Boxology. Vitaly Ischenko. Charter 7 - Sphere.

 Book “Boxology”

Chapter Seven. The Sphere

1. Introduction
The sphere in nature and thought has always been an image of completeness, harmony, and stability. In
Grigori Perelman’s proof of the Poincaré conjecture, we find confirmation: a closed three-dimensional space
without holes is equivalent to a sphere. This opens the possibility of a new geometric theory, where the
sphere—not the point or the line—becomes the elementary unit of world construction.
We will call this system Ishchenko’s Geometry of Spheres. Its axioms and methods form the core of a new
coordinate system that describes self-unfolding space.

2. The Self-Unfolding World
Space can be imagined as filled with a multitude of equal or nearly equal spheres that:
are held by internal and external forces (fields, frequencies, matter density);
possess a shell that maintains equilibrium;
contain a center, which at once may serve as the center of geometry, mass, and energy, while deviations
become criteria of a body’s state.
Such space has no absolute emptiness: it is formed of stable cells—spheres, whose unfolding sustains the
integrity of the Universe.

3. Natural Example: Water
Water molecules are a vivid model of spherical packing. Hydrogen bonds create relative stability of the
cluster, where each molecule tends toward sphere-like equilibrium.
Here, fluidity and elasticity unite: water demonstrates dynamic sphericity, which becomes a natural archetype
of the new geometry.

4. The Geometric Apparatus: Ishchenko’s Chords
Unlike classical geometry, where the radius serves as the basic measure of the sphere, boxology introduces
the principle of equatorial division:
Ishchenko’s chord = ⅓ the length of the sphere’s equatorial circle.
Ishchenko’s half-chord = ⅙ the length of the equator.
Thus:
6 half-chords = 3 chords = 1 equator.
The whole sphere is described as a harmonic system of divisions.
This shifts the focus:
The radius is no longer the primary measure.
What matters is the proportion of divisions, not absolute distance.
The sphere becomes rhythm, not merely a geometric body.

5. A New Axiomatics of Geometry
1. Space consists of spherical cells that unfold from within and mutually balance one another.
2. Measurement is carried out by chords and half-chords, not the radius: length is defined in relation to the
circumference.
(In the figure: a fractal completely filled with “half-chords,” serving as a graphic standard).
3. The binding energy of the sphere is proportional to the harmonic ratio of the chord’s length (three
chords = circumference) and half-chord to local figures—segments and shapes that arise in the
development of the field pattern.
4. The density of matter is expressed through the packing density of spheres and the ratio of their
equatorial divisions.
5. The unfolding of the world is the process of adding spheres to the system without violating global
closure (analogous to the three-dimensional Poincaré sphere).

6. Consequences for Science
Physics of matter: atomic and molecular structures can be described through chordal proportions rather than radial values.
Hydrodynamics: water flows become a clear model of spherical geometry.
Cosmology: the Universe can be understood as a self-unfolding system of spheres, closed in the sense of Poincaré.
New coordinate system: instead of three orthogonal axes—a grid of spherical cells, where motion is expressed as transition from sphere to sphere.

7. Conclusion

The sphere is not just a body of rotation but a universal cell of the world, which holds equilibrium of forces and energies. Ishchenko’s geometry of spheres sets forth a new coordinate system based on proportion, harmony, and density.
Boxology asserts: space is an orchestra of spheres, each sounding with its chord and half-chord, but together creating the stable symphony of the Universe.
Continuation of Chapter VII. The Sphere

8. From Point to Sphere: Local Forms as Foundation
In Boxology we begin from basic local forms:
1. The point — the primary center, the unit of being.
2. The line — the connection of points, expression of relation, motion, and extension.
3. The tetractys — a simple system of proportions (Plato), a harmonic structure of 10 points forming a
triangle and setting the first principles of proportion.
These local forms form the skeleton of geometric intuition, which can then be unfolded in 2D and 3D:
In 2D, local forms combine into an optimized grid, where each cell is a miniature sphere, its equator and chord reflecting proportion and density.
Global unfolding (Poincaré’s conjecture + Perelman’s proof) allows uniting these local elements into a single closed structure.

9. Transition to a Three-Dimensional System: Spheres, Equators, and Chords
Imagine a sphere equipped with:
centers of mass or energy inside each sphere;
equators, defining the circumference of the sphere;
Ishchenko’s chords and half-chords, defining local proportions.

10. Ishchenko’s Coordinate System of Spheres
Each sphere has:
a center (point), an equator and chords, defining local structure, potential directions of flows of energy or matter.
Coordinates within global space are set by the combination of the center and the local chordal system:
— where the center vector, and local division coefficients by chord and half-chord, together define position.
Thus is formed a 3D grid of harmonic spheres, each being a local realization of the common structure.

11. Visual Concept of Circle and Petals Inside the Sphere
1. Sphere’s center: exists intuitively at point 0, but placement of circles is not strictly bound to it.
2. Circles (equators): a multitude of circles with relatively equal equatorial lengths that can be inscribed within a sphere. Each circle has a circumference, which we call the equator in a general sense.
3. Division of the circle into petals:
Each circle is divided into 12 “petals”:
6 along the equator (circumference),
6 from the circle’s center to the boundary (radial).
Large petal: formed if Ishchenko’s chord is doubled on both sides, creating a petal shape inside the circle.
Small petal: constructed relative to the sphere’s boundary and the circle’s center, using half-chords.
As a result, highly diverse fractals appear (see Chapter 5).
4. Dimensions:
Chord length = 1 conventional unit,
Half-chord length = 0.5 c.u.
5. Intra-spherical arrangement:
Petals and chords are arranged like sinusoids, ensuring harmonic distribution of energy and density
across the surface and within the sphere.
6. Sphere = multitude of circles:
Each point inside the sphere may belong to several circles, ensuring depth of structure and
multilayering.
Circles intersect and overlap, forming a grid of petals that shapes a three-dimensional body with internal harmonic divisions.