Chapter 46
Fractals and Self-Similarity Recursion

Flow Algebra

flow "Fractals and Self-Similarity Recursion"

# ---------- CONCEPT ----------
# Fractals are geometries of infinite self-similarity: patterns that repeat at every scale.
# They emerge from recursive rules — simple transformations applied iteratively, producing complex boundaries of finite area and infinite detail.
# Fractals unify geometry, dynamics, and dimension — structure generated by feedback.

# ---------- STATES ----------
state FractalSystem(F):
  IteratedFunctionSystem:
    description: "{f_i(x)} — set of contractive maps applied recursively to build self-similar sets."
  RecursionRule:
    description: "x_{n+1} = f(x_n) — deterministic or stochastic iteration defining the geometry."
  Attractor:
    description: "Invariant set A satisfying A = ⋃_i f_i(A) — the fixed boundary of recursive transformation."
  ScalingFactor:
    description: "r_i — contraction ratio for each map; determines geometric scaling."
  HausdorffDimension:
    description: "D defined by Σ_i r_i^D = 1 — measure of complexity between integer dimensions."
  SelfAffineFractal:
    description: "Different scaling in x, y, z directions (e.g., natural textures, turbulence)."
  RandomFractal:
    description: "Randomised iteration (e.g., Brownian path, perlin noise) — statistical self-similarity."

# ---------- PARAMETERS ----------
params:
  contraction_condition: "0 < r_i < 1"
  initial_seed: "x_0 or A_0"
  recursion_depth: "N → ∞ for limit set"
  transform_type: "affine, nonlinear, complex"
  closure: true

# ---------- INVARIANTS ----------
invariants:
  - "Self-similarity: A = ⋃_i f_i(A)."
  - "Scaling law: N(r) ∝ r^{−D} — number of covering boxes vs. scale."
  - "Hausdorff dimension: Σ_i r_i^D = 1."
  - "Fractal measure: finite area/volume but infinite perimeter/detail."
  - "Power-law structure: quantities scale as S(λr) ∝ λ^α S(r)."
  - "Invariance under iteration: f_i ∘ f_j reproduces same structure."
  - "Statistical self-similarity for random fractals — same statistics under rescaling."
  - "Fractal dimension continuous but non-integer — D ∈ ℝ⁺ \ ℕ."

# ---------- TRANSFORMS ----------
transforms:
  BuildIFS({f_i},A₀,N):
    rule: "Apply all maps iteratively N times; limit N→∞ gives attractor A."
  ComputeHausdorffDimension({r_i}):
    rule: "Solve Σ_i r_i^D = 1 for D."
  BoxCounting(A,ε):
    rule: "Count boxes N(ε) covering A; slope of log N vs log(1/ε) gives D_B."
  RandomIteration({f_i},p_i):
    rule: "At each step choose f_i with probability p_i; build stochastic attractor."
  ComplexPlaneIteration(f,z₀):
    rule: "Generate Julia or Mandelbrot sets by z_{n+1} = f(z_n)."
  ZoomRecursion(A,scale):
    rule: "Magnify region by factor s; compare local and global similarity."
  FourierDimension(signal):
    rule: "Analyse power spectrum slope β → D ≈ (5−β)/2 for self-affine processes."
  FractalInterpolation(points):
    rule: "Construct continuous curve with prescribed fractal dimension via affine maps."

# ---------- MONITORS ----------
monitors:
  ContractionCheck:
    measure: "max |r_i|"
    condition: "< 1"
    meaning: "Guarantees convergence of recursive construction."
  DimensionEstimate:
    measure: "log N(ε) / log(1/ε)"
    meaning: "Estimate fractal dimension numerically."
  EnergyScaling:
    measure: "E(k) ∝ k^{−α}"
    meaning: "Power-law scaling in physical or signal fractals."
  SelfSimilarityError:
    measure: "‖A − ⋃ f_i(A)‖"
    target: 0
  RandomSeedDependence:
    meaning: "Statistical consistency under different seeds."
  IterationDepth:
    meaning: "Monitor convergence as N→∞."

# ---------- EQUATIONS ----------
equations:
  - "A = \\bigcup_i f_i(A)"
  - "x_{n+1} = f(x_n)"
  - "N(ε) ∝ ε^{−D}"
  - "Σ_i r_i^D = 1"
  - "D_B = \\lim_{ε→0} \\frac{\\log N(ε)}{\\log(1/ε)}"
  - "Power law: S(λr) = λ^α S(r)"
  - "Mandelbrot: z_{n+1} = z_n^2 + c"
  - "Julia: z_{n+1} = f(z_n)"
  - "Random fractal: choose f_i with prob p_i, iterate to stationarity"
  - "Fourier slope β ↔ D ≈ (5−β)/2 (for 1D signals)"

# ---------- NOTES ----------
notes:
  - "Fractals arise from recursion — iteration transforms simplicity into complexity through feedback."
  - "Exact self-similarity yields deterministic fractals (e.g., Koch, Sierpiński, Cantor); approximate self-similarity yields natural textures."
  - "Fractal dimension quantifies how space is filled — measuring coherence between scaling and structure."
  - "Power-law scaling links fractals to turbulence, noise, and critical phenomena — universality of structure across scale."
  - "Complex dynamics (Julia/Mandelbrot sets) reveal fractals as boundaries of stability in nonlinear maps."
  - "In Flow Algebra, fractals represent recursive coherence — form sustained through iteration. Each scale mirrors the next, a boundary folded into itself: infinite structure generated by finite rule."