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Published January 7, 2025 | Version v4
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"Harmonic Geometry" - A Multidimensional Fractal Approach

Description

This report presents a unified fractal framework combining number theory, fractal geometry, and cosmology. It embeds Fibonacci and Lucas sequences, prime-based sets (semiprimes), and non-trivial zeros of the Riemann zeta function into 5D/6D manifolds, revealing deep harmonic structures across numerical and cosmic scales. By examining how minor deviations of zeta zeros from the critical line disrupt fractal coherence, the study provides a geometry-based argument supporting a conditional Riemann Hypothesis. It further draws parallels with cosmic phenomena—such as entropy-driven cycles and galactic fractality—suggesting that universal harmonics underlie both discrete mathematics and large-scale cosmology.  
The document includes:  
A theoretical overview linking prime distributions and zeta zeros with fractal geometries.  
Methodological details for generating, normalizing, and clustering integer-based sequences.  
Evidence and results demonstrating fractal alignment and the high sensitivity to zeta-zero placement.  
A concluding discussion on broader impacts—philosophical, ethical, and scientific—alongside future directions and applications. 
A Python script, with a 5D version of the model, where you can explore the clusters in a simplistic way.  
This comprehensive report aims to stimulate interdisciplinary dialogue and to challenge conventional knowledge, proposing that harmonic fractal principles may connect micro-level prime expansions to macro-level cosmic evolutions. 

This version includes a supplementary document, sharing insights on philosophical reinterpretations and thoughts. This additional document gives an overview on paradoxes, reinterpretation of trinity, comparision with sacred geometry and thoughts on possible usages of the Riemann Zeta Function in cosmology and physics.

I'm a self taught independent researcher, therefore I would value your opinion on the topic. Engagement of the scientific community is needed for further development.

Files

Harmonic Geometry.pdf

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Additional details

Related works

Is supplemented by
10.5281/zenodo.14607642 (DOI)

Dates

Issued
2025-01-07
Multidimensional Fractal Geometry

References

  • 10.5281/zenodo.14607642