Section E
Varieties of Matter
In the preceding sections, we have considered both photons
and atoms merely as general classes of objects. This is sufficient so
far as the photons are concerned, as there are no individual differences
in this class of objects other than in frequency. There is, however, a
large amount of variability in the atoms of matter, and our next undertaking
in the exploration of the theoretical universe of the Reciprocal System
will be to examine the nature of this variability and the reason for its
existence.
This investigation will be concerned largely with the
magnitudes of the various motions involved, and some points concerning
these magnitudes should benoted before proceeding with the development.
As stated in Section B, the natural datum,
or reference level, for physical phenomena is unit speed, not zero. The
true magnitude of any absolute quantity (one that is not arbitrarily related
to some selected reference datum) is therefore the deviation from the
unit value, rather than the mathematical total. In the case of the combinations
of rotational motion that constitutes matter, the magnitudes with which
we will be primarily concerned are the rotational speeds.
But inasmuch as we will be dealing with units of deviation from unit
speed, rather than with speeds measured in the usual manner from the mathematical
zero, it will be desirable to utilize some different terminology to avoid
confusion. We will therefore refer to this deviation as a displacement
of the spacetime ratio from the normal unit value. When the speed, s/t,
is 1/n we will say that there is a displacement in time (or "time
displacement") of n1 units. Conversely, when the speed is n/1, and n
units of space are associated with each unit of time, we will say that
there is a displacement in space (or "space displacement") of
n1 units. In this connection it should be noted that in the region of
displacements in time (speed = 1/n) a higher displacement value (a greater
deviation from the unit speed that constitutes the natural datum) corresponds
to a lower speed as customarily measured.

In the context of a stationary, threedimensional
reference system, coincident translational motion in more than one
dimension is impossible, as each omtion alters locations in a different
manner, and such motion would result in the same absolute locaiton
occupying two or more different positions in the reference system.
Rotational motion, on the other hand, does not alter the location
in a reference system of this kind, and coincident rotational motion
an all three dimensions is therefore possible.

It is not possible, however, for a onedimensional
object, such as a photon, to have rotational motions of the same
kind in all three dimensions. Rotation of the photon cannot take
place independently around the line of vibration as an axis. Such
a rotation would be indistinguishable from no rotation at all. The
photon may, however, rotate around its midpoint. One such rotation
generates a twodimensional figure, a disk. Rotation of the disk around
a diameter generates a threedimensional figure, a sphere. Since no
fourth dimension is available, this process cannot be continued farther.
The basic rotation of the photon is thus twodimensional.

With this twodimensional rotation in existence, the
photon may rotate around the third axis in the opposite scalar direction.
This is a rotation of the sphere generated by the basic rotation.
Since the twodimensional rotation is distributed over all three dimensions,
the additional rotation in the third dimension is not required for
stability of the structure, and the total rotation of the atom therefore
consists of a twodimensional rotation of each photon, with or without
an oppositely directed onedimensional rotation. For convenience,
we will refer to the onedimensional rotation as electric
rotation, and the twodimensional rotation as magnetic rotation.
At the present stage of development, there are no electric or magnetic
forces in the structures under consideration, but the identification
of "electric" with "onedimensional" and "magnetic" with "twodimensional"
will be of advantage when electric and magnetic phenomena are introduced
later in the development.

The speed of the electric rotation is independent
of that of the magnetic rotation, except to the extent that probability
considerations favor the magnetic rotation, and the speeds in the
two magnetic dimensions are partially independent, inasmuch as this
rotation may be distributed spheroidally rather than spherically.
Consequently, there are a number of different combinations of rotational
speeds, which give rise to corresponding differences in physical behavior:
differences in the properties of the various rotational combinations,
we may say. The theoretical universe thus contains many different
kinds of atoms with different properties. These can be identified
as the chemical elements, each element corresponding to a
specific combination of rotations.

The number of such combinations that can actually
exist in limited by the probability principles, the validity of which,
in application to the theoretical universe, is specified in the postulates.
The most significant limitation results from the principle that small
numbers of units are more probable than large numbers.

Geometrical considerations indicate that two photons
can rotate around the same central point without interference if the
rotational speeds are the same, thus forming a double unit. For a
given number of units of effective motion, such combinations result
in lower displacement values, and the probability principles therefore
give them precedence over single units with higher displacement values.
All rotating units with sufficient net total displacements to enable
forming double units therefore do so.

The electric rotations of the two photons of a double
unit can, and therefore do, take place in different dimensions. Each
such rotation involves only one photon. Similar independence of the
magnetic rotations is not possible because each is distributed over
all available dimensions. Each magnetic rotation therefore involves
movement of both photons. As a result, a unit of magnetic rotation
in an atom is equivalent to 2n² units of electric rotation, where
n is the effective magnetic displacement.

In the normal outward progression each unit of motion,
s/t = 1/1, is succeeded by a similar 1/1 unit, yet another, and so
on, the total up to tany specific point being n units. In a combination
structure, involving a series of displacements, the sequence
is 1/1, 1/2, 1/3,... 1/n, or the reciprocals of these values, 1/1,
2/1, 3/1... n/1. Here, n is the last unit, not the total, and in order
to arrive at a total a summation of the individual values is required.
To obtain the total electric equivalent of a magnetic displacement,
we must similarly sum up the individual 2n² terms.

Since the simple motions that have been considered
thus far are inherently scalar, addition of another displacement of
the same kind of an existing displacement would simply alter the scalar
magnitude, without changing the nature of the motion. In order that
there may be motion of the original motion—rotation
of a photon, for example—it is necessary for the added displacement
to be of an opposing nature. We have previously noted that the basic
twodimensional rotation of the photon can be rotated in the opposite
scalar direction, but this is possible only because the magnitude
of the onedimensional rotation is less than that of the twodimensional
rotation, and the net rotational displacement of the combination
is still negative, as it must be to oppose the positive vibrational
motion. This possibility is not open in the case of the original rotation
of the photon, but the necessary dissimilarity between the vibration
and the rotation can be attained by means of the divergence of displacements
in space from displacements in time. A photon with a vibrational displacement
in time can acquire a rotational displacement in space, and vice versa.

For the present we will be dealing only with those
atoms whose vibrational displacement is in space and whose net rotational
displacement is in time. The terms "matter", without any qualification,
will hereafter refer to aggregates of atoms of this nature. Where
it is desired to differentiate specifically between this and the inverse
type of matter, in which the displacement of the vibration is in time
and the net displacement of the rotation is in space, we will use
the term "ordinary matter".

While we will include all rotational combinations
with net rotational displacement in time under the classification
"matter", we will hereby restrict the term "atom" so that it applies
only to those combinations which include two rotating systems.

Since the magnetic rotational displacement is numerically
smaller than the equivalent electric displacement, it is correspondingly
more probably, and the magnetic rotation consequently takes precedence
over the electric rotation wherever both would otherwise be possible.
It will therefore be appropriate to begin our identification of the
specific rotational combinations by considering those which have no
effective electric rotation.

If a unit of space displacement is added to a motion
with n units of time displacement, the new unit and one of the time
displacement units constitute a full unit of motion (displacement
zero) and since every such unit is independent, according to the postulates,
this new unit separates from the remainder, leaving a residue of n1
units of time displacement. Adding space displacement is therefore
the equivalent of subtracting time displacement, and vice versa.

A structure in which the rotation is limited to one
unit of magnetic displacement may be represented by the symbol 000,
where the first two numbers represent the displacements in the magnetic
dimensions and the third represents the electric displacement. In
accordance with the principle expressed in item 13, the one unit of
rotational time displacement merely neutralizes the one unit of vibrational
space displacement, and brings the new total to zero. The 000 structure
is therefore the rotational equivalent of nothing at all: the rotational
base, we will call it.

By the operation of probability, added units of
magnetic displacement go alternately to the two magnetic dimensions.
A second such unit therefore brings the structure up to ½½0. As
has been stated, we are restricting the term "atom" to those combinations
with two rotating systems, which requires effective rotational
displacements in both magnetic dimensions. The ½½0 combination does
not qualify as an atom under this definition. The question as to just
what it actually is will be considered in Section
F.

The next combination, 210, is the first of the purely
magnetic rotational combinations that qualifies as an element. As
has been noted, each magnetic displacement unit is equivalent to 2n²
electric displacement units, and the total displacement of this atom
above the rotational base, in electric equivalent, is 4 units.

Inasmuch as the electric displacement unit is the
smallest rotational unit that exists, and therefore the smallest amount
by which one rotational combination can differ from another, the possible
combinations form a series in which the total equivalent electric
displacement of each successive member is one unit greater than that
of its predecessor. We will identify the position in this sequence
as the atomic number of the element, and because the first
two units of displacement have been excluded from the atomic classification,
this atomic number can be described as the net total equivalent electric
displacement, less two units. On this basis, the atomic number of
the 210 combination is 2, and we will identify this structure as
the element Helium.

The 210 combination is one unit above the rotational
base in each magnetic dimension. Addition of another magnetic unit
therefore requires 2 x 2², or 8, equivalent units. The result is 220,
atomic number 10, which we identify as the element Neon. Another magnetic
addition produces 320, atomic number 18, the element Argon. Similar
additions complete the series of inert gases, a group of
elements whose distinctive properties results from the fact that these
are the only chemical elements without effective rotation in the electric
dimension.
Atomic Number

Element

Displacements

2

Helium

210

10

Neon

220

18

Argon

320

36

Krypton

330

54

Xenon

430

86

Radon

440

The reason why the series terminates at 440 rather
than continuing on to higher values will emerge later in the development.

In view of the greater probability of the magnetic
displacement, the role of the electric displacement is confined to
filling in the gaps between the combinations listed in the foregoing
table. For example, helium is followed by these four elements:
Atomic Number

Element

Displacements

3

Lithium

211

4

Beryllium

212

5

Boron

213

6

Carbon

214


The next combination in this sequence would be 215,
but another factor enters into the situation at this point because
electric rotation can take place with displacement in space as well
as with displacement in time. As previously noted, the rotational
displacement of the atom as a wholethat is, the net total displacementmust
be in time in order to constitute rotation of the photon.
But as long as the larger component of this total, the magnetic displacement,
is in time, the smaller component can be in space. In this case, the
addition of space displacement reduces the net total time displacement.
The 7unit net effective time displacement that corresponds to the
structure 215 can therefore be attained in an alternate manner by
adding 3 units of displacement in space to the 220 combination.
To distinguish space displacements from time displacements, we will
enclose the space values in parentheses. On this basis, the alternate
7unit structure is 22(3), and by reason of the greater probability
of the smaller electric displacement, this structure exists in preference
to 215.

The other members of the second half of the group
of elements between helium and neon are subject to the same considerations,
and this sequence is as follows:
Atomic Number

Element

Displacements

6

Carbon

22(4)

7

Nitrogen

22(3)

8

Oxygen

22(2)

9

Fluorine

22(1)

The probabilities of the two possible structures are
nearly equal for carbon, midway between the two inert gases, inasmuch
as the electric displacement is 4 in both cases. This element can
therefore take either structure, and it is shown in both tabulations.

Each of the other gaps between inert gas elements
is similarly filled by a series of combinations in which there is
an increasing electric displacement in time up to the midpoint of
the series, followed by a decreasing electric displacement in space
in conjunction with the next higher magnetic displacement. Availablity
of electric displacement in space, as a component of the rotational
combinations, also permits the existence of an element below helium.
This is hydrogen, atomic number 1, which has rotational displacements
if 21(1).

All of the foregoing conclusions with respect to the
effect of probability are based on a consideration of the characteristics
of the elements as they exist in isolation. When they are interacting
with other elements, as in chemical compounds, additional probability
factors may be involved, and the net effect of all of the probability
factors mauy be involved, and the net effect of all of the probability
factors may favor some combination other than that which would exist
if no external forces may be to favor 215 rather than 22(3), or
22(5) rather than 213.
