CONTENTS: Fractal Taylor Series Visual Fractal Taylor In this diagram, the fractal extension of the classical Taylor expansion is visualized, where derivative terms become scale-dependent through self-similar modulations. Fractal Laplace Transform Visual This graph shows the fractal extension of the classical Laplace transform: damping behavior on the amplitude axis and self-similar resonances on the frequency
Fractal series expansions are the redefined forms of classical Taylor, Maclaurin, and Fourier series using the principle of self-similarity. The aim here is to capture not only the local behavior of functions but also their fractal resonances that repeat at every scale.
Fractal Beginning Axiom System 1. Beginning Constant Axiom ∀𝑋 ∈ 𝒰, ∃! 𝐵(𝑋) Every system has a singular, unmultipliable, and irreducible beginning. 2. Reduction Axiom 𝑥/0 = 1 ⇒ 𝑥 ↦ 𝐵 Every mathematical expression is reduced to the beginning. 3. Fractal Evolution Axiom 𝐵(𝑋) ⇒ {𝑌1, 𝑌2, … , 𝑌∞} The beginning is singular,
As one of the most influential thinkers of the 20th century, Jean-Paul Sartre is known for his existentialist philosophy that centers on human freedom and responsibility. In his analysis, the individual is “condemned to be free”; meaning they cannot escape making choices and must bear the consequences of these choices.
In quantum fractal analysis, potential functions are the extension of classical quantum potential energy with fractal scale dependence. The aim is to model energy resonances at both micro and macro levels by analyzing the probability waves of particles within a fractal space-time structure.
While defined by self-similarity and scale invariance in classical mathematics, the quantum fractal exponential function combines this structure with quantum wave functions, revealing fractal resonance in probability distributions. The side-by-side graphs in the visual show a comparative view of the deterministic repetition of the classical fractal exponential function and the wave-particle interactive, luminous fractal structure of its quantum version.
7- Let’s expand the fractal analysis chain with fractal probability distributions (𝑷𝒇). This is a motif-repeating, multi-scale version of classical probability theory and provides entirely new definitions for uncertainty, risk, and variational systems. Classical Probability Distribution The classical probability density for a random variable 𝑋: 𝑃(𝑥) ≥ 0, ∫-∞∞ 𝑃(𝑥) 𝑑𝑥 = 1 It is a
The logarithmic multifractal model describes the structure of the universe not as uniform, but in the form of multi-scale spiral–fractal rings.
To establish fractal theology, we can build a structure inspired by mathematical self-similarity but carrying the meaning to the metaphysical and sacred plane. I present a motif-oriented, hierarchical, and universal framework:
Fractal transformation is the process that mathematically reveals how a motif repeats itself at different scales. Let me explain this in detail: