Entertaining Geometry: Harmonic Space and Boxology

 

Entertaining Geometry: Harmonic Space and Boxology

by Vitaliy Ishchenko, LII Lab, Kiev – buchalive@gmail.com

Abstract
Modern science aspires to achieve perfect precision in measurements and metric systems. Paradoxically, in an age of high-speed data exchange and advanced computation, we continue to rely on Euclidean geometry—a framework that is fundamentally inaccurate for describing natural, multidimensional, and quantum phenomena.
This article introduces Harmonic (Curvilinear) Geometry, a new approach that seeks to overcome the limitations of linear approximations. Built upon petal-like structures and “woviks”, derived from curved chords intuitively connected to a circle’s center and diameter, this geometry aims to provide natural precision, eliminating long decimal coefficients and fractional approximations.
We present a conceptual framework called Boxology, which evolves from basic spatial elements to a quantum representation of space, bridging ancient intuition, modern mathematics, and quantum science.

Introduction: The Limitations of Euclidean Geometry
Euclidean geometry emerged as a powerful tool for describing flat, static, and localized measurements.
However, when attempting to model curved motion, growth patterns, and multidimensional dynamics, this system falls short. We encounter:
Infinite coefficients (e.g., π) leading to approximation errors;
Arithmetic fractions that accumulate inaccuracies;
Complexity that grows with attempts to correct these deviations.
Ultimately, we ask: “What is what, and what is not?”, yet the results are fragmented, burdened with generalized and imprecise formulas.

Curvilinear (Harmonic) Geometry
To transcend these limitations, we define a 2D curvilinear space, naturally expanding into 3D. Its properties:
Harmonic curves replace straight lines as fundamental metrics;
Whole-number proportions replace infinite decimal expansions;
Measurements emerge from curved chords linked to intuitive centers of circles and diameters.
This foundation allows a more precise and natural description of motion, energy flow, and spatial interactions—from planetary orbits to quantum fluctuations.

Boxology – A New System of Understanding
Boxology is structured into progressive chapters, each unveiling a deeper layer of harmonic geometry:
1. Point – Elementary unit of space.
2. Pixel – Bridge between continuous and digital space.
3. Line – Path of harmonic motion.
4. Tetractys – Sacred numerical triangle guiding proportions.
5. Spiral – Universal growth and motion code.
6. Area – Measuring surfaces in a curvilinear system.
7. Circle – Perfect equilibrium and resonance. Fractal World (book).
8. Yin-Yang – Duality and balanced forces.
9. Astral Lines – Nonlinear connections beyond coordinates.
10. Sphere – Natural enclosure of multidimensional space.
11. Torus – Circulation of energy in closed systems.
12. Encoding and Decoding – Information flow within harmonic geometry.
13. Geometric Experiment – Practical exploration of the new metrics.
14. Space-Time Continuum – Integrating geometry with temporal dimensions.
15. Numbers of the Tetractys and John Searle – Magical Ratios – Exploring balanced compositions of numbers in triangular, square, and circular forms, revealing harmonic proportions bridging ancient knowledge and modern curvilinear mathematics.
Early Chapters and Hidden FiguresThe early chapters may appear childlike at first glance. This is because they dive into lesser-known figures— petals and woviks—from which curvilinear space is constructed. Unlike straight lines, these forms are curved chordal elements, rooted in the circle’s natural geometry. They form the true foundation of harmonic

measurements and open a path to non-Euclidean, quantum-aware space.

From Boxology to Quantum Space
Step by step, Boxology moves beyond classical mathematics and physics, entering a quantum representation of space:
Measurements arise from harmonic resonance, not linear approximations;
True precision is achieved via balanced ratios, not extended decimal coefficients;
Space and time form a unified curvilinear continuum.
Through this system, science gains not only greater accuracy but also inner harmony, reshaping our understanding of geometry and bringing humanity closer to the true structure of the universe.
Would you like me to also design a visual abstract (diagram) for this article – showing the petals, woviks, curved chords, and how Boxology evolves into quantum space?