THE INVERSE SEMI-CUBE OF DISTANCE

 

 

THE INVERSE SEMI-CUBE OF DISTANCE:

THE VOLUMETRIC GEOMETRY OF

GRAVITY AND THE RELATIVISTIC

CONVERGENCE OF 1.75

 

Otacilio Alves Meirelles

Independent Researcher

E-mail: otacilio.meirelles@gmail.com

DOI 10.5281/zenodo.18458651

https://doi.org/10.5281/zenodo.18458652 

January 31, 2026

Abstract

 

 

ABSTRACT

This study proposes a fundamental revision in the propagation metric of radiant and

gravitational fields. While area is sufficient to measure light tangentially striking

opaque objects without penetrating them, estimating gravity—which acts entirely

upon every atom of the involved bodies—leads us to deduce that static volume

might offer the solution. However, deeper investigation reveals that static volume is

also insufficient. The exclusivity of the Inverse Square Law (1/r2)—based on two-

dimensional surfaces—is questioned in favor of a three-dimensional model termed

the “Inverse Semi-Cube of Distance.” It is demonstrated that by considering energy

propagation as a continuous flux filling spherical shells, density dilution follows a

geometric factor of 1/7 when doubling the radius, rather than 1/4 (area) or 1/8

(total volume). The involvement of spacetime and motion attenuates the effective

volumetric density, reducing its total charge by 12.5% relative to the 87.5% larger

space in the second half of the radius, reflecting the fact that gravity between two

bodies encompasses them entirely (volumetrically), and not merely superficially. The

study concludes by presenting a robust mathematical correlation: the ratio between

the proposed volumetric dilution (7) and the classical surface dilution (4) yields

the factor 1.75—precisely matching the gravitational deflection value predicted by

General Relativity and confirmed by the Eddington expedition in 1919.

 

Keywords: Inverse Semi-Cube. Gravity. Volumetric Geometry. Inverse Square

Law. General Relativity. Light Deflection.

 

 

1.      INTRODUCTION: THE LIMITATION OF THE SURFACE

Historically, Newtonian physics established that the intensity of light and gravity de-

creases with the square of the distance. This premise is based strictly on the geometry of

the area of a sphere (A = 4πr2). However, this classical approach treats physical propa-

gation as a mere phenomenon of surface interception, ignoring the intrinsically volumetric

nature of space.

The present thesis postulates that light and gravity are not merely “shells” touch-

ing two-dimensional surfaces, but fluid contents in evolution that occupy and fill three-

dimensional spacetime. To gauge the true energetic density of these entities—although

light flowing volumetrically may touch objects superficially—regarding gravity, it is im-

perative to abandon area geometry in favor of volume geometry (V ∝ r3) and consider

the continuous flux determined by the speed of light (c). In this study, it is demonstrated

that the factor 1/7 accurately expresses the spherical volume in continuous flux. When

confronting the data with the number derived by Einstein and Eddington, we find 1.75

for the curvature of light from the Hyades near the Sun.

 

 

 

 

2.      THE SOURCE BODY AND THE PROPAGATION OF ENTITIES

When emanating from the source body—in this case, specifically gravity—these enti-

ties flow from the center of the bodies as if starting from a singular point.

1. After 1 second: The entity will fill the space forming a sphere with a radius of

300,000 kilometers around the source body.

2. After 2 seconds: The first layer moves to a second zone (between 300,000 and

600,000 km), allowing a new layer of 300,000 km to occupy the initial space.

3. After 3 seconds: A third layer appears, occupying the first radius of 300,000

kilometers.

4. The second layer, previously in the first space, expands to occupy the second zone,

giving way to the third.

5. The first emitted layer, having passed through the second zone, now occupies the

third, situated in the range of 600,000 to 900,000 kilometers from the source body,

and so on.

6. From the central point, when doubling the distance, the first half of the total radius

will present a gravity 7 times more intense, concentrated in a space 7 times smaller.

7. In the second half of the radius, gravity will be 7 times less intense, dispersed in a

space 7 times larger.

8. This gravity will envelop the bodies entirely—in a mutual sense—exerting force

according to the mass and distances of the bodies orbiting each other, or even in

multi-body systems with gravitational interrelations.

 

 

 

 

 

 

3.      SPACE KINETICS: MOTION AS A VOLUME GENERATOR

To understand the real density of a field, one cannot treat space as a pre-existing static

container. The space occupied by light or gravity is generated intrinsically, in real-time,

by the dynamic propagation of the source. The radius vector (r) is not a line drawn in

a vacuum; it is the physical manifestation of constant motion (c) acting integrally over a

time interval (t).

Consequently, the solar irradiation necessary to reach Earth exceeds predictions based

on the Inverse Square Law (1/r2). From this perspective, it is demonstrated that the

solar energy flux, traditionally calculated by the surface dilution of the inverse square,

underestimates the real dissipation. To compensate for the proposed volumetric dilution,

the magnitude of emission at the source must be rescaled by a factor of 1.75 (or +75%),

correcting the discrepancy between spherical shell geometry and surface geometry. This

principle is fundamentally extensible to gravitational interaction.

Although under the perspective of Newton/Soldner light would curve 0.87 arcseconds

when passing by the Sun, and under Einstein/Eddington it would curve 1.75 arcseconds,

what is affirmed here is based not on estimates, but on the real response that the in-

verse square of the distance delivers when confronted with the Inverse Semi-Cube of

Distance, which asserts that entities flow not in static volume, but in constant flux.

The Fundamental Equation of the Kinetic Radius is defined as:

r = c · t

(1)

Consequently, Volume (V ) ceases to be merely a static spatial function to become a

direct function of time and motion:

V (t) = 4

3π(c · t)3

(2)

3.1

Continuous Flux and Dynamic Dilution

This kinetic formulation reveals that volume grows with the cube of the flow time. If

we consider the motion of light or gravity flowing from an instant t1 to an instant t2 = 2t1:

• The entity travels with constant linear velocity c.

• However, the Volume Creation Rate is accelerated. The motion needs to “sweep”

and fill an increasingly larger amount of cubic space for every additional second it

travels.

When passing from the first to the second time interval (t → 2t), the linear motion

merely doubles the radial distance, but the need for volumetric filling multiplies. The

“Factor 7” (the basis of the Inverse Semi-Cube) arises from the tension between Constant

Linear Velocity (c) and Cubic Volumetric Expansion (t3).

Motion is the agent that,

by advancing, dilutes the density of the entity, as the energetic content of the source

cannot multiply in the same explosive proportion that the volume generated by the motion

demands.

 

 

 

 

4.      GEOMETRIC FOUNDATION: THE EXPANSION FACTOR 7

Deepening the kinetic analysis from a geometric perspective, we consider isotropic

propagation. For every unit of time, the source emits a quantity of energy that must

occupy a volumetric spherical shell.

When analyzing the evolution of volume between a unit radius r and its double 2r,

we observe the quantitative phenomenon:

1. The total accumulated volume of a sphere of radius 2r is eight times larger than the

sphere of radius r (23 = 8).

2. However, the energy emitted in the second interval does not occupy the total volume

from the origin, but flows to fill only the newly created layer (the physical space

between r and 2r).

The physical volume of this second layer (∆V ) is defined by the difference:

∆V = V2r − Vr = 8Vr − 1Vr = 7Vr

(3)

Geometric Corollary:

It is concluded that the energy contained in the first unit volume, upon expanding to

occupy the second layer, must fill a physical space seven times larger. Therefore, the Vol-

umetric Energy Density undergoes dilution by a factor of 7. This is the mathematical

principle of the “Inverse Semi-Cube of Distance.”

 

 

 

 

5.      THE NATURE OF GRAVITY: PERMEABILITY AND INTEGRAL INTERACTION

The distinction between metrics becomes crucial when analyzing the nature of the

interaction.

While the Inverse Square Law (area) works adequately for opaque light

sensors (which intercept radiation superficially), it fails conceptually when describing

gravity.

Gravity is a phenomenon of absolute permeability. It does not recognize surfaces, shad-

ows, or shielding. The gravitational field traverses mantles, cores, and crusts, interacting

with the totality of the internal mass of bodies.

• Volumetric Immersion: Upon reaching Earth, the Sun’s gravitational field (the

expanded volumetric shell) does not merely touch the planet’s surface. The total

volume of Earth is immersed and engulfed by the volumetric density of the solar

field.

• Mutual Resonance: Reciprocally, Earth’s field, although weaker, fully permeates

the Sun’s volume.

Since gravity is a force that acts upon mass (density × volume) and permeates the

interior of bodies, the metric to calculate its intensity must be, obligatorily, volumetric

(Semi-Cube) and not superficial (Square).

 

 

 

6.      THE CONVERGENCE OF 1.75: THE LINK TO RELATIVITY

The most robust validation of this model lies in the emergence of a fundamental

numerical constant that connects the Semi-Cube geometry to modern physics. When we

confront the two dilution metrics upon doubling the distance:

1. Classical Model (Newton/Surface): Predicts dilution by a factor of 4 (22).

2. Proposed Model (Volume/Semi-Cube): Predicts dilution by a factor of 7 (23−

1).

The ratio between the volumetric reality (the space to be filled) and the classical

superficial expectation reveals the necessary correction factor:

Convergence Factor = Volumetric Dilution (7)

Surface Dilution (4)

= 1.75

(4)

6.1

The Meaning of 1.75

Historically, 1.75 arcseconds was the exact value predicted by Albert Einstein for the

deflection of light when passing through the Sun’s gravitational field, confirmed by the

celebrated expedition of Arthur Eddington in 1919. Newtonian physics, based on the

inverse square, predicted only half of this value (0.87, Soldner’s calculation), failing to

explain reality.

6.2

Thesis Conclusion

The Inverse Semi-Cube model suggests that the “extra curvature” of spacetime de-

scribed by General Relativity is, geometrically, the manifestation of the difference between

a universe observed in areas (factor 4), static, and a real universe of volumes (factor 7)

of entities in motion. The factor 1.75 is not an anomaly, but the mathematical signature

of volumetric density flowing and imposing itself over plane geometry. Gravity acts with

a magnitude “1.75 times different” from the classical expectation because it operates in

the fullness of the three-dimensional volume. Thus, this model offers the geometric and

mechanical substrate for Einsteinian spacetime curvature.

References

[1] EINSTEIN, A. The Foundation of the General Theory of Relativity. Annalen der

Physik, 49, 1916.

[2] NEWTON, I. Philosophiæ Naturalis Principia Mathematica. London, 1687.

[3] DYSON, F. W.; EDDINGTON, A. S.; DAVIDSON, C. A Determination of the De-

flection of Light by the Sun’s Gravitational Field, from Observations made at the

Total Eclipse of May 29, 1919. Philosophical Transactions of the Royal Society of

London, 1920.

[4] SOLDNER, J. G. On the Deflection of a Light Ray from its Rectilinear Motion.

Berliner Astronomisches Jahrbuch, 1801.

 

 

                                                                       DOI 10.5281/zenodo.18458651

                                                             https://doi.org/10.5281/zenodo.18458652