FURTHER MATHEMATICS OF THE RECIPROCAL SYSTEM
This paper will present in the most concrete, explicit manner the mathematics of spacetime, radiation, and matter of the Reciprocal System. Readers without special knowledge of the Reciprocal System are first urged to study Larsons books, especially Nothing But Motion¹ before undertaking the study of this paper.
I. Mathematics of SpaceTime
A. Rectangular Coordinates
Starting from any reference point x0, y0, z0, t0 in the 0system, the spacetime progression is a spherical expansion. In rectangular coordinates the equation is
where c is the speed of light. If we choose the reference point to be x0 = 0, y0 = 0, z0 = 0, t0 = 0, then the equation is simply
Now consider a second system, the 0´system, moving translationally with respect to the 0´system in the xdirection. What is the equation for the progression in the 0´system? From the inverse Lortentz transformations,
Upon substitution, we obtain
But since the left side of the equation equals zero, so must the right side:
Thus the progression as determined by 0' is also spherical. And so the equation for the progression is invariant under a Lorentz transformation.
B. Polar Coordinates
In polar coordinates the equation is simply
In the Reciprocal System the speed of light is the natural unit of velocity and so r and t must take equal natural values. The spacetime progression is thus ¹/1, ²/2, 3/3, etc. Thus one unit of space is equivalent to one unit of time. If there are an infinite number of space units, there must be an infinite number of time units; if there are a finite number of space units, there must be a finite number of time units.
II. Mathematics of Radiation
In the Reciprocal System radiation is the combined motion of a simple harmonic oscillation in one dimension and a uniform translation in a perpendicular direction.
A. Simple Harmonic Oscillation
The equation for a simple harmonic oscillation in one dimension (say the y direction) is
where A is the amplitude and fos is the frequency. Since the oscillation takes place over one natural space unit, the amplitude must be onehalf a natural space unit and thus is
for all photons. In observation from the timespace region this value is reduced by the interregional ratio142.222 to 1.6027 x 1010m = 1.6027 Å
The other variable to be determined in eq.(10) is the frequency, fos. In one cycle the oscillation travels one space unit up and one space unit down, for a total of two units. The average velocity of the oscillation is th
The natural unit of frequency must occur when the average velocity is c.
the Rydberg frequency. (Actually, Larson derived the natural unit of time from the Rydberg frequency, but I think it was instructive to do the reverse, and this method will be used to calculate rotational and rotational vibration frequencies as well. Of course, this method assumes that the natural unit of time can be found by some other means.) Because of the discrete nature of the Reciprocal System, it is only possible to have integer multiples or reciprocal integer multiples of the Rydberg frequency.
Putting the values of A and fos in eq.(10) we have
B. Perpendicular Translation
Perpendicular to the oscillation is a translation at unit velocity (the speed of light). Let x be perpendicular to y. Then
C. Combined Motion
From eq.(17) t can be found in terms of x and c and put in eq.(15). The result is
if x is given in meters. This is the equation for a monochromatic wave of radiation in the Reciprocal System.
III. Mathematics of Matter
Particles of matter consist of rotating photons. Subatoms have one rotating photon; atoms have two rotating photons (both photons rotate about the same central point). The rotational motion has a translational effect, which will be discussed after the mathematics of the rotation has been worked out.
1. single systemsparticles
A photon can rotate around either of two horizontal axes passing through its midpoint, and also around itself. In the Reciprocal System the true physical zero is motion at unit speed. Anything physical must have a motion either greater or less than unit speed. This deviation is called a speed displacement by Larson. The first particle has 1 speed displacement around one horizontal axis of the photon and is called the rotational base. Actually there are two rotational bases: one with one speed displacement above unity, the other with one speed displacement below unity. As will be discussed later, the one displacement unit neutralizes the translational motion of the photon in the original dimension, but the progression now continues in the remaining dimension, so the effective displacement is zero. In the ground state condition, the photon that is being rotated is one vibrational displacement away from unity (either 2R or (1/2)R). Here is a table giving the photon frequency, the rotational displacement, the effective rotational displacement, and rotational speed of the cosmic rotational base and the material rotational base:
In the above table the speeds are calculated from the displacements as follows. For displace
ments of np, ns,
and nE, the speeds are (np+1), (ns
+ 1), and (nE + 1) for a cosmic particle, and 1/(np
+ 1), 1/(ns + 1), and 1/(nE + 1) for
a material particle. (Of course material particles could have high
speed electric displacement, and cosmic particles could have low speed
The natural frequency of rotation must occur when the speed is c.
where R is the Rydberg frequency,
as before. In these terms, then, the cosmic rotational base is a
photon that has a vibrational oscillation of 1.6440288 x 1015
cycles/sec and is rotating at 4.1864848 x 1015 revolutions/sec
around one axis, and 2.0932424 x 1015 revolutions/sec around
the other two axes. Likewise, the material rotational base is a
photon that has a vibrational oscillation of 6.576115 x 1015
cycles/sec and is rotating at 1.0466212 x 10 15 revolutions/sec
around one axis, and 2.093242 x 1015 revolutions/sec around
the other two axes.
Many more permutations appear to be possible, but the probability principles keep eccentricity to a minimum. Since none of the above particles has an effective displacement of 1 or more, they are all massless (aside from the mass contribution of an electric charge). The diameter of all the particles is one natural space unit, reduced by the (onephoton) interregional ratio, or 3.2054 Å. However, because these particles do not exert any force in the uncharged state, a particlemeasuring probe would not be able to detect any size of these particles at all.
2. intermediate systems
Intermediate particles have two rotating photons, but one of the two sets has no effective displacement and thus contributes no primary mass. The two intermediate particles are the neutron and the mass one hydrogen isotope (and their cosmic analogs). There are only two kinds of rotations that can combine to form this kind of particle, the proton type and the neutrino type We identify the combination of the material proton rotation and the material neutrino rotation as the mass one hydrogen atom; the combination of the material proton rotation and the cosmic neutrino rotation as the neutron; the combination of the cosmic proton rotation and the cosmic neutrino rotation as the mass one atom of cosmic hydrogen; and the combination of the cosmic proton rotation and the material neutrino rotation as the cosmic neutron. The proton is a single system with displacements 21(1), effective displacements 11(1), speeds ½-½-2, and rotational frequencies 2R/3pR/p4R/p. Then we would have the following table for the neutron and mass one hydrogen.
The new notation makes clear the two photons involved and the five rotations (to be discussed next).
3. Atomic cycles
Atoms have two rotating photons, but here both systems have effective displacements and both systems ordinarily have the same velocities. Let the first photon be called A and the second be called B. A and B are mutually perpendicular. We have the following five rotations:
(i) the rotation of A about B produces disk a;
(ii) the rotation of B about A produces disk b;
(iii) then disk a can be rotated about A;
(iv) then disk b can be rotated about B;
(v) finally the whole structure can be rotated in the electric dimension.
This last rotation is in the scalar direction opposite to that of the previous rotations. Cosmic atoms have speeds above unity for the first four types of rotations, whereas material atoms have speeds below unity for the first four types. The electric rotation may be above or below unity for both cosmic and material atoms.
The first particle with two effective rotating systems is deuterium, the second is helium, etc. A table similar to that for the intermediate particles can be made.
All other atoms can be given appropriate values in the same manner. In the solid state, however, the values that govern the physical properties are not the actual rotations, but the relative rotations, and the different values there are not due to inherent differences in the rotational speeds, but to differences in the orientations of the interacting atoms, and this will be discussed further later.
4. Electric charges and magnetic charges
According to the Reciprocal System an electric charge is a rotational vibration about the electric axis, and the magnetic charge is a rotational vibration about one of the magnetic axes. Both charges have the same natural frequency, calculated as follows. In one cycle the motion covers a distance of ¹ * snat one way and p * snat back, for a total of 2p * snat. So we have
At the unit level, vch = c = snat/tnat, so
Solving for fch nat and recalling that tnat = 1/(2R),
This frequency is onehalf that of a full rotation and can thus be considered to be effective in one direction only half the time. One negative electric charge is a rotational vibration of R/2p = 5.233106 x 1014 cycles/sec. One positive electric charge is a rotational vibration of 2R/p = 2.093242 x 1015 cycles/sec. Similarly one unit of magnetic charge is a rotational vibration of 2R/p = 5.233106 x 1014 cycles/sec, whereas one unit of isotopic charge is a rotational vibration of R/2p = 5.233106 * 1014 cycles/sec. The isotopes of atoms result from the addition of isotopic charges.
The rotational motion of particles has a translational effect. The maximum inward translation is two full units, giving one net inward unit. In terms of rotation we can have 2³ = 8 onedimensional rotational electric displacements or 4 twodimensional rotational magnetic displacements. Note that since 1³ = 1, the first magnetic rotational displacement, which is ½ unit rotational speed, produces one unit of inward translation and thus neutralizes the original translational motion of the photon, but the progression still continues in the third dimension. Thus the rotational base and all the single system massless particles previously discussed move at the speed of light. Additional magnetic and electric displacements produce a net inward motion, and the inward motion of a group of atoms is termed gravitation.
For atoms with magnetic displacements
of less than 4 and electric displacements of less than 8, the frequency
of the rotating photons is normally one displacement above unity, or 2R
(the frequency of photons in cosmic atoms is (½)R).
When the magnetic displacement reaches 4 or the electric displacement
reaches 8, the rotation must be extended to a second vibrational displacement
unitwhich means that the frequencies of the photons are now 3R
(or (¹/3)R for cosmic atoms). As Larson points
out, though, it is possible to have these higher frequency photons even
when the rotational displacements are less than 4 or 8, in which case
we can say that the atom is excited.
And the speeds corresponding to electric displacements can be listed as follows:
solid state, the values for electric rotation can be further altered.
Larson states that a combination of one atom of electric displacement
x with another atom of electric displacement 8x results in a neutral
bond. This bond gives rise to an electric speed of ¹/10
for vibration one, and ¹/5 for vibration two.
Also there can be a combination of two 8x atoms, which Larson calls
a secondary positive bond. In this case the rotational speed comes
Lower Group Atomic Table in Solid Stat
The reader can continue the table all the way to element 118. Again, one must first determine the kind of bond involved before the electric rotational speed can be determined.
Since different atoms have different rotational speeds and thus different rotational forces, a particle probe of equal energy shot at atoms of different elements would penetrate to different depths. Thus experimenters have concluded that nuclear size is proportional to atomic weight. Actually what they are measuring is atomic size, and according to the Reciprocal System this is constant (2.914 Å diameter)but the force is proportional to the atomic weight of the atom. Also, where interatomic distances are less than 2.914 Å, the atoms are partially merged; where distances are greater than 2.914Å, the atoms are separate.
1. Dewey B. Larson, Nothing But Motion (North Pacific Publishers: Portland, Oregon, 1979)