^{-1113}_{3}(Determination
of the values r and d defining the implosion waves

In analyzing, within the scope
of the new hypothesis, the implosion waves, I find necessary to determine the value of certain relevant space-time dimensions
that determine and measure their intensity an location at different points of the field. These dimensions are, the distance
of the wave from the generating source indicated by r and the dimension d defining its wavelength at a distance from the source
equal to r. In order to determine the values of r and d at different points in space, we will indicate the terms involved
in the mathematical equations as follow.

re
= Electron radius

r1 = radius defining
the distance of the first implosion wave from the source center.

ri = radius defining the distance of the first gravity wave from the source center

rn = radius defining the distance of the nth implosion wave

dl = a dimension defining the implosion wavelength at the distance rl

dn = a dimension defining the implosion wavelength at the distance rn

n= a whole number from one to infinity

Vi1 Volume
of th first implosion wave

V = the volune
of space representing in size not a substantial body, but a three-dimensional domain limiting in space the energy generated
field disturbances creating a source.

Ve
= the spherical volume of the Electron source. = 4.188 x re ^{3}

Vi1= the volume of the first implosion wave surrounding the source. Vs1defined in space by a volume limited by the
two spherical surfaces of radius r1 and re, the two surfaces being separated by the distance d1.

V2 = volume of the second implosion wave. V2 = V3 - V2 = 4.188 x re^3

^{}

Vn = volume of the nth implosion wave.

Vil = Vs2 = Vsn = Ve
(see sketch) = 4.188 x re^3

Vi1 = V1- Ve
Vi1 = 4.188 x re^3 - 4.188 x re^3 = 4.188 x (r1^3- -re^3

Vi1
= 4.188 x 2 re^3 re = 1.61 x 1.2599 = 2.028 xre^3

In
order to determine the actual dimension of r and d, we have to establish the value of re, the radius of the Electron source.

The exact value of re could not be established
experimentally, but was estimated by some scientists to be less then 10^{-18}
m.

In the new atomic model I propose that
the dimension re must have a value equal to 10^{-24} m , ( See chapt 14). This dimension is much smaller than
first estimated, but much bigger than in the quantum physics conception of the eElectron that defined it as an infinitely
small point in space calIed a singularity. I have chosen this dimension for the Electron source radius because I have found
that in my hypothesis the Planck constant and the Universal gravity constant can be determined. considering an Electron
source possessing that radius. (Seealso chap.I3)

V1 = V2 = Vn = Ve Ve
= 4.188 x re ^{3 } re =3 v2=1.25992x
re^{ }

V1=
4.188 x re^{3} r1 =3√4.188
x re = 1.61 x re

Ve = 4.188 x re^{3}

V1= 4.188 x (r1^{3} - re^{3}) =
( 1.61 x -re - re ) = 1.61 x 2re^3

Since V1 = Ve = 4.188 x re^3-(- re^3) = 4.188 x 2 re ^3

V1 = 2 re ^3

V2 = 4.188 x 3 x (r1^{3} - re^{3}) = 3√3 x re

Vn = 4.188 x [rn^{3} - (rn-1)^{3 }

^{ }

V1 = V2 = Ve therefore 4.188
x (r1^{3} - re^{3}) r1^{3} = re^{3}
+ re^{3 } r1^{3} = 2re^{3V2} x re^{3}
= 4.188 x 3re^{3}

r2^{3} = re^{3} + 2re^{3} = 3re^{3}

r2 = 3√3 x re

rn = 3√(n+1) x re ^{
}

di1 = 2 re ^3 -r1 - re^3

di2 =3 re3

din = 3√(n+l)
- 3√n x re ^3

In assuming the electron source radius dimension at 10^{-24} m we can evaluate the dimension dl of the electron
first implosion wave from the formula

r
n = 3√(n + 1) x re

dsn = 3√(n +1) - 3√n x r e

An important linear dimension is di, representing the wavelength of an implosion wave generated by an electron at
a distance from its source center that I assumed to be approximately equal to 10^{-10} m the dimension of an atomic
radius. The value of the wavelength di, produced by an implosion wave at the distance, ri, detennines the minimum possible
dimension of the field substance domain. r . I have assumed that the implosion wavelength dimension will remain constant
from that distance on. The wave at the distance ri and above will continue to radiate in space at a velocity equal to C with
a negative intensity value that from that point on will increase, tending from zero to the value h of the field at rest. To
it's vanishing negative intensity can be assigned a value measured in a fraction of the Planck constant now in a function
of the inverse square of its distance from the center of the source. At a distance ri and over where the implosion waves change
their character in maintaining a constant wavelength dimension di, I will identify them as gravity waves. The value of
dsi can be found with the mentioned fonnula that determines r n and d n. In this

^{ }

di = 3√( n+ 1) - 3√n ] x re
This equation being to big for my mathematical knowledge and for my computer capacity I estimated at 10^{-53 m.}

In a proton with a mass and a volume 1836 larger then the electron and therefore a radius 12 times larger, each following
implosion wave is possessing a wavelength proportionally larger r n than r i. (See chap. 11) In suggesting a new
atomic model in line with the new hypothetical structure of matter I made, I will have to make another assumption regarding
the previously described sources dynamic characteristics. When the Electron energy implodes around the X axis, a residual
of the source volume will fail to implode. I determined that the dimension dl of the electron first imploding wave is equal
to 0.2599. If we attribute to the residual non imploding spherical volume a radius 0.0099 time smaller than re, the value
of the electron implosion less the non imploding residual volume will produce a wave dl exactly 0.25 x re or = 1/4th the electron
source radius.

Vev = non imploding volume
of the electron.

Ee
= 0.51 x 10^{6} eV

Ve 4.188 x 10^{-72} m^{3}

Vev = 4.188 x (0.0099 x 10^{-24})^{3}
m^{3}

Vev = 4.188 x 10^{-6}
x 10^{-72} m^{3}

Vev =
4.188 x 10^{-78} m^{3}

Ve/Vev
= 10^{6}

The non imploding volume
Vev has a volume that is 1000000 times smaller than the size of the electron. The volume of the electron Vev embodies an energy
equal to:

Eev = 0.51 x l0^{6}
x l0^{-6 } Eev = 0.51 eV = 0.51 x 10^-9 GeV

In
describing an atomic model based on the new hypothesis regarding the constitution of matter we had to use dimensions in the
direction of the infinitesimal much smaller then are allowed by the principle of uncertainty of Heisenberg in the quantum
mechanics theory. Heisenberger principle states that it is a mathematical impossibility to define with certainty the
position and the momentum of a particle simultaneously_ The mathematic expression of this fact is stated in the fonnnia (DELTA
P)x(DELT A X) is approximately equal to h. (DELTA P) represent the uncertainty of position, (DELTA X) the uncertainty of momentum
while their product is equal to the Planck constant h. Heinsenberger principle limits what can be learned about a particle
with dimensions below the value of the Planck constant.

In my new hypothesis we are talking about waves created in the medium of the field not of particles as defined by
particle physics the infinitesimal dimension limit can be found in a particle of the field substance acting in a spherical
domain with a radius equal to 1 GeV^-3.