Expanding Zoom-Out Recursive Fractal Algorithm
Creators: name: Scott Monroe
Publisher: name: Zenodo
Year: 2025
Subjects: subject: Quantum physics
subject: Cosmology
subject: Loop Quantum Gravity
subject: Universal Expansion

description: Expanding Zoom-Out Recursive Fractal Algorithm and Its Cosmological Implications:

Overview:

This fractal algorithm models an expanding, zoom-out recursive fractal, governed by cosmological parameters such as the Hubble parameter (H(t)), an exponentially decaying rotation function (B(t)), and an escape radius tied to cosmic expansion. The structure grows recursively in three-dimensional space, incorporating rotation, scaling, and time-dependent transformations.

Key Components and Their Cosmological Significance

 1. Scaling Function:R_n = R_0 * (H_0 / H(t))^n      The fractal’s scale evolves with the Hubble parameter, mimicking the universe’s expansion. 2. Beta Function (Rotation as a Function of Cosmic Time):B(t) = B_0 * exp(-t / tau)      Rotation decreases over time, similar to damping of primordial perturbations or anisotropy reduction in large-scale cosmic structures. 3. Escape Radius Dependent on Hubble Parameter:R_esc(t) = c / H(t)      Defines a cosmic boundary akin to the observable universe’s limit, suggesting fractal growth is constrained by cosmic expansion. 4. Recursive Growth Rule in 3D Space:X_{n+1} = X_n + f(R_n, B_n, t)Y_{n+1} = Y_n + g(R_n, B_n, t)Z_{n+1} = Z_n + h(R_n, B_n, t)      Introduces recursive spatial transformations, resembling the hierarchical formation of cosmic structures. 5. Recursive Fractal Expansion Law:Z_{n+1} = Z_n + (α_n * Z_n) + (β_n * R_n) + γ_n      •   α_n: Time-dependent expansion coefficient (linked to cosmic evolution).      β_n: Scaling factor governed by exponential decay, similar to dark energy-dominated expansion.      γ_n: Additional transformation (e.g., quantum fluctuations or perturbations). 6. Transformation Functions (Spherical Evolution):f(R, B, t) = R * cos(B) * sin(t)g(R, B, t) = R * sin(B) * cos(t)h(R, B, t) = R * cos(B) * cos(t)      Suggests a spherical recursive pattern, resembling cosmological shell structures in galaxy distributions. 7. Recursive Tiling and Expansion Function:P_n = P_{n-1} + Rotation(B) * Scale(R) * Direction_Vector      Ensures self-similar hierarchical expansion, mirroring:         Hierarchical clustering of galaxies         Fractal-like matter distribution in the universe         Inflationary conditions for large-scale structure

Cosmological Implications: 1. Mimicking Cosmic Expansion:      The fractal’s growth follows the Hubble parameter, reflecting how cosmic structures evolve. 2. Fractal Nature of the Universe:      The hierarchical, self-similar nature aligns with real-world cosmic structures, supporting the idea that the universe has fractal-like properties at large scales. 3. Quantum Corrections and Perturbations:      The term γ_n introduces stochastic or quantum fluctuations, similar to early-universe density perturbations that seed galaxies. 4. Constraints from the Hubble Sphere:      The escape radius (R_esc(t) = c / H(t)) introduces a natural cosmic boundary, making the model a useful tool for simulating the observable universe’s limits.

Conclusion:

This expanding zoom-out recursive fractal algorithm encodes cosmological expansion, hierarchical structure formation, and rotational evolution into a mathematical framework. It aligns with real-world astrophysical phenomena, from inflationary growth to galaxy distribution. With further refinement, it could serve as a computational model for simulating large-scale cosmic structure evolution.

Expanding Zoom-Out Recursive Fractal Algorithm and its Cosmological Implications © 2025 by Scott Monroe is licensed under CC BY-NC-ND 4.0