Part IV: Biological Space-Filling and Thermodynamic Geometry¶
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Status: chapter draft
Part IV is an unreviewed chapter draft that synthesises published results and reports no new measurements. Its section headings are numbered 1.1 to 5.4 within the chapter.
Architectures of the Infinite: Self-Affine Fractals, Complex Dimensions, and the Thermodynamic Geometries of Biological Space-Filling
Organisms must capture diffuse resources while minimising the metabolic cost of the tissue that reaches them, and this chapter argues that evolution solves the problem with self-affine rather than self-similar architectures. It extends the West-Brown-Enquist model, whose radial and longitudinal branching ratios \( \beta_r = n^{-1/2} \) and \( \beta_l = n^{-1/3} \) differ, to the complex dimensions that arise from finite-scale fractality. Three space-filling mechanisms are examined: the four affine maps of the Barnsley fern as a compressed growth algorithm, diffusion-limited aggregation in lichens, algae and kelp with fractal dimensions near 1.6 to 1.7, and competitive canopy packing through crown shyness, weighted Voronoi tessellations and Apollonian circle packing. Log-periodic oscillations in growth data are read as signatures of discrete scale invariance, leading to the proposal that organisms act as zeta-function machines whose stable forms are eigenstates of their own spectral geometry.