Imatge de la portada

Guardat en:
Dades bibliogràfiques
Autor principal: Bolduc, Son David
Format: Recurso digital
Idioma:anglès
Publicat: Zenodo 2025
Matèries:
causal structure of φ
spectral theory
optimization
biological morphogenesis
phyllotaxis
golden spiral
Fractals
fractals
E8 spectrum
quasicrystals
Dynamical systems
dynamical systems
eigenvalues
golden triangles
pentagons
golden rectangles
golden geometry
optimal irrationality
continued fractions
Fibonacci numbers
harmonic division
recursive growth
autosimilarity
self-similarity
phi constant
golden ratio
causality theory
Accés en línia:https://doi.org/10.5281/zenodo.17834405
Etiquetes: Afegir etiqueta
Sense etiquetes, Sigues el primer a etiquetar aquest registre!
Taula de continguts:
  • <p><em>The Causal Trinity of φ</em> presents a complete mathematical and causal analysis of the golden ratio,<br><span><span>ϕ=1+52\phi = \frac{1+\sqrt{5}}{2}</span><span><span><span>ϕ</span><span>=</span></span><span><span><span><span><span><span><span><span>2</span></span><span><span>1<span>+</span><span><span>5</span><span></span></span></span></span></span><span></span></span></span></span></span></span></span></span>, across all major scientific and structural domains. The document unifies the appearance of φ in geometry, algebra, number theory, continued fractions, eigenvalue problems, differential equations, fractals, dynamical systems, quasicrystals, spectral theory, optimization, and biological morphogenesis.</p> <p>The paper establishes φ as the <strong>unique fixed point of self-similar transformation</strong> <span><span>x=1+1xx = 1 + \tfrac{1}{x}</span><span><span><span>x</span><span>=</span></span><span><span>1</span><span>+</span></span><span><span><span><span><span><span><span><span><span>x</span></span></span><span><span>1</span></span></span><span></span></span></span></span></span></span></span></span>, and therefore the universal metric of recursive growth, harmonic division, extremal smoothness, and autosimilar structure. It catalogues all formulas involving φ — golden rectangles, pentagons, Binet’s formula, Fibonacci limits, eigenvalues, growth ODEs, logarithmic identities, irrationality bounds, golden spirals, quasicrystal spectra, E8 ratios, and phyllotaxis laws — and organizes them into a unified causal framework.</p> <p>Across mathematics, physics, and biology, φ appears wherever systems maintain identity under scaling, minimize interference, or optimize recursive expansion. The document concludes by formalizing φ as the <strong>causal metric of autosimilarity</strong>, governing all structures whose form is preserved under self-similar transformation.</p>

Ítems similars