WTSM 100L: Patterns in Nature

Introduction to Fractal Dimension: Length and area of the Koch curve

Length:

Stage 0: length of piece: 1

number of pieces: 1

total length:1x1=1

Stage 1: length of piece: 1/3

number of pieces: 4

total length: 4x1/3=4/3

Stage 2: length of piece:

number of pieces:

total length:

Stage 3: length of piece:

number of pieces:

total length:

Come up with an expression for the total length at stage N:

What is the length of the Koch curve (remember that the Koch curve is the limit of the figure as the number of stages, N, goes to infinity)?

Area:

We'll now try to compute the area of the Koch curve by covering it with isosceles triangles. (from FMN website)

First, cover the Koch curve with a single triangle.

This triangle has base length 1 and altitude sqrt(3)/6 (from the Pythagorean theorem),

hence area A₀ = sqrt(3)/12. Certainly, the area of the Koch curve is less than A₀.

Now replace this single triangle with four smaller triangles. For each small triangle the base has been shrunk by 1/3 and the altitude by 1/3, so the area of each is 1/9 that of the original triangle.

Thus these four triangles have total area A₁ = (sqrt(3)/12)*(4/9).

What is A₂?

As the process continues to infinity (getting closer and closer to the Koch curve), what limit does A approach?