Fractal Dimension and Applications in Landscape Ecology Jiquan Chen University of Toledo
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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)
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Sierpinski Carpet generated by fractals
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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
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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.
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Triadic Koch Island
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DN N
r rn n
n n
ln ( / )
ln ( / )1
1
1) r1=1/2, N1=22) R2=1/4, N2=4
D=0
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http://mathworld.wolfram.com/Fractal.html
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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•…
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Wiens et al. 1997, Oikos 78: 257-264
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DP
Aij
ij
2 * ln ( )
ln ( )
Vector-Based Raster-Based
DP
Aij
ij
2 0 2 5* ln ( . * )
ln ( )
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Figure 11: The Sierpinski hexagon and pentagon
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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
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