---
slug: oip-node-c10-scale-invariance-fractals-allometry
title: Node C10: Scale Invariance / Fractals / Allometry
canonical: https://miscsubjects.com/a/oip-node-c10-scale-invariance-fractals-allometry
---

# Node C10: Scale Invariance / Fractals / Allometry

C10 — Scale Invariance / Fractals / Allometry
{
  "id": "C10",
  "claim": "The same quantitative rule governs structure across many orders of magnitude; branching transport networks, coastlines, and metabolic rates follow power-law scaling with characteristic exponents.",
  "domain": ["mathematics", "condensed matter physics", "biology", "geography", "urban science", "cosmology"],
  "pattern": ["fractal", "scaling_law", "allometry", "power_law", "self_similarity"],
  "mechanism": "Fractals: objects with non-integer Hausdorff dimension, exhibiting self-similarity across scales. Allometry: metabolic rate B ~ M^(3/4) (Kleiber's law), explained by West-Brown-Enquist through optimal branching network geometry minimizing energy dissipation. Renormalization group: critical exponents are universal — same values across different microscopic Hamiltonians.",
  "scale": "molecular → cosmic",
  "claim_tier": "T1",
  "sources": [
    "Mandelbrot, B.B. (1982). The Fractal Geometry of Nature. W.H. Freeman.",
    "Kleiber, M. (1932). 'Body Size and Metabolism.' Hilgardia, 6(8), 315-353.",
    "West, G.B., Brown, J.H. & Enquist, B.J. (1997). 'A General Model for the Origin of Allometric Scaling Laws in Biology.' Science, 276, 122-126.",
    "Wilson, K.G. (1971). 'Renormalization Group and Critical Phenomena II.' Phys. Rev. B, 4(9), 3184-3205."
  ],
  "dual": "Characteristic-scale systems — objects with a single intrinsic scale (like a sphere of fixed radius).",
  "falsifier": "A branching transport network (circulatory, river, fungal) that violates the 3/4 metabolic scaling exponent or the fractal dimension predictions under controlled conditions.",
  "rival_frame": "Scaling laws are dimensional necessity, not deep structure. The 3/4 exponent emerges from geometric constraints (space-filling + minimal energy), not from a 'grain' of nature. Fractals are descriptive tools, not explanations — they say 'it looks similar at different scales,' not why.",
  "independence_check": "HIGH. Mandelbrot (mathematics, IBM, 1982) derived fractals from study of noise and cotton prices. Wilson (physics, Cornell, 1971) derived scaling from renormalization group in quantum field theory. WBE (biology, Santa Fe, 1997) derived allometry from optimal transport network theory. Kleiber (agricultural biology, Davis, 1932) found the 3/4 law empirically decades before theory. Four origins, same pattern: scale-independent rules.",
  "pattern_type": "mathematical",
  "maps_to_axiom": ["A7"]
}

---

## Corpus map
- Same node, other planes: [Encyclopedia C10](/a/convergence-encyclopedia-c10) · [Inventory invariant](/a/oip-invariant-6-36-scale-invariance-fractals-allometry)
- Edges touching C10: [convergence edge 4](/a/oip-convergence-edge-4) · [convergence edge 8](/a/oip-convergence-edge-8) · [disconfirming edge 5](/a/oip-disconfirming-edge-5)
- Catalogue hub: [Public Article](/a/oip-convergence-public-article) · [Schema](/a/oip-convergence-schema)
