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Conclusions

The main points of this paper are to establish:

  1. Natural fractals are mostly self-affine and require special descriptors: Many fractals found in nature, particularly in biological systems, are self-affine rather than self-similar. This distinction is crucial for accurately measuring and understanding their scaling properties.

  2. The spectral \( 1/f^\beta \) noises and fractal dimensions of organisms can be described as self-affine behavior: Fractal geometry in living systems, particularly in hierarchical branching networks like vascular systems, is driven by self-affine scaling, where different dimensions grow at different rates. This anisotropic scaling provides a more accurate description of these biological networks.

  3. The mechanism of all these processes is the maximization of entropy through mass transfer and enzyme kinetics: The fundamental processes behind these self-affine fractal patterns include fractional Brownian motion (fBm), Fickian diffusion, and convective mass transfer mechanisms, all of which work to maximize entropy production. These processes reflect the optimization of energy and resource distribution in biological systems.

We also reaffirm the predictions of Metabolic Scaling Theory (MST) that the geometry of organisms is fractal, rather than Euclidean. Although Mandelbrot (1982) suggested that self-affinity lacked a basic, universally valid reduction to principles, we argue that the fundamental mechanisms driving self-affine fractals in biological systems are well-explained by mass transfer processes and enzyme kinetics.

In ecology, the Zipf-Mandelbrot law has been used to describe relative abundance distributions (RAD) and species abundance distributions (SAD). Ecologists should be aware of the differences between self-similar and self-affine fractals and apply the appropriate methods for deriving affine-fractal dimensions. Additionally, fractal lacunarity can serve as a useful measure for quantifying the phase state of ecosystems, especially when analyzing forest disturbance history.

Further research into the fractal geometry of organisms and ecosystems may be necessary to fully understand whether entropy production is the driving force behind self-affine fractal dimensions. The unification of thermodynamic theories, metabolic scaling, and evolutionary principles could provide a broader understanding of fractal behavior in biological systems.