Horizon Expansion Fractal
Creators: name: Scott Monroe, Scott
Publisher: name: Zenodo
Year: 2015
Subjects: subject: Fractals
subject: fractal geometry
subject: recursive fractal
subject: Mathematics
subject: FOS: Mathematics
subject: Algorithms
subject: Mathematical Concepts

description: Horizon Expansion Fractal: A Novel Mathematical Construct

Abstract

The Horizon Expansion Fractal (HEF) is a novel mathematical construct that challenges traditional fractal theory by increasing in complexity as one zooms out. Unlike classical fractals, which exhibit self-similarity at finer scales, HEF generates new structures dynamically at larger scales. This behavior suggests a potential new class of fractals governed by expansion rather than recursion.

This paper presents the mathematical formulation, key findings, and implications of HEF, along with conjectures for future research.

Introduction:

Fractals have traditionally been studied as self-similar structures that exhibit recursion and intricate details upon magnification. However, the Horizon Expansion Fractal (HEF) defies this pattern by introducing greater complexity when zooming out. This paper presents a rigorous analysis of the HEF, exploring its mathematical properties and implications.

Mathematical Formulation:

The iterative function governing the HEF is:

Z_(n+1) = Z_n^d + (S_n * C) / (|Z_n|^(p + beta * S_n) + epsilon)

Where:

- S_n = 1 / (|Z_n|^alpha + epsilon) ensures numerical stability.

- The escape condition |Z_n| > R determines whether a point diverges to infinity.

Key behaviors include:

- **Exponential complexity growth**: The fractal expands at a rate of 4^n.

- **Lack of classical self-similarity**: Each zoom level introduces novel structures rather than repeating patterns at different scales.

- **Escape-time dynamics**: The function T(C) follows an exponential growth pattern, leading to rapid divergence for many points.

Key FindingsThrough numerical simulations and theoretical analysis, the following observations have been made:

- HEF does not conform to traditional fractal self-similarity, as complexity increases with zooming out fractal systems.

- The fractal's Hausdorff dimension is dynamic and may vary depending on the zoom level.

- The escape function exhibits exponential behavior, distinguishing it from classical escape-time

- HEF's growth pattern suggests connections to cosmological expansion and recursive dynamical Potential Conjectures for Future Study.

Several questions remain open for future mathematical exploration:

- Can a formal proof be established that HEF does not exhibit classical self-similarity?

- What are the precise Hausdorff and fractal dimensions of HEF at various scales?

- Could HEF represent a new classification of expanding fractals with broader applications?

Conclusion:

The Horizon Expansion Fractal introduces a novel paradigm in fractal geometry, demonstrating an increase in complexity at larger scales rather than finer details at smaller scales. This uniqueproperty suggests a fundamentally new classification of fractals. Further mathematical andcomputational research is necessary to fully characterize its properties and potential applications incosmology, complexity theory, and dynamical systems.