ELEMENTARY PARTICLES AND UNITS OF
MEASUREMENT
In order to see how the minimal quantum extent and
content of our smallest possible elementary distinction relates to known
elementary particles, we develop equations that can be used to describe the
combination of up and downquarks to form the proton and neutron of the
Hydrogen atom. We choose the Hydrogen atom to start with because it is the
simplest, most stable, and most abundant known element in the universe. If all
forms of substance are quantized, then in order for quarks to combine in stable
structures, they must satisfy the Diophantine (integer) forms of the equations
of Dimensional Extrapolation conveying the logic of the transfinite substrate
into the spacetime domain of our experience. This family of Diophantine
equations is represented mathematically by the expression
Σ^{n}_{i=1} (X_{n})^{m} = Z^{m}.
The Pythagorean Theorem equation, the Fermat’s Last
Theorem equation and other important equations are contained within this
general expression. We mention this fact here because these theorems play key
roles in the geometry and mathematics of Dimensional Extrapolation and the
combination of elementary particles to form stable physical structures. Because
the various forms of this expression as m
varies from 3 to 9 conveys the geometry of 9dimensional reality to our
observational domain of 3S1t, we call this expression the “Conveyance
Expression”, and individual equations of the expression ”Conveyance Equations”.
When n = m = 2, the expression yields the
equation
(X_{1})^{2} + (X_{2})^{2 }= Z^{2}
which, when related to areas, describes the
addition of two square areas, A_{1 }and A_{2} with sides equal
to X_{1 }and X_{2 }respectively, to form a third area, A_{3},
with sides equal to Z. When these
squares are arranged in a plane with two corners of each square coinciding with
corners of the other squares to form a right triangle, we have
a geometric representation of the familiar Pythagorean Theorem demonstrating
that the sum of the squares of the sides of any right triangle is equal to the
square of the third side (the hypotenuse) of that triangle.
We use this simple equation in Dimensional
Extrapolation to define the
rotation and orthogonal projection from one dimensional domain into another, in
the plane of the projection. There are an infinite number of solutions for this
equation, one for every conceivable right triangle, but in a quantized reality,
we are only concerned with the integer solutions. Considering the Pythagorean
equation as a Diophantine equation, we find that there exists an infinite
subset of solutions with AB = X_{1},_{ }BC = X_{2 }and
AC = Z equal to integers. Members of this subset, e.g. (3,4,5), (5,12,13),
(8,15,17), etc. i.e., (3^{2} + 4^{2} = 5^{2}, . 5^{2}
+ 12^{2} = 13^{2}, 8^{2} + 15^{2} = 17^{2},
… ) are called “Pythagorean triplets”.
To describe the combination of two
threedimensional particles, we have the Conveyance equation when n = 2 and m = 3:
(X_{1})^{3} + (X_{2})^{3 }= Z^{3}.
When we define X_{1},
X_{2} and Z as measures of volumes, just as we
defined them as measures of areas when n
= m = 2, we can apply this equation to quantal volumes in a
threedimensional domain. Using the minimal quantal volume of the electron as
the unit of measurement, and setting it equal to one, we have a Diophantine
equation related to our hypothetical elementary particle with minimal spinning
volume containing uniform substance: if it is spherical, we can set its radius
equal to r_{1}, and if there
is a second uniform spinning particle rotating at maximum velocity, with radius
r_{2}, we can describe the
combination of the two particles by the expression 4/3π(r_{1})^{3} + 4/3π(r_{2})^{3}.
If this combination produces a third spinning spherical object we have:
4/3π(r_{1})^{3} + 4/3π(r_{2})^{3} =
4/3π(r_{3})^{3},
where r_{3}
is the radius of the new particle. Dividing through by 4/3π, we have:
(r_{1})^{3} + (r_{2})^{3} = (r_{3})^{3},
which is a Diophantine equation of the form of the Fermat equation,
X^{m} + Y^{m} = Z^{m} when m =3.
Notice that the factor, 4/3π cancels out, indicating that this equation is obtained
regardless of the shape of the particles, as long as the shape and substance is
the same for all three particles. (This is an important fact because we found
in investigating the Cabibbo angle that the electron, while symmetrical, is not necessarily spherical.) Note also,
that the maximum rotational velocity and angular momentum will be different for
particles with different radii, because the inertial mass of each particle will
depend upon its total volume. In a quantized reality, the radii must be integer
multiples of the minimum quantum length. Since this equation is of the same
form as Fermat’s equation, Fermat’s Last
Theorem tells us that if r_{1}
and r_{2} are integers, r_{3} cannot be an integer.
This means that the righthand side of this equation, representing the
combination of two quantum particles, cannot be a symmetric quantum particle.
But, because Planck’s principle of quantized energy and mass tells us that no
particle can contain fractions of mass and/or energy units, the righthand side
of the equation represents an unstable asymmetric spinning particle. The
combined highvelocity angular momentum of the new particle will cause it to
spiral wildly and fly apart. This may lead us to wonder how it is that there
are stable particles in the universe, and why there is any physical universe at
all. Again, we are faced with Leibniz’s most important question: why is there something instead of nothing?
The answer turns out to be relatively simple, but
is hidden from us by the limitations of our methods of thinking and observation
if we allow them to be wholly dependent upon our physical sense organs. For
example, we think of a sphere as the most perfect symmetrical object; but this
is an illusion. Spherical objects can exist in a NewtonLeibniz world, but we
actually exist in a PlanckEinstein world. In the real world, revealed by
Planck and Einstein, the most perfectly spherical object in three dimensions is
a regular polyhedron. (polyhedron =
multisided threedimensional form; regular; all sides are of equal length.)
The most easily visualized is the sixsided regular polyhedron, the cube. In
the NewtonLeibniz world, the number of sides of a regular polynomial could
increase indefinitely. If we imagine the number of sides increasing without
limit while the total volume approaches a finite limit, the object appears to
become a sphere. But in the quantized world of Planck and Einstein, the number
of sides possible is limited, because of the finite size of the smallest
possible unit of measurement (which we are defining here) is relative to the
size of the object. And because the “shape” factor cancels in the Conveyance
Equation for n = 3, Fermat’s Last Theorem tells us that, regardless of the
number of sides, no two regular polyhedrons composed of unitary quantum volumes
can combine to form a third regular polyhedron composed of unitary quantum volumes.
To help understand the physical implications of
this, suppose our true quantum unit exists in the shape of a cube. Using it as
a literal building block, we can maintain particle symmetry by constructing
larger cubes, combining our basic building blocks as follows: a cube with two
blocks on each side contains 8 blocks; a cube with three blocks on each side
contains 27 blocks; a cube with four blocks on each side contains 64 blocks;
etc. Fermat’s Last Theorem tells us that if we stack the blocks of any two such
symmetric forms together, attempting to keep the number of blocks on all sides
the same, the resulting stack of blocks will always be at least one block
short, or one or more blocks over the number needed to form a perfect cube.
Recall that if these blocks are elementary particles, they are spinning with
very high rates of angular velocity, and the spinning object resulting from combining two symmetric objects
composed of unitary quantum volumes will be asymmetric, causing its increasing
angular momentum to throw off any extra blocks until it reaches a stable, symmetrically
spinning form.
This requirement of symmetry for physical stability
creates the intrinsic dimensionometric structure of reality that is reflected
in the Conveyance Expression. It turns out that there can be stable structures, because when n = m =3, the Conveyance Expression yields the equation:
(X_{1})^{3} + (X_{2})^{3 }+ (X_{3})^{3}=
Z^{3},
which does have integer solutions. The first one (with
the smallest integer values) is:
3^{3} + 4^{3 }+ 5^{3}= 6^{3}
^{}
^{}
It is important to recognize the implications of Σ^{n}_{i=1} (X_{n})^{m} = Z^{m}.
When n, m, the X_{i} and
Z are integers, is an exact Diophantine expression of the form of the logical structure of the
transfinite substrate as it is communicated to the 3S1t domain. For this
reason, we call it the Conveyance Expression. It should be
clear that the Diophantine equations yielded by this expression are appropriate
for the mathematical analysis of the combination of unitary quantum particles.
When the Diophantine expressions it yields are equations with integer solutions, they represent stable
combinations of quantum equivalence units, and when they do not have integer
solutions, the expressions are inequalities
representing asymmetric, and therefore, unstable
structures.
In
the quantized ninedimensional domains of TDVP, the variables of the Conveyance
Equations are necessarily integers, making them Diophantine equations, because
only the integer solutions represent quantized combinations. When n = m = 2, we have the Pythagorean
Theorem equation for which the integer solutions are the Pythagorean Triples.
When n = 3 and m = 2, the Conveyance Equation yields the inequality of Fermat’s
Last Theorem, excluding binomial combinations from the stable structures that
elementary particles may form. On the other hand, the Diophantine Conveyance
Expression when n = m = 3, integer
solutions produce trinomial combinations of elementary particles that will form stable structures. This
explains why there is something rather than nothing, and why quarks are only
found in combinations of three.
Embedded within the transfinite substrate
are three dimensions of space and three dimensions of time that are temporarily
contracted during observations, and condensed into the distinctions of spinning
energy (energy vortices) that form the structure of what we perceive as the
physical universe. In the humanly observable domain of 3S1t, this spectrum
ranges from the photon, which is perceived as pure energy, to the electron,
with a tiny amount of inertial mass (0.51 MeV/c^{2} ≈ 1 x10^{47}
kg.) to quarks ranging from the “up” quark at about 2.4 MeV/c^{2}, to
the “top” quark at about 1.7 x10^{5 }MeV/c^{2}, to the Hydrogen
atom at about 1x10^{9} MeV/c^{2} (1.67 x10^{27}kg.),
to the heaviest known element, Copernicum (named
after Nicolaus Copernicus) at 1.86 x10^{24}kg . So
the heaviest atom has about 10^{23} times, that is, about
100,000,000,000,000,000,000,000 times heavier than the inertial mass of the
lightest particle, the electron. All
of the Elements of the Periodic Table are made up of stable vortical
distinctions that are known as fermions, “particles” with an intrinsic angular
spin of 1/2, or they are made up of combinations of fermions. Table One, above, lists the fermions
that make up the Hydrogen atom and their parameters of spin, charge and mass
based on experimental data.
Bohr’s
solution of the EPR paradox, validated by the Aspect experiment and many
subsequent experiments refined to rule out other possible explanations, tells
us that newly formed fermions do not exist as localized particles until they
impact irreversibly on a receiver constituting an observation or measurement.
In the TDVP unified view of reality, every elementary particle, every distinct
entity in the whole range of particles apparently composed of fermions, is
drawn from the continuous transfinite substrate of reality when it is
registered as a finite distinction in an observation or measurement. Our
limitations of observation and measurement and the dimensional structure of
reality result in our perception of fermions as separate objects with different
combinations of inertial mass and energy. What determines the unique mix that
makes up each type of observed particle? To answer this question, we must
continue our investigation of the rotation of the minimum quantal units across
the four dimensions of space, time and the additional dimensions revealed by
the mathematics of TDVP.
One
of the most important invariant relationships between dimensional domains is
the fact that each ndimensional
domain is embedded in an n+1
dimensional domain. This means that all distinctions of extent, from the
ninthdimensional domain down, and the distinctions of content within them, are
inextricably linked by virtue of being sequentially embedded. Because of this
intrinsic linkage, the structure of any distinction with finite extent and
content, from the smallest particle to the largest object in the universe,
reflects patterns existing in the logical structure of the transfinite
substrate. Such a distinct object will always have in its content, combinations
of the forms reflecting those patterns. In a quantized reality, the
dimensionometric forms of such objects will be symmetric and a multiple of the
smallest unit of measurement,
STABLE VORTICAL FORMS AND TRUE QUANTAL
UNITS
Chemists trained in the current
paradigm think of the combination of elementary particles and elements as forming
atoms and molecules by the physical bonding of their structures, and model
these combinations in tinkertoy fashion with plastic or wooden spherical
objects connected by single or double cylindrical spokes. This is helpful for
visualizing molecular compounds in terms of their constituents prior to
combining, but that is not necessarily what actually happens. Inside a stable
organic molecule, volumetrically symmetric atoms are not simply attached; their
subatomic spinning vortical “particles” combine, forming a new vortical
object. Elementary particles are rapidly spinning symmetric vortical objects
and when three of them combine in proportions that satisfy the threedimensional
Conveyance Equation, they do not simply stick together  they combine to form a
new, dimensionally stable, symmetricallyspinning object. Because they are
spinning in more than one plane, these objects are best conceived of as closed
vortical solitions.
The
triadic combinations of elementary vortical objects, like up and downquarks,
form new vortical objects called protons and neutrons; the combinations of
electrons, protons and neutrons form new vortical objects called elements; and
the triadic combinations of volumetrically symmetric elements form new vortical
objects called organic molecules. Thus, the dimensional forms of
symmetricallyspinning objects formed by the combining of smaller vortical
objects form closed vortices in 3S1t with new physical and chemical characteristics,
depending upon both their internal and external structure. We will take the
volume of the smallest possible quantized vortical object as the basic unit of
measurement as the true quantal unit.
THE TRUE UNIT, THE CONVEYANCE EQUATION AND
THE THIRD FORM OF REALITY
Conceptually,
the true quantum unit in TDVP is therefore a subquark unitary extent/content entity
spinning in the mathematically required nine dimensions of quantized reality.
When we choose to measure the substance of a quantum distinction, the effects
of its spinning in the three planes of space register as inertia or mass, spin
in the timelike dimensional planes manifests as energy, and spinning in the
additional planes of reality containing
the space and time domains, requires a third form of the stuff of reality, in
addition to, but not registering as either mass or energy, to complete the
minimum quantum volume required for the stability of that distinct object.
Because this third form of the stuff of reality is unknown in current science,
we need an appropriate symbol to represent it. Every letter in the English and
Greek alphabets has been used as a symbol for something in math and science, so
we have gone to the historically earlier PhoenicianAramaicHebrew alphabet. We
will represent that potential third form of reality here with the third letter
of the Aramaic alphabet, ג
(Gimel), and we will call the subquark unitary
measure of the three forms of reality the Triadic
Rotational Unit of Equivalence,
or TRUE Unit.
The
mix of the three forms, m, E and ג, needed
to maintain symmetric stability, present in any given 3S1t measurement,
will be determined by the appropriate Conveyance Equation, as demonstrated
below. When n = m = 3, Σ^{n}_{i=1} (X_{n})^{m}
= Z^{m} yields:
(X_{1})^{3} + (X_{2})^{3
}+ (X_{3})^{3}= Z^{3}
^{}
^{}
The integer solutions of this
Diophantine equation in TRUE units represent the possible combinations of three
symmetric vortical distinctions forming a fourth threedimensional symmetric
vortical distinction.
THE PRIMARY LEVEL OF SYMMETRIC STABILITY – QUARKS
With the appropriate integer values of X_{1},^{ }X_{2}, X_{3}, and Z, in TRUE units, this equation represents the stable combination of three
quarks to form a Proton or Neutron. There are many integer solutions for this
equation and historically, methods for solving it were first developed by
Leonhard Euler ^{ref}. The smallest
integer solution of this Conveyance Equation is 3^{3} + 4^{3 }+ 5^{3}= 6^{3}.
Trial Combination of Two UpQuarks and One DownQuark, i.e.
The Proton, With Minimal TRUE Units
Particle

Charge^{*}^{}

Mass/Energy

ג

Total TRUE
Units

MREV^{**}

u_{1}

+ 2

4

1

3

27

u_{2}

+ 2

4

0

4

64

d

 1

9

4

5

125

Total

+ 3

17

5

12

216=6^{3}

* For consistency in a quantized reality,
charge has also been normalized in these tables.
^{** }Minimum Rotational Equivalent Volume (MREV)
If we attempt to use the smallest integer solution,
3^{3} + 4^{3 }+ 5^{3}=
6^{3}, to find the appropriate values of ג
for the Proton, we obtain negative values for ג
for the first upquark and the downquark and zero for the second upquark. It
is conceivable that some quarks may contain no ג
units, but negative values are a problem, because a negative number of total ג units would produce an entity with
fewer total TRUE units than the sum of mass/energy units of that entity,
violating the conservation of mass and energy, destroying the particle’s
equilibrium and identity. When we try to use the smallest integer solution of
the conveyance equation to describe the combination of one upquark and two
downquarks in a neutron, all of the quarks have negative ג units. See the table below:
Trial Combination of One UpQuark and Two DownQuarks in TRUE Units
Particle

Charge

Mass/Energy

ג

Total
TRUE
Units

MREV

u

+ 2

4

1

3

27

d_{1}

 1

9

5

4

64

d_{2}

 1

9

4

5

125

Totals

0

22

10

12

216=6^{3}

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