Fractal Mechanical Interpretation of the Photoelectric Effect
The photoelectric effect is the emission of an electron when a photon strikes a metal surface. Quantum mechanics explains this with the formula: 𝐸 = ℎ𝜈 − 𝑊
Spiral–Fractal Cell Theory
A new foundational biological theory that describes the living cell through spiral flows, fractal motifs, and multi-scale resonances.
Fractal Biology – The Multiscale Geometric Theory of Life
Fractal biology connects the following properties of living systems to a single principle: geometry, function, evolution, energy flow, information processing capacity.
Unification of Spiral–Fractal Mechanics with Quantum Field Theory
In this study, a spiral–fractal number system that goes beyond classical analysis and arithmetic is unified with quantum field theory. The aim is to transform the particle–wave duality into a motif–resonance duality and to redefine quantum dynamics on spiral coordinates.
Spiral Number System
Spiral numbers are a functional and fractal extension of classical complex numbers: S=a+bθ+if(θ)
Fractal Atom Theory: Spiral Flow, Motif, Orientation, Resonance, Scale, and Cycle-Based New Atomic Model
This study defines the atom not as a particle-based structure but as a multi-scale process formed by spiral-fractal flow modes. Proton, neutron, and electron correspond respectively to out-spiral (S⁺), equilibrium spiral (S⁰), and in-spiral (S⁻) flow modes. The atom’s geometry is expressed as a spiral-fractal manifold determined by motif functions, orientation field, resonance modes, scale fractality, and cycle periods. This approach transforms quantum mechanics into process physics, converts the periodic table into a motif-based fractal map, and redefines atomic interactions via spiral flow coherence.
Reconstruction of the Hodge Conjecture from the Perspective of Fractal Analysis
This study reformulates the classical Hodge Conjecture within the framework of Fractal Analysis. Fractal Analysis is a paradigm in which the topological structure of algebraic varieties is represented by multi-scale fractal resonance modes, and the algebraic subvarieties are represented by geometric motifs. This approach reinterprets Hodge decomposition as scale decomposition, harmonic forms as minimal energy resonances, and Hodge classes as rational-phase symmetric resonance modes.
Birch – Swinnerton – Dyer Conjecture
For an elliptic curve E/Q, the Birch – Swinnerton – Dyer Conjecture expresses the correspondence between two different worlds: – Arithmetic world: the structure of rational points on E(Q) → rank – Analytic world: the behavior of the function L(E,s) at s=1 → order of the zero
Fractal Analysis
This article defines a new mathematical paradigm that I call Fractal Analysis. Fractal Analysis is built upon three fundamental components in order to explain the multi-scale nature of algebraic, topological, and analytic structures: Fractal Motif, Fractal Resonance, and Fractal Flow. This triadic structure unifies geometric, topological, and dynamical properties—traditionally studied in separate disciplines of classical mathematics—within a single integrated framework. The paper formally presents the axiomatic foundation of Fractal Analysis, its structural components, and the relationships between these components. In addition, the relationship of Fractal Analysis with Hodge theory, algebraic geometry, and multi-scale analysis is discussed.
Fractal Arithmetic – A New Structure for Number Theory
This study presents a new framework called Fractal Arithmetic, which reformulates classical number theory through the concepts of fractal structure, motif, scale, direction, and resonance. Fractal Arithmetic treats natural numbers not merely as algebraic objects, but as fractal arithmetic wave functions. Each number is characterized by its prime factor structure, magnitude scale, directional flow within sequences, and resonance density within arithmetic patterns. Prime numbers are modeled in Fractal Arithmetic as resonance points with maximum motif purity, while composite numbers are modeled as structures carrying motif diffraction. Modular arithmetic is reinterpreted as resonance orbits. This paper presents the formal axiomatic foundation of Fractal Arithmetic and proposes a new structural/topological perspective on classical problems of number theory (especially prime distribution and modular structure).