ABSTRACT
Obviously, if m approaches to infi nity, both the number of fractal copies and the length of fractal boundary will diverge, while the area of fractal body will tend to zero.
Where the growing process is concerned, the scaling relations between fractal-part number and boundary length can be derived by taking the logarithm of equations (10) and (12) and by eliminating m-1, and the result is
a mm LLN =∝ . (16)
The scaling exponent a=ln(5)/ln(5/3) ≈ 3.151. The scaling of fractal-part number vs boundary length is a positive allometry. Conversely, the scaling of fractal-boundary length vs fractal-part number is a negative allometry. The allometric scaling relation between the fractal boundary length and fractal-part area is in the following form
b mm AAL =∝ . (17)
The scaling exponent b=ln(5/3)/ln(5/9) ≈ -0.869. The scaling phenomenon with an exponent b<0 is so-called inverse allometric growth.
