The Larson Research Center

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,

v_s = ds³/dt⁰

where v_s 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 = v_s * (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 2^1/2, 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 * (2^1/2)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.”

One of the important topics of discussion in ISUS, since the days of Larson, was the “factor 3” he used, in the calculations of the gravitational and Planck’s constant. His calculation was challenged over and over again, because of this “factor,” which he seems to pull out of no where. He felt it was certainly connected with dimensions, but he wasn’t sure just how. He writes in Chapter 14 of Basic Properties of Matter:

The ratio of the natural unit of mass in the cgs system to the arbitrary unit, the gram, was evaluated in Volume I as 2.236055 x 10^-8. It was also noted in that earlier volume that the factor 3 (evidently representing the number of effective dimensions) enters into the relation between the gravitational constant and the natural unit of mass. The gravitational constant is then
3 x 2.236055 x 10^-8 = 6.708165 x 10^-8 (with a small adjustment that will be considered shortly).

In calculating Planck’s constant, he refers to the same “factor 3.” He writes in Chapter 33 of The Structure of the Physical Universe:

If we call the energy of the photon E and the ionizing energy k, the maximum kinetic energy of the ejected electron is E - k. Energy, E, is t/s, but in the time-space region where velocity is below the unit value, the effective value of s in primary processes is unity, hence E = t. The unit value of s has a similar effect on frequency (velocity) s/t, reducing it to 1/t. The conversion factor which relates frequency to energy is therefore t divided by 1/t or t². Since the interchange in the photoelectric effect is across the boundary between the time-space region and the time region it is also necessary to introduce the dimensional factor 3 and the regional ratio, 156.44. We then have

E = 3 / 156.44 t²v (139)

In cgs units this becomes

E = (3 / 156.44) × ((0.1521×10^-15)²/6.670×10⁸) v = 6.648×10^-27 v ergs (140)

The coefficient of this equation is Planck’s Constant, commonly designated by the letter h.

Later, Satz rederives Planck’s constant, without recourse to the factor of three. In “A New Derivation of Planck’s Constant,” he writes:

Larson¹ was the first to attempt to derive Planck’s constant from the Reciprocal System. Because of the change in the calculated natural values of mass and energy in the second edition of his work², the original derivation has been invalidated. The factor of three that was used is dimensionally incorrect since the photon is a one-dimensional vibration. And the use of the cgs gravitational constant in such an equation is wrong since the result cannot be converted to a different system of units such as the Sl (mks) system. The remainder of Larson’s original equation (including the use of the interregional ratio and the square of the natural unit of time) will be shown to be correct.

This paper assumes that the factor of three is definitely connected to dimensions, even though Larson evidently wasn’t convinced. On this basis, Satz eliminates it, because, as he indicates above, in Larson’s development the photon of radiation is one-dimensional, but he probably got his clue from Sammer, as the following letter to Larson, dated 1986, shows:

Jan Sammer     560 Riverside Drive   Apartment 3Q   New York, NY 10027

    
 September 3, 1986          

Dear Dewey:

Re: revisions to chapter 4 of Basic Properties of Matter, “Compressibility.”

As I understand from Ron, the change involves the natural unit of pressure, which has increased by a factor of three, because the mass unit was increased by this same factor. The change in the mass unit affects enrgy, momentum, pressure, and force. But this increase is compensated for by the fact that pressure is one-dimensional, rather then three-dimensional. Thus the final figures should be close to the ones you calculated on the basis of the old value of the mass unit.

Regards,

Jan

However, the factor 3 that appears in the gravitational constant would not be affected by these one-dimensional considerations since mass is three-dimensional. Indeed, Larson’s uncertainty about the precise meaning of the dimensional character of the factor 3 can be seen in his discussion of one-dimensional electrical phenomena too, where the factor again shows up. He writes in Chapter 16 of The Basic Properties of Matter:

The same considerations apply to the size of the unit of this quantity. Since the charge is not defined independently of the equation, the fact that there is only one force involved means that the expression QQ’ is actually Q¹/2Q’¹/2. It follows that, unless some structural factor (as previously defined) enters into the Coulomb relation, the value of the natural unit of Q derived from that relation should be the second power of the natural unit of t/s². In carrying out the calculation we find that a factor of 3 does enter into the equation. This probably has the same origin as the factors of the same size that apply to a number of the basic equations examined in Volume I. It no doubt has a dimensional significance, although a full explanation is not yet available.

The natural unit of t/s², as determined in Volume I, is 7.316889 x 10^-6 sec/cm². On the basis of the findings outlined in the foregoing paragraphs, the value of the natural unit of charge is

Q = (3 x 7.316889 x 10^-6)² = 4.81832 x 10^-10 esu.

There is a small difference (a factor of 1.0032) between this value and that previously calculated from the Faraday constant. Like the similar deviation between the values for the gravitational constant, this difference in the values of the unit of charge is within the range of the secondary mass effects, and will probably be accounted for when a systematic study of the secondary mass relations is undertaken.

But as we discussed in the previous post, the difference between the algebraic calculations and the geometric ones, has to be taken into account. At the LRC, the photon (bosons) and the matter particles (fermions) are combinations of three-dimensional entities, not one-dimensional vibrations that can then be rotated two-dimensionally and one-dimensionally, as in Larson’s development. In the LRC development, the initial vibration is a 3D vibration, so the factor 3 can’t be eliminated on the grounds of reduced dimensions in the case of photons, pressure, and charge.

It turns out though that the dimensional meaning of the factor 3 may have more to do with the inherent characteristics of Larson’s cube than those of physical entities and forces. The reason for this conclusion is purely mathematical, however. Recall that, as we have been able to assign geometric dimensions to natural numbers, by modeling the the 3D progression with Larson’s cube, where each dimensional component of the scalar progression pertains to a natural series, the algebraic series cannot correspond to the geometric series; that is, the numbers which progress algebraically, are not the same as the numbers which progress geometrically, or scalarly, because a scalar progression must occupy none or all of the possible dimensions to be scalar, by definition.

Thus, the RST’s time progression is scalar, because time has no direction in space. In other words, it has zero dimensions, while its space progression is scalar, or what we call pseudoscalar, because it expands in every direction, defined by all three spatial dimensions of Euclidean space. Hence, the RST’s scalar space|time expansion, from any given moment of time or space, algebraically produces Larson’s cube in one unit of time. The three dimensional components of this expansion are shown in table 1 below.

Table 1. The Algebraic Number Series of the RST Expansion 

Referring to Table 1, we can see that in one unit of time, 1⁰, the 1D expansion expands to 2 linear units in each of 3 directions (one unit left, right, up, down, forwards, backwards), while the 2D component expands to 4 square units in each of 3 directions (three orthogonal planes, each with 4 square units), and the 3D expansion increases to 8 cubic units, one in each of 8 diagonal directions. In the subsequent units of time, the numbers increase as shown in the table.

However, these algebraic numbers cannot be produced by a physical scalar expansion, because such an expansion expands in every direction simultaneously, producing an expanding ball, not an expanding cube. This brings us to consideration of the efforts to square the circle, as discussed in the previous post below. Recall that the inner circle of figure 1, with radius equal to 1/2 of the number of the 1D series, has a π*r² area, which, at one unit of time, is equal to an area of π, since 1² = 1. The area of this circle is 1/2 the area of the outer circle, after one unit of expansion, which has a larger radius equal to the square root of 2, making the area of the larger circle 2π. Figure 1 is reposted below for convenience:

 

Figure 1. The Geometric Expansion vs. Algebraic Expansion

In other words, the geometric expansion is a fraction of the algebraic expansion. However, the two are related integer values, related by π, but π in terms of expansion area per unit of time, not in terms of rotation per unit of time. This is made clear by arranging the geometric calculations of four units of progression in table 2, below.

 

Table 2. The Geometric Number Series of the RST Expansion

As can be seen by comparing the two tables, besides the π factor, the coefficients of the two tables are remarkably different. What’s really remarkable, however, is that the geometric series reproduces the numbers of the periodic table of elements! In the progression of the area of the inner circle, A_ic, we see the 0D expansion of the algebraic series squared, which corresponds to the four, 1/4, periods of the periodic table. In the progression of the area of the outer circle, A_oc, we see the reproduction of the 1/2 periods of the periodic table, corresponding to the number of elements between the noble elements, where the geometric numbers are the product of the 0D algebraic numbers and the 1D algebraic numbers of table 1.

When the progression of the surface areas of the inner and outer spheres (A_is & A_os) are calculated for the same four units of expansion, the progression of A_is surface area corresponds to the number of elements in each successive period of the periodic table (recall that the first period actually ends with deuterium, having three precursor entities before it (including hydrogen). The strange series of numbers in the last row is the series corresponding to the surface area of the outer sphere, A_os, but what do these numbers correspond to?

We don’t know yet, but notice the startling fact that they are a factor of three larger than the A_is numbers! What we have here is a purely mathematical, multi-dimensional, scalar, progression matching nature’s pattern of elements arranged by relative mass. The fact that combining these with their inverses, in the manner that we are studying at the LRC, in the form of SUDR and TUDR combinations, provides a number of “slots,” corresponding to these numbers, is simply breathtaking.

For instance, the 4π “slots” can be filled with four 1/4 magnitudes, the 16π slots with sixteen 1/16 magnitudes and so on. As regular readers of this site know, our major goal is to calculate the atomic spectra, using the new system of theory. One major milestone that we anticipated in this effort was reproducing the marvelous work of Larson in explaining the periodic table of elements, as a 4n² pattern of physical magnitudes, rather than the 2n² pattern of quantum mechanics. As can be seen from the right most column of table 2, we are closing in on this milestone, but we hardly expected to find the mysterious factor of 3 in the process.

In Larson’s work, all physical constants disappear except for the constant c, and the constant R. For example, the constants G and h, are derived in the RSt, in principle. In practice, however, it has not worked out as well as might be hoped. Both Nehru and Satz have attempted to improve on Larson’s derivation of the Planck constant, though no one has attempted to rederive G, as far as I know.^1,2

It appears that, In the work of the LRC, however, these calculations will have to be redone, because here it must be recognized that the dimensional property of the space|time progression has been misunderstood in Larson’s work; that is, mathematically speaking, the geometric values of the physical dimensions of the progression, s/t (t/s), in Larson’s development, are s¹/t¹ (t¹/s¹), but they are s³/t⁰ (t³/s⁰) in our development. This means that the function of space per unit of time is non-linear: At time t₁, one unit removed from time t₀, space has progressed (1x2)³ = 8 cubic units; At time, t₂, the space progression is (2x2)³ = 64 cubic units, and so on.

This does not effect the constant c, because it is measured in only one direction from the source, as the spherical expansion, when measured in polar coordinates, proceeds according to the equation,

r - r₀ = c(t₁ - t₀).

Clearly, when

r₀ = t₀ = 0, then

r/t = c,

where r is the radius of the expanding sphere, regardless of the direction measured from the origin to the surface of the sphere. In other words, the volume of the expanding ball of space (the algebraic pseudoscalar) increases non-linearly, even though the radius of the expansion increases linearly. 

At first glance, this might seem to be an immaterial observation (no pun intended), but it becomes relevant the moment we introduce space (time) direction reversals in the progression. When space (time) direction reversals are introduced in Larson’s development, the s/t progression oscillates in only one dimension, but, in the LRC development, this is impossible, since it would reduce the dimensions of the pseudoscalar and violate the conservation of motion. The space (time) reversals in the s³/t⁰ progression must be a 3D oscillation, expanding by a magnitude of (1x2)³ = 8, 3D, units in the outward direction, then contracting by a magnitude of (1x2)³ = 8, 3D, units in the inward direction, for each cycle.

Since, unlike c, the G and h constants necessarily arise in the context of the scalar vibrations, the 16, 3D, units of change per cycle must be taken into account. We can’t just focus on one dimension, ignoring the dynamics of the 3D, pseudoscalar, oscillation. Moreover, in the case of the gravitational motion, evidently only a portion of the total change is directed inward, giving rise to the gravitational constant of proportion, G. If this is the case with the Planck constant, as well, it follows that the proper physical dimensions of the vibrational motion of radiation are actually the dimensions of velocity, s/t, not the artificial dimensions of frequency, 1/t, just as Larson deduced.

The reason that using the dimensions of frequency, 1/t, as a parameter in the energy equation, works in spite of this is that the dimensions of Planck’s constant, similar to the dimensions of the constant G, which is, aside from the higher dimensions involved, similar to spatial distance, or motion multiplied by some interval of time, are actually inverse motion multiplied by some interval of space, yielding a value analogous to temporal distance. The difference, in this case, is that the constant h is an analog of spatial distance; that is, in the gravitational force equation,

Gm/s² = ((s³/t³ * t) * t³/s³)/s² = t/s²,

G is motion times time (a form of spatial distance), and in the energy of radiation equation,

hv = ((t²/s² * s) * 1/t = t/s,

h is inverse motion times space ( a form of temporal distance), because inverse motion times space is time.

Since, in our development, unlike in Larson’s development, the initial combinations of speed displacement units are combinations of 3D motion, it appears that matter (3D time speed-displacement) precedes radiation, if radiation is taken to be motion in less than three dimensions. However, we have concluded that the SUDR|TUDR combination, 3D time speed-displacement plus 3D space speed-displacement, constitutes radiation, because it is this combination that propagates at unit speed relative to oscillating space and oscillating time (SUDRs & TUDRs). Yet, it is clear from observation that all radiation originates from matter, so the implication seems to be that the original conclusion was not correct, and that radiation forms as dimensionally reduced matter.

However, this line of thought also raises several other issues, especially issues with particle pair production and particle annihilation, where nothing is left behind in the conversion of radiation into matter and matter into radiation. If, in the latter case, the 3D scalar motions, constituting the mass of the electron and the positron, are transformed into the dimensionally reduced motion of radiation, and the motion is conserved according to the equation,

E = mc²,

the question becomes how, or why, the collision of the two particles changes the dimensions of the motion from 3D to 1D? But in the former case, the opposite occurs, so the question becomes how, or why, the 1D motion of the radiation is transformed into the 3D motion of mass, and the motion conserved according to the equation,

m = E/c².

Of course, the particle pair production cannot occur spontaneously, that is, photons don’t decay into particle pairs spontaneously. For the phenomenon to occur, there must be an interaction of radiation with matter, and even then it cannot occur unless the energy of the radiation is at least twice the equivalent energy of the mass of the particle produced. These are our clues as to the physics involved, along with the fact that the properties of the produced pair of matter particles are always opposite, except for their masses and spins.

For example the mass and spin of the electron and positron are the same, but they have opposite charge. In LST theory, the reason why particle pair production cannot occur spontaneously has to do with conservation of momentum. Photons have the equivalent energy of the particle pair, according to E = mc², but not the momentum; that is, the photon is massless, and consequently restless (it cannot be brought to rest in a spatial reference system), while the particles are massive and therefore restful (they can be brought to rest in a spatial reference system). How, then, can the created pair’s quantity of 1D translational motion, or momentum (t³/s³ * s/t = t²/s²), be created from the photon’s 1D vibrational motion, or ν (reckoned as 1/t)?

The LST answer is that the photon has to interact with matter in a way to bring the photon to rest, which is to say it has to interact in a way that doesn’t result in mere Compton scattering, reducing the energy of the photon through absorption of some of its energy by the scattered electron, but in a way that converts all of its energy into the moving mass of the particle pair.

Since, in the LST, the photon has angular momentum of spin 1, it is presumably conserved in the angular momentum of the spin 1/2 of the particle pair, but the photon’s motion of propagation, relative to matter, at light speed must be converted into linear momentum, or rest mass times velocity, according to the m = E/c² equation.

Even so, before we can even begin to make these calculations, there is the immediate problem of the geometric versus algebraic disconnect, affecting the size of the units involved; that is, though the number of volume and area units is algebraically attainable, from the dimensional considerations, the actual size of the units must be geometrically determined.

As long as one-dimensional, linear, units are considered, the number and size of the units is straightforward, as only the radius of the expansion/contraction is relevant, but when the oscillation is a three-dimensional expansion/contraction, the number of orthogonal directions (i.e. the mathematical dimensions) is eight units (the eight diagonals from the origin to each corner of a 2x2x2 stack of cubes), not the one linear unit of the radius.  Moreover, the size of these volume units is not the 1x1x1 algebraic unit, in each of the eight directions, but something less than this due to the geometric expansion from the origin in all directions simultaneously.

The change in the physical situation is shown in figure 1 below.

Figure 1. Squaring the Circle  

Referring to figure 1 above, the algebraic value of one 1x1x1 cube is calculated using the formula of a prism (V_p = a x b x e), while the geometric value of the unit radius expansion within this cube is less and must be determined by the formula of a ball (V_b = 4/3π x r³). But in three dimensions there are eight 1x1x1 algebraic cubes, so the total volume of the ball is contained in the algebraic volume and is consequently something less than the 2x2x2 algebraic volume, with the radius d. For a unit volume, this turns out to be V₂, which is equal to 4/3π, as shown. The area of the surface of the ball, the area of the sphere, is A₂, which is equal to 4π. 

However, given that the measure of the speed of the expansion, from the origin to a point selected on the surface of the sphere, is a one-dimensional line of one unit, the radius d, the question arises as to how to properly calculate this value in terms of scalar motion, rather than vector motion, if its proportional area and volume are relevant physical factors, which they must be.

For instance, in the case of a material explosion, the momentum of a particle, traveling from the origin to the surface in a one-dimensional line, would represent only a fraction of the total momentum and energy contained in the expanding volume, and, in fact, only a fraction of the momentum and energy contained in the fraction of the volume of the section contained within one of the eight 1x1x1 algebraic cubes of the total volume.

Yet, while the total scalar motion of the actual 3D unit of space|time has no associated mass or momentum, per se, it definitely has a three and two-dimensional component, not just the one-dimensional motion of one radius, and clearly, therefore, it must be taken into account in any calculation.

However, a definitive understanding as to how to proceed in this manner is not available at this point in the development, even though it seems clear enough that previous treatments, such as Nehru’s, which relies on a one-to-one identity between natural units of energy E and natural units of speed S/T, might have to be re-examined in the light of the three-dimensional units discussed here.

References:
1. K.V.K. Nehru, “Theoretical Evaluation of Planck’s Constant,” http://www.reciprocalsystem.com/rs/cwkvk/planck.htm
2. R. W. Satz, “A New Derivation of Planck’s Constant,” http://www.reciprocalsystem.com/rs/satz/constant.htm 

The online development of RST-based physical theory here at the LRC is an experiment in itself. The reason for taking this path is to provide maximum visibility into the process for students and donors. The idea is to develop the Structure of the Physical Universe Document (SPUD), as an interactive, living, document that both documents the development of the theory and provides a basis for discussion and teaching of the principles involved.

Consequently, the development of the SPUD should be our main focus. However, it’s been somewhat neglected in our initial efforts, while most of the work has gone into writing the three blogs. Unfortunately, this has resulted in a fragmented commentary, as the topics of the blog entries tend not to follow a linear path of development.

However, the theoretical development has progressed to the point where we really need to get it documented in the SPUD, so we will begin doing that next. Meanwhile, the blogs will be a running commentary on the SPUD work, and aspects of the implications, the backtracking, the mistakes, the breakthroughs, the problems, the rumination, etc, associated with the development effort.

In this way, even though the commentary is necessarily fragmented, the logical, linear, development of the theory (the product we might say) will always be available in an organized form for study and analysis.

Currently, we are concentrating on the “Material Combinations” section of the SPUD, where we are working on describing the S|T units as the preons of standard model entities. In the course of this effort, we are refining the ideas about the preons, first discussed in this blog (here), in some interesting ways. One of these has to do with the dimensions of scalar magnitudes.

Recall that, as we follow the logic from the unit space|time progression to the time speed-displacements (SUDRs), and the space speed-displacements (TUDRs), we have three dimensions of SUDR|TUDR combo units (S|T units), which are our preons of standard model entities: the time-like, the space-like, and the light-like, dimensions of progression, where we assigned primary colors to each: Time-like progressions (with net time displacements) are red, space-like progressions (with net space displacements) are blue, and light-like progressions are green.

Adding one or more SUDRs to the initial S|T unit,

ds/dt = 1|2 + 1|1 + 2|1 = 4|4 num

alters the light-like (green) progression of this S|T unit in two ways,

(1|2 + 1|1 + 2|1) + (1|2) = (2|4 + 2|1 + 2|1) = 6|6 num 

With two red SUDRs and one blue TUDR the resultant S|T unit is “redder” than the initial S|T unit; that is, it has more “red” time speed-displacement than “blue” space speed-displacement, so the color of the S|T unit, which we can represent with the 2D color magenta, when the number of 1D red and blue components in the unit are equal, shifts to a redder shade of magenta, when there are more SUDRs than TUDRs, and a bluer shade of magenta, when there are more TUDRs than SUDRs. 

However, while a shift in the 2D color of the unbalanced S|T unit, either to the redder end of the spectra, or to the bluer end of the spectra, indicates the relative number of time and space speed-displacements in a given S|T unit, the question arises, does it also indicate a shift in the “direction” of the unit’s progression; In other words, while the progression of the magenta 4|4 unit is light-like (green), with equal space and time progression, is the progression of the 6|6 unit shifted either in the time-like (red) “direction” or the space-like  (blue) “direction?” 

We can plot these three cases of S|T units, as shown in figure 1 below, assuming that the shift in color equates to a shift in the “direction” of the progression:

Figure 1. Three Instances of S|T Unit Progression

Of course, a shift in the red “direction” corresponds to a slower speed of progression, while a shift in the blue “direction” corresponds to higher speed of progression, contrary to observation, where photons of varying frequency always propagate at c-speed, so what is going on? 

Well, this is an interesting point, because as it turns out, even though the number of SUDRs and TUDRs in a given S|T combo unit may be equal or unequal, the total ratio of space to time progression is always 1:1; that is, 4|4 = 6|6 = n|n = 1|1. This means, that the plots in figure 1 above are incorrect, because while the color of the arrows change, depending upon the relative number of SUDRs and TUDRs in an S|T unit, the unit value of the total progression is always conserved. This means that adding SUDRs and/or TUDRs to S|T units changes the color of the unit, but not the “direction” of its progression in the world line chart, just as the frequency of photons can vary, without changing their speed of propagation.

Consequently, we can represent a two-dimensional result of combining two, orthogonal, one-dimensional values of progression, as a frequency change, but without affecting the speed of progression, which always remains at unity. Still, while 4|4 = 6|6 = 1|1, the quantity of progression of a 4|4 S|T unit is not the same as the quantity of progression of a 6|6 unit, for a given number of steps of progression, since the constituent number of SUDR and TUDR oscillations are happening in parallel so to speak

For example, after three cycles of progression (2 units per cycle) in a 4|4 S|T unit, there are a total of 3 x 4|4 = 12|12 total nums, while in the 6|6 unit there are 3 x 6|6 = 18|18 total nums. Moreover, while there is only one S|T unit with the 4|4 magnitude, there are two S|T units corresponding to the 6|6 magnitude, or, in general, to the n|n magnitude, when n > 4, depending on which “direction” the unbalance occurs. Thus, in the 6|6 unit, the two values are

2|4 + 2|1 + 2|1 = 6|6, and

1|2 + 1|2 + 4|2 = 6|6,

The red and the blue magnitudes, we might say. Switching to the continuous reference system, also see the difference, as the SUDR value, at 1/2 = .5, is 1/4 the size of the TUDR value, at 2/1 = 2. Hence, in this system, the magnitude of the red 6|6 unit is

2(1/2) + 2 + 2 = 5, and the blue 6|6 unit is

1/2 + 1/2 + 2(2) = 5.

And there is only one continuous value for the 4|4 unit:

1/2 + 1/1 + 2/1 = 3.5

I have no idea yet what all this implies, but it’s clear to see that it parallels the idea of a ground state of energy, since there is nothing less than 1|1 = 0, or 1/1 = 1, and if we incorporate this “green” dimension into our plots on the world line chart, clearly, the range of n|n values would be orthogonal to both the time-like and space-like dimensions.

Interesting.

The discrete and continuous nature of space|time ratios in our RST-based theory of SUDRs and TUDRs is a new feature, but the vibration and wave analogs of these is not. We have seen that the expansion/contraction of a sphere, or spring, is a binary function, easily confused with the dual functions of sine/cosine, when the latter are represented as two, independent, springs, whose oscillations are given the proper phase relation, as commonly demonstrated in textbooks.

Nevertheless, when the expansion from point (local) to sphere (non-local) is analyzed, it is quite clear that it is an analog of binary rotation, or, maybe more correctly, binary rotation is an analog of the expansion/contraction function. In our case, this was discovered accidentally, when we happened to use GA to try to graphically illustrate the SUDR, TUDR, expansion/contraction here. Even more remarkably, however, it now turns out that when Hamilton discovered the quaternions, he immediately realized that the quaternion rotation was binary, but he ignored this fact, and, according to Simon L. Altmann, it wasn’t until 1958 that the mathematics community began to recognize it:

The first explicit statement to the effect that something is wrong here which I have been able to find is as recent as 1958 by Marcel Riesz…: ‘Hamilton and his school professed that the quaternions make the study of vectors in three-space unnecessary since every vector can be considered as the vectorial part.. .of a quaternion. ..this interpretation is grossly incorrect since the vectorial part of a quaternion behaves with respect to coordinate transformations like a bivector or “axial” vector and not like an ordinary or “polar” vector.’ However damning this statement is, it is only half the story, since the pure quaternion is not anything like a vector at all: we shall see that it is a binary rotation, that is a rotation by π.

Altmann believes that Hamilton’s motivation for disregarding the binary nature of the rotation of a pure quaternion is found in his need to “understand the physical, or geometrical, meaning of equating the square of the imaginary, or quaternion, unit, i², with -1.” Hamilton understood from Argand that “i² should be a rotation by π, which, duly enough, multiplies each vector of the plane by the factor -1,” writes Altmann, who then goes on to explain the crux of the matter:

For this reason, Hamilton always identified the quaternion units with quadrantal rotations, as he called the rotations by π/2. Clifford associates himself with this interpretation which he presents with beautiful clarity. The sad truth is that, however appealing this argument is, to identify the quaternion units with rotations by π/2 is not only not right, but it is entirely unacceptable in the study of the rotation group: we shall see, in fact, that they are nothing else except binary rotations.

Altmann is a recognized expert on the rotation group. He shows conclusively in this 1989 paper that Hamilton deliberately forced quaternions to fit his preconceived convictions of I² = -1, as a 180 degree rotation, even though he knew that the so-called “pure quaternion” (where the real part is 0) is a binary, or double, rotation, which presumably would make i² = -1 equate to a 360 degree rotation. Altmann quotes Hamilton’s admission of this in his 1853 Lectures on Quaternions:

…the SYMBOL OF OPERATION q( )q-l. where q may be called (as before) the operator quaternion, while the symbol (suppose r) of the operand quaternion is conceived to occupy the place marked by the parentheses.. .[can be regarded as] a conical transformation of the operand round the axis of the operator, through double the angle thereof, …

The parentheses are Altmann’s. It appears, though, that, in the special case of “pure quaternions,” either rotation, single or double, may be understood, and Hamilton’s choosing the single angle is interpreted by Altmann to be due to his focus on algebra and his notion of vectors, a notion that was not the same concept as what eventually came to be meant by his word vector:

…the status of vectors in this scheme is highly dubious, …in Hamilton’s approach rotations become subservient to the algebra, which opens the door to a variety of misinterpretations.

Altmann’s thesis, as indicated by his paper’s title, Hamilton, Rodrigues, and the Quaternion Scandal, is that Rodriques understood the binary nature of quaternion rotations, before they were discovered by Hamilton, but this was recognized only belatedly, and its significance is therefore unappreciated. He writes:

We must now discuss again the significance of the quaternion units. Because they are pure quaternions they must now be identified with binary rotations (rotations by π). This, for Hamilton, must have been absurd: the relation i² = -1 must still be satisfied. But the product of two rotations by r about the same axis is a rotation by 2π. This is clearly the identity operation, i.e., one which does not change any vector, whereas we are now saying that it is equal to - 1, i.e., that it changes the sign of all vectors in space. I believe that this is the reason Hamilton was forced to accept his parametrization, since this agreed with his picture of quaternion units as quadrantal rotations. Rodrigues, practical man as bankers must be, knew better than to worry about this strange result of his geometry-he did not carry, like Hamilton, all the world’s problems on his shoulders. Nature and history, alas, were playing games with Hamilton. How was he to know that Cartan was going to discover in 1913 objects (spinors) which are indeed multiplied by - 1 under a rotation by 2π, exactly as Rodrigues’s parametrization requires?

Moreover, when the topology of the rotation group became understood in the 1920s through the work of Hermann Weyl, it became natural to accept that the square of a binary rotation multiplies the identity by - 1 and thus behaves like the quaternion units. Though this should have shown the enormous importance of quaternions in the rotation group, they were by that time somewhat discredited, so that other much less effective parametrizations of the rotation group were in universal use. It must be stressed that the Rodrigues approach to rotations, by emphasizing their multiplication rules and by regarding them entirely as operators, fully reveals the group properties of the set of all orthogonal rotations, the full orthogonal group S0(3), as it is now called. The set of all normalized quaternions (in the Rodrigues parametrization) is a group homomorphic to SO(3) and it is its covering group.

Although I cannot go into the mathematical significance of this statement, its practical importance in quantum mechanics, e.g., can be easily understood: it permits the study of the transformation properties of the wave functions of the electron spin. It is for this purpose that quaternions are superb, because their use in dealing with rotations makes the work not only simpler but also more precise than with any other method.

But for us, of course, the significance of this confusion goes even further, because we have identified the scalar expansion/contraction of the SUDRs and TUDRs with a binary transformation, equating to 2π rotation. Hence, the mathematical expansion of a point to a sphere, and the contraction from a sphere to a point, can be represented as a full cycle, or 2π rotation, as shown in the figure below:

      

 

Figure 1. Sine Curves of Two, Reciprocal, Rotations

In the figure above, the phase of the curves is shifted 90 degrees relative to the rotations, from what they would normally be. The 0 points are at the top and bottom and the 1/1 point is at the center, so, when the rotations start, at the 0 point, the curves should start at the top and bottom, respectively. Also, the timing is off a little, but this will have to do until I can find the time to fix it. 

When a SUDR and TUDR are joined together in an S|T combo, the two, reciprocal, oscillations, represented as two, counter, rotations, must be represented as two, inverse, pure wave forms, but these are not the so-called positive and negative frequencies of the real sinusoid waveform that is so fundamental to physics, the canonical example of which is the mass-spring oscillator.

The new interpretation of the mass-spring oscillator, which we are employing, where the vectorial displacement of the mass, due to the restoring force of the spring relative to the equilibrium position, is not the important consideration, but rather the scalar expansion/contraction of the springs, in the two-spring configuration, is our focus. When we do this, we find that the two components of the real sinusoid take on a new meaning. Of course, as might be guessed, the new meaning has to do with a change from Hamilton’s quadrantal unit to Rodrigues’s binary unit, in the quaternions.

If the true quaternion unit is a binary unit, not a quadrantal unit, as Altmann points out, and this discovery is so germane to the study of groups and quantum mechanics, this appears to us to be an extremely important connection in our study of these scalar motions.

It shows that the representations of the 90 degree phase relation of the positive and negative components of the sinusoid,  the crucial element of Fourier transforms, are only part of the story.  It is important to recognize that the 180 degree, inverse, relation of two scalar magnitudes, the compressed and expanded states of the springs, or the S|T unit oscillations, that is associated with simple harmonic motion, can no longer be ignored. The S|T unit is analgous to a combination of sines and cosines, but now there are two, reciprocal, sinusoids, which we must consider that are composed of these.

Clearly, however, combining these two, 2π, units, does not result in constructive and destructive interference, as in the case of the quadrantal units, but rather in two, reciprocal, aspects of one function, as seen, for example, in the envelope of a modulated carrier wave, with a modulation index of 1, as illustrated below.

Figure 2. AM and FM Modulation of a Carrier Wave.

Interestingly, in the case of frequency modulation of a pure tone, we can clearly see that the information carried by the high frequency is the reciprocal of the information carried by the low frequency, which is a perfect analog, if not proof, of our compressed/expanded spring concept in representing these scalar oscillations as binary (2π) rotations. Of course, what’s not shown in the above graphic, is that there are always two side bands in a modulated carrier wave, one of which is regarded as redundant, and normally discarded.

Nevertheless, in treating each sinusoid, we use the usual equation of the function

x(t) = A (sin ωt + φ),

and then of course if the phase, φ, is shifted 90 degress, we get the cosine function

x(t) = A (sin ωt + π/2) = cos(ωt) 

both of which comprise the sinusoid signal.

But what equation do we use in treating both sinusoids of a modulated carrier? We need an equation for a single function of two binary rotations, and recalling that the double binary rotation (meshed gears) is the relation of two, reciprocal, aspects of two individual S|T units, joined in an S|T triplet, we can recognize something else, something that is profound: Equating the rotation of each gear to a sinusodial, one the reciprocal of the other, means that, since each of the two sinusodials is composed of a sine and a cosine, or phase quadrature, function, the analogy between our RST-based theory, with its discrete units of motion (SUDRs, TUDRs, and S|T combos), and the science of complex-number-based digital signal processing, with its discrete Fourier theory, is suprisingly and breathtakingly close.

It seems clear that, to the extent that we can take advantage of the well established principles in the science of digital signal processing, we can proceed to uncover the true relations of vector and scalar magnitudes, the continuous and the discrete, as they are reflected in the fundamental tetraktys of numbers and its associated algebras.

While this thought is as daunting as it is provoking, what choice do we have but to continue to follow our noses? Stay tuned.  

Update: I edited the text and added a graphic to clarify this post Oct 31, 2007