinity. One of the most notable appearances of fractals in pure mathematics is the Sierpinski Triangle which turns up in Pascal’s Triangle. Pascal’s Triangle is a triangular array of addition; each number in the triangle is the sum of
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