Fractal Dimension and Applications in Landscape Ecology

Jiquan ChenUniversity of Toledo

Feb. 21, 2005

The Euclidean dimension of a point is zero, of a line segment is one, a square is two, and of a cube is three. In general, the fractal dimension is not an integer, but a fractional dimensional (i.e., the origin of the term fractal by Mandelbrot 1967)

Sierpinski Carpet generated by fractals

So what is the dimension of the Sierpinski triangle? How do we find the exponent in this case? For this, we need logarithms. Note that, for the square, we have N^2 self-similar pieces, each with magnification factor N. So we can write:

http://math.bu.edu/DYSYS/chaos-game/node6.html

Self-similarityOne of the basic properties of fractal images is the notion of self-similarity. This idea is easy to explain using the Sierpinski triangle. Note that S may be decomposed into 3 congruent figures, each of which is exactly 1/2 the size of S! See Figure 7. That is to say, if we magnify any of the 3 pieces of S shown in Figure 7 by a factor of 2, we obtain an exact replica of S. That is, S consists of 3 self-similar copies of itself, each with magnification factor 2.

Triadic Koch Island

DN N

r rn n

n n

ln ( / )

ln ( / )1

1

1) r1=1/2, N1=22) R2=1/4, N2=4

D=0

http://mathworld.wolfram.com/Fractal.html

A geometric shape is created following the same rules or by the same processes – inducing a self-similar structure

•Coastal lines•Stream networks•Number of peninsula along the Atlantic coast•Landscape structure•Movement of species•…

Wiens et al. 1997, Oikos 78: 257-264

DP

Aij

ij

2 * ln ( )

ln ( )

Vector-Based Raster-Based

DP

Aij

ij

2 0 2 5* ln ( . * )

ln ( )

Figure 11: The Sierpinski hexagon and pentagon

                                           

                                                                                       

                                           

n mice start at the corners of a regular n-gon of unit side length, each heading towards its closest neighboring mouse in a counterclockwise