In RST-based physical theory, the universe is composed of two, reciprocal, sectors, the material sector, where we live, and the cosmic sector, where the time and space magnitudes are inverted in the equation of motion, forming an anti-particle, and an anti-element, for every type of particle and element in the material sector.
In the material sector, the particles and elements are formed from what Larson calls the “time-displaced” compounds of scalar motions. As these motions are compounded, the successive atoms of each element are formed, each having, theoretically, 1 unit of mass more than the preceding element.
The pattern, or periodicity, of the material elements is very interesting, from a mathematical point of view, as we have been pointing out, in the previous post. Not only do the number of the periods make sense as concentric areas derived algebraically and now even geometrically, but, unlike the periodic table of quantum mechanics, the pattern emerges as 4n2, not 2n2, and it is limited at n = 4.
When one looks at the table in the wheel format, it is tempting to wonder what the inverse of the wheel would be; that is, what comes after the last element in the material wheel? The logical answer would seem to be that the first element of the cosmic wheel would correspond to the last element of the material wheel, since the inverse of the heaviest material element should be the lightest cosmic element. But how can the top of the material wheel be tied to the center of the cosmic wheel? It really confuses the mind to try to visualize how to invert the wheel.
It turns out though that through something called “inversion geometry” a lot can be learned about the inverse of a circle. I don’t know a lot about it yet, but in studying it, I’ve come to appreciate how fundamental it is. It turns out that, if we want to be able to equate the legacy system of discrete oscillations (i.e. the four quantum numbers, the principle energy levels in terms of h, the angular momentum of probability amplitudes and magnetic moments (orbitals), and quantum spin) to the new system of discrete oscillations (pseudoscalar expansions/contractions), we need to find a mathematical correlation between rotation and expansion/contraction.
In our investigations to date we found a lot to be encouraged about. We found that the expansion/contraction is equivalent to a binary rotation, just like the quaternions, rather than the usual quadrantal rotation of sine and cosine functions used in the legacy system. We also realized that we could compare the 3D oscillations with the counter rotations of two meshed gears, which are reciprocally related; that is, one is always the counter rotation of the other. If one rotation is clockwise, the other must be a counter-clockwise rotation, which, in a sense at least, are two, reciprocal, binary motions.
Of course, since our spatially expanding/contracting SUDR is the reciprocal oscillation of the temporally expanding/contracting TUDR, in the new RST-based development, then the combination of the two, as a space|time, or S|T, unit is also a combination of two, discrete, reciprocal, motions. However, though rotational motions may be considered analogs of expansion/contraction motions, they are not the same thing. Physical rotation and the corresponding equations of rotation and frequency are part of the vector system of legacy physics, the principles of which are quite distinct from those of the RST.
Maybe for this reason, as unfortunate as it is for us, oscillation, as a pseudoscalar expansion/contraction, has not been studied much per se. As Larson put it, “After all, nobody is very much worried about the physics of expanding balloons. But that situation was changed very drastically by the development of the theory of the universe of motion, because scalar motion plays a very important part in that theoretical structure.” 1
Nevertheless, because Larson’s investigation of the mathematics of the scalar oscillations focused on one-dimensional vibrations, rotated two-dimensionally, his work is not very applicable to our investigations of 3D oscillations, where rotation, as defined in the legacy community, is replaced by expansion/contraction.
As explained in previous posts below, and in posts on the New Math blog, we note the interesting correlation between the ancient “mediato/duplatio” method of reckoning and the combinations of the 3D pseudoscalar oscillations. This is especially important, since, using the operational interpretation of number in the new Reciprocal System of Mathematics (RSM), we find the same “half/double” principle emerging as the central concept of operation in the system’s arithmetic; that is, 1/2 is the discrete unit in the negative direction, while its inverse, 2/1, is the discrete unit in the positive direction.
As we apply the new mathematics concepts to the new physics concepts, we find that we must add dimensions to the operationally interpreted numbers, since a physical expansion in all directions of space over time, is a reciprocal relation of 3D units to 0D units, or the 3D pseudoscalar over the 0D scalar, giving us the system’s equation of motion,
vs = ds3/dt0
where vs is the rate of volume change, rather than linear change, as in the legacy system equation. So, the natural algebraic unit of spatial volume, V, is simple to calculate:
V = vs * (t0 - t1)
= 23
= 8 cubic units
This is because, in one unit of scalar time, the pseudoscalar expands one unit of space in both directions of each of the three available dimensions, simultaneously, which produces Larson’s cube. However, since the three dimensions of the expansion are only the basis needed for describing the expansion in any given direction, they cannot be used for calculating the geometry of actual physical expansion, in all directions. The physical expansion is spherical, not cubic, and therefore we have to confront the age-old challenge of squaring the circle, in order to find the actual spatial volume in cubic measure.
Of course, as discussed previously, we know that squaring the circle is not possible, given that π is a transcendental number, not a rational one. Yet, in terms of relative units of π, we find a rational proportion between the inner sphere, contained by Larson’s cube and the outer sphere that contains the cube. Fortunately, it turns out that these two spheres are related by inverse geometry. In fact, in terms of relative values of π, inverse geometry shows that the outer sphere turns out to be the unit sphere, the identity element, if you will, while the inner sphere is half of the unit value, with a ratio of 1/2, while the inverse of the inner sphere is double the unit sphere value!
Since it’s much easier to make 2D diagrams than 3D ones, we will show how this works in 2D for now. Referring to figure 1 below, we see the plan view of the familiar combination of the sphere of radius 1, the sphere of radius 21/2, superimposed on one quadrant of Larson’s cube, containing its portion of the inner sphere, and at the same time just contained by the outer sphere. The largest sphere is the inverse of the inner sphere (making the outer sphere, the middle sphere in the diagram), according to the principles of inverse geometry (we will ignore the lines of the outer quad for now).
Figure 1. Three Concentric Circles of Unit Expansion
Proportionally, we know that the area of the inner sphere’s cross section, which is equal to π * r2 = π * 12 = π, is twice the area of the outer sphere’s cross section, which is equal to π * r2 = π * (21/2)2 = 2π, by the Pythagorean theorem. Now, the Greek, Apollonius, proved that the radius of the inner circle, OP, (O = origin) times the radius of the largest circle, OP”, is equal to the radius of the outer circle, OP’ squared, or
OP * OP” = OP’ 2,
which, numerically, is the inverse of OP. Using the usual notation of geometry, this is the same as that shown in figure 2 below:
Figure 2. B is the inversion of A with respect to C (and vice versa), by r2 = CA * CB
With this much understood, we can see that if we normalize the areas of the circle, setting the area of the outer circle with radius OP’ equal to 1 (i.e. 2π = 1), then the area of the inner circle is 1/2 of this value (i.e. 1π = .5*2π), and the area of its inverse circle is double this value (i.e. 4π = 2*2π). In terms of π then, we have three circles, the areas of which are numerically, or proportionately, equivalent to three ratios,
1/2, 1/1, 2/1,
which is the basis of the RSM and our theoretical, RST-based, development. But, what is more, is that these ratios also correspond to the 2π rotation of legacy physics! In other words, 1π of physical expansion, in the inner circle, is the inverse of 4π of physical expansion in the largest circle, so a total of 2π motion (one expansion/contraction cycle) is the inverse of a total of 8π motion (one, inverse, expansion/contraction cycle), so the ratio of one to the other is 2π/8π = 1/4, and 8π/2π = 4/1. Taking the latter case, the dimensions of the S|T combo motion would be 2D energy per unit of 2D velocity, or
(dt2/ds0)/(ds2/dt0) = dt2/ds2,
which is dimensionally correct for Planck’s constant, in the energy equation for radiation, E = hv, if it is understood that the dimensions of frequency, 1/t, should actually be the dimensions of velocity, s/t, in the equation, as Larson maintained.
The fact that the dimensions and the magnitude (8π/2π = 4π) of the S|T combo are correct, in this analysis, and that they accord with the findings of legacy physics, is very encouraging. Of course, we need to consider the volume of the spheres, not just their cross sections. We’ll discuss that more fully another time, but it should be noted that the 4π value of the S|T ratio can be understood in terms of uncertainty, because while a point is 100% localized, it’s 100% non-localized when expanded into a sphere, until it’s measured.
When measured after 1π, or 4π, expansion, the location of the original point is indeterminate, but can be described to within the parameters of the expansion. This reminds us of Heisenberg’s concern with the epistemology of quantum theory, as described in a paper by W. A. Hofer:
If quantization is only appropriate for interactions, i.e. measurement processes, then the results of quantum theory can only hold for actual measurement processes. But since the formalism of quantum theory is based on single eigenstates, meaning states of isolated particles, this logical structure is not accounted for by the mathematical foundations of quantum theory. While, therefore, the mathematical formalism suggests a validity beyond any actual measurement, it can only be applied to specific measurements. What it amounts to, in short, is a logical inconsistency in the fundamental statements of quantum theory.Even though the precise equations are yet to be determined, the new approach is giving us tantalizing hints that we are on the right track. If so, this inconsistency in the fundamentals of quantum theory promises to be completely resolved by the concept of 3D, or pseudoscalar, oscillations. Heisenberg’s discovery, so perplexing to scientists and philosophers alike, ever since, turns out to be simple child’s play: Expand a point and then measure its location at some point on the expanded surface, it will appear to have moved to the location on the surface at which the measurement was taken, and the next measurement will most likely define a new path. The probability amplitude is a function of time, the greater the expansion, the more possibilities there are on the greater surface area.
Thus, in the words of Heisenberg, “The path comes into existence only when we measure it.”