Self-Similar Structure

Self-Similar Structure is the nested, scale-repeating form that recursive emergence produces, in which each generation magnifies structural depth while keeping the same proportions, characterized by a non-integer fractal dimension. The book develops it within recursive scaling: wavelength contraction under the Wavelength Scaling Law, λ_n = λ_0/n, and π-based exponential growth of emergence arcs, EA_n = EA_0 × π^(n/2) × Φ(n), extend the arcs fractally across generations. Emergence therefore does not grow linearly but through nested self-similarity, and with π as the intrinsic scaling constant the dimensionality settles between one and two dimensions. The Fractal Dimension characterizes it: D_f = log(π)/log(2) ≈ 1.65, where D_f is the fractal dimension [∅], log the logarithm function [∅], π the mathematical constant pi [∅] and 2 the binary base [∅]. The result is dimensionless: [∅] ÷ [∅] = [∅], valid. The book reads this non-integer dimension as the mark of the self-similar structure of recursive transitions, with π creating natural fractals. It connects to Fractal Dimension, BPT Recursive Scaling, Wavelength Scaling Law, Emergence Arc Function and Recursive Density Evolution.