To say the least, we are quite excited. This is the very breakthrough that enabled the LST community to advance to what was eventually called the old quantum physics and, then, when Bohr’s model of the atom failed for the Helium spectra and Heisenberg, Schrodinger et. al. discovered how to apply the non-commutative mathematics of matrices to the problem, the new quantum mechanics was launched and the rest is history.

K.V.K. Nehru called the inability of the RST community to calculate the observed atomic spectra a “great lacuna” in the theoretical development of the Reciprocal System. In his 2002 paper, “Quantum Mechanical Approach Inevitable?” he suggested that the only course open to us was to use the vector motion wave equation after all. Nothing ever came of it however.

More recently, Ronald Satz published a paper in May of 2012 entitled, “Theory of Atomic Spectra and Ionization Energies,” in which he claimed to explain “both simple and complex spectra.” Nevertheless, I have not been able to verify it’s validity, and I know of no one else who has either. In my case, I find it too difficult and tedious to follow his unfamiliar and cumbersome notation to evaluate it adequately.

Of course, the very existence of the LRC is due to the difference in opinion over Larson’s conclusion that rotational motion can be considered scalar motion of magnitude only. Since the motion of rotation necessarily requires a change of position of something to define it, even the changing position of a rotating linear vibration doesn’t change the fact that such a change does not constitute scalar motion, but vector motion, and, since Satz’s theoretical development follows that of Larson’s, it incorporates the same error, and therefore cannot be considered an RST-based theory of scalar motion.

Consequently, even if he has correctly derived the equations in his paper, it makes no difference in the end, given the concepts behind those equations are vector motion magnitudes, rather than scalar motion magnitudes. It must be understood that scalar motion is not change of position motion, but change of scale motion, or a change of space size over time and a change of time “size” over space.

The foundation of the LRC’s RST-based theoretical development is the three-dimensional progression posited by its first fundamental postulate, where three-dimensional motion, with two reciprocal aspects, space and time, progresses in discrete units; that is, it constitutes an eternal increase of discrete units, and from this 3D progression of space and time, all physical entities and phenomena emerge, as motion, combinations of motion and relations between them.

Plotting the orthogonal increase of these discrete units of space and time, leads naturally and logically to the fundamental entities of observed radiation and matter, only when 3D “direction” reversals are introduced into this uniform progression, as developed in the LRC’s theory.

Formulating these 3D scalar reversals, in a simple closed form equation, shows that the most elementary motion combination, emerging from the theory, consists of a minimum four units of scalar motion. Combining multiple instances of these units, in combinations of four or more units generates various entities of scalar motion, which correspond to the entities of matter found in the first family of the standard model of the LST community’s particle physics.

The properties of motion, inherent in these combinations, relate to one another in exactly the same manner as the observed entities of the standard model; that is, they exist in pairs that have magnitudes of opposite polarity, and each pair is mirrored in a reciprocal, or orthogonal manner.

The various magnitudes of these scalar motion entities fit together in a manner exactly required to form the higher order combinations observed in the corresponding entities of the standard model, which are called bosons and fermions, with the exact number of observed fermions and bosons, each having the correct polarities, dimensions and magnitudes, which not only identify them with the observed entities, but also gives them the properties that permit them to combine in the same manner that the observed entities combine, forming combinations identified with the observed protons and neutrons and electrons of matter observed in the laboratory.

However, this remarkable achievement of the LRC’s theoretical development has been stymied for years by the same “glaring lacuna” Larson’s own development was. Even though the scalar motion combinations corresponding to the observed entities of the standard model easily combine into the elements of the periodic table, we have been unable to calculate the atomic spectra of the elements and show how the observed periods of the periodic table are a consequence of the energy levels of these combinations.

In other words, until now we have not been able to translate the scalar motion magnitudes of these various scalar motion combinations into observed values of conventional units of energy and radiation, which are characteristic of each element, beginning with the first element, Hydrogen.

Now, however, noticing the importance of the number 4 in the LST community’s history of the spectroscopic developments, which formed the foundation of the development of its quantum theory of physics, we have discovered the underlying connection, which has been so elusive.

It turns out that Johannes Rydberg used Johann Balmer’s empirical equation to formulate his own equation, which led to the LST’s Bohr model of the atom. Balmer discovered the important role of the number 4, or 22, in the sequence of rational numbers, which, when multiplied by a constant he had obtained by trial and error, equaled the observed wavelengths of the spectra of Hydrogen.

The trouble is, however, this important role of the number 4 in Balmer’s work has been obscured by the interpretation of the n terms in his and Rydberg’s equations, as simply the electronic energy levels of the Bohr model of the atom. It has been so easy to take for granted that the number, 22, explicit in Balmer’s equation, but disappearing in Rydberg’s reworking and generalizing of its mathematics, is to be identified with the n=2 orbit of the single electron of the Hydrogen atom.

Nevertheless, what really happened is that Rydberg noticed that Balmer’s constant was proportional to the ionization energy limit of Hydrogen. Indeed, it is exactly four times the wavelength of that energy. Consequently, Rydberg simply divided the number 4 by Balmer’s constant and inverted it! Therefore, Rydberg’s constant is nothing more than the inverted wavelength of Hydrogen’s ionization energy.

It’s not that this is a revelation to students of spectroscopy, but for non-specialists, this fact is glossed over in teaching how the Bohr model works and it’s implications have been buried in the history of the development of the LST’s quantum physics, where the emphasis is on the so-called new quantum physics, in which the Bohr model’s electron orbits, based on classical physics, are replaced by the orbitals of the weird physics of probability and the non-commutative mathematics of matrices and wave equations.

Not that the new shouldn’t have replaced the old quantum mechanics. No one is arguing that, but the fact that the number 4, which played such an important part in Balmer’s original work, as a crucial number, working perfectly in an equation for calculating the Hydrogen spectra, might not be tied to the n=2 orbit of an invalidated atomic model, is something that definitely should be considered!

In our scalar motion atomic model there are no electrons changing positions, so not only are they not rotating around the nucleons, in a classical, planetary-like orbit, with definite angular and spin momentum, at any given time, they are also not waves of probabilities either, based on equivalents of magnitudes of rotation, in moving or standing waves.

Again, rotation is just as much a vector motion as is linear motion. It’s only linear motion in two dimensions, as defined by changing sines and cosines. Sure, the probability of finding an electron in a given position along a classical orbit is 100 percent assured, as WWI fighter pilots soon learned from the Germans, but just because a particle is also a wave, whose position and momentum are mutually exclusive properties, doesn’t necessarily imply that the electron actually changes positions relative to the nucleons!

It’s a fact that it could also be changing size, as an oscillating volume of space and time.

If we stop and consider the position of a point, before it expands into a ball, we recognize that its position is clearly defined, but subsequently that position becomes undefined, as the radius of the ball increases along with the volume of the ball and the surface area of its sphere.

How is the position of the point to be determined at some time t, during its 3D expansion? The fact is that any location on its surface will be a valid location, since they are all located at the same distance from the origin, but when such a location is detected and considered to be a change of the point’s position, the probability of it being a given point on the surface decreases as the surface area grows.

Thus, without the knowledge of 3D scalar oscillations, the LST mathematicians and physicists evidently have done the only thing they knew how to do: They have constructed a rotational analog of 3D scalar oscillation and called it wave mechanics, giving birth to the new quantum physics, and all the angst that has surrounded it ever since.

Will we here at the LRC be able to progress from the calculations of the Hydrogen spectra, in our scalar motion model, to the calculations of the Helium spectra and beyond? I can’t say at this point, but I am confident, given that the theory yields bosons and fermions, and that the bosons consist of W minus and plus bosons, perfectly explaining beta minus and beta plus decay, while preserving the scalar motion of the entities involved, and that the theory’s fermions consists of quarks and leptons, which fit together perfectly as protons, neutrons and electrons, to form the observed elements of the periodic table.

To say the least, my confidence has now grown by leaps and bounds, given that the atomic spectra relations of these combinations of scalar motion based bosons and fermions have now been found to follow just as rigidly and inevitably as the previous ones have.

The only other explanation that I can see is that the fact that this rigid logical and mathematical development, of the consequences of the Reciprocal System postulates, as far as we have been able to carry them out, corresponds exactly to observation, is just a coincidence.

Who’s going to assert that at this point?

]]>In that article, he theoretically derived the force equations for electrostatic and magnetic charges, by showing that the space/time dimensions for these equations are correct, when the space/time dimensions for permittivity and permeability are correctly understood.

He shows that the correct space/time dimensions for the electric charge force equation are,

t/s2 = (s2/t)(((t/s)(t/s))/s2),

thanks to the dimensions of the permittivity term, (s2/t), which are those required on the right hand side of the equation to equal the space/time dimensions of force on the left hand side.

Satz gives us his rationale for using the space/time permittivity dimensions of the RSt, which differ from those of the LST, in that they are the inverse of the RSt’s space/time dimensions. Of course, such a clarification of space/time dimensions of permittivity, enabling this impressive derivation of the equation, which is missing in the LST community, went unnoticed then and is still obscure, even to this day.

Nevertheless, Satz went on in the article to derive the equation for the magnetic force law, as well, using the space/time dimensions of permeability, in the same manner. He shows that the correct space/time dimensions for the Coulomb law of magnetostatics are,

t/s2 = (s4/t3)(((t2/s2)(t2/s2))/s2),

but, as Satz explains in the article, the concept of permeability is more correctly understood as a magnetic analog to electric resistance, which we might as well consider “impermeability.” If we do this, we must invert the term, putting it in the denominator of the equation, so that the “impermeability” term of the equation becomes (1/t3/s4).

While this brilliant derivation of the force equations in terms of space/time dimensions still cannot make the headlines it deserves in the LST community, it is now confirmed, by the work of Xavier Borg of Blaze Labs, where he has, independently of Larson’s work, shown that the SI units of the LST are easily reduced to space/time dimensions, and that those for permittivity and permeability agree completely with those of the RSt, as can be seen from the table and explanation here.

Indeed, we can now extend Satz’s work to the gravitational force law and derive its equation as well, using the space/time dimensions of the gravitational constant, G, given in Borg’s table. These space time dimensions of G are (s6/t5) and inserting them into the force equation for gravitational mass, we get:

t/s2 = (s6/t5)(((t3/s3)(t3/s3))/s2),

Something that physicists might have really been interested in, especially given the energy equations, E= mc2 and E = h*v*, which are also expressable in terms of space/time dimensions,

t/s = (t3/s3)(s2/t2) = (t2/s)(1/t).

Though the implications of these insights might be lost on theoretical physicists busy “battling for the heart and soul of physics,” as we have blogged about on our “The Trouble with Physics” blog today, they should not be lost on followers of Dewey Larson.

However, I caused a real dust-up in conversations with online ISUS discussion groups, when I suggested using space/time dimensions with the force law, F = ma, years ago. I still don’t know exactly what the problem was, but I remember it caused Ronald Satz a great deal of heart burn.

Since then, of course, we’ve gone our separate ways, and I no longer have to worry about what anyone thinks of my ideas, even though I think I’ve made a fool of myself on more than one occasion. Yet, nothing ventured, nothing gained, as they say. If I happen to write what I think and what I think is not thought through enough, no one but me suffers for it, and I learn and grow in the process.

So, with that in mind, let me share what I have been thinking lately. As the readers of the three blogs on this site know, the recent entries of the New Math blog have dealt with a new multi-dimensional, scalar number system. By the usual definition and understanding of the term “scalar” in mathematics and physics, this may seem to be an oxymoron, but this is only the case, if rational numbers and motion are connected to the concept of direction.

Normally, scalars are used in the sense of denoting magnitude only. This is why Hestenes caused such a stir, with his geometric product at the heart of his geometric algebra. It mixes scalars and vectors in a way never contemplated in vector algebra.

But the idea of multi-dimensional scalar math is much simpler. It posits that numbers themselves have multiple dimensions. Not just in an operational sense, where a quantity of factors in an operation denotes dimensionality, but in the sense of the unit scalar itself.

Just as 11, 12 and 13 denote 1, 1x1 and 1x1x1 operationally, and mathematically are equivalent, but a unit line, a unit square and a unit cube are definitely not equivalent, so too are the unit scalars in the new math. They are equivalent is some sense, but not in another.

The difference is in the orthogonality of the factors. When each term in the algebraic operation of more than one term is orthogonal to the others, the result is quite different than when they are not. Geometric algebra has a way of dealing with this, but it mixes vectors and scalars in such a way that it becomes a much more powerful language than vector algebra.

However, in the world of scalar motion, we are dealing with something quite different. One-dimensional scalar motion produces a one-dimensional length, with magnitude in two “directions” (a line). Two-dimensional scalar motion produces a two-dimensional area, with magnitude in four “directions” (a circle), while three-dimensional scalar motion produces a volume, with magnitude in eight “directions” (a ball).

As the LRC’s investigation of these multi-dimensional scalar motions has proceeded over the years, it has been discovered that the units of the corresponding numbers used to denote these 2, 4 and 8 “directions” of scalar magnitude differ.

For the two, one-dimensional magnitudes of the length, the units are the familiar units we designate with the symbol 1, but for the four, two-dimensional magnitudes of the area, and the eight, three-dimensional magnitudes of the volume, the unit is not 1, but the square root of 2 and the square root of 3, respectively.

For the details, please see the entries on the New Math blog. The challenge before us on this blog is how to use the New Math to advance the LRC’s RSt. Recall that the bottom line of Larson’s RST is that the universe is composed of nothing but motion, combinations of motion and relations between them.

Given that, in our RSt, the scalar motions begin as three-dimensional vibrations, or pulsations, called Space Unit Displacement Ratios (SUDRs) and Time Unit Displacement Ratios (TUDRs), and, because of their unique properties, they may combine into SUDR|TUDR combinations (or S|T units), and then further combine into one, two and three dimensional combos, identified with the LST community’s fermions and bosons, we require a multi-dimensional system of mathematics to investigate their properties and interactions properly.

The standard, or legacy system of mathematics is based on concepts analogous to vector motion, i.e. motion with direction, while we need a system of mathematics based on concepts analogous to scalar motion, i.e. motion of magnitude only, but magnitudes with “direction,” the 2. 4 and 8 “directions” referred to above.

Whether or not we will succeed in this endeavor, remains to be seen, but it couldn’t be any worse than the awful situation the LST community now finds itself in.

]]>For instance, the discussion on one threaded forum goes like this (edited for clarity):

Commentor A:

[The question is] why a particle’s spin doesn’t remain invariant when you rotate it by 2pi, using the corresponding rotation operator. The easy answer is that spin doesn’t live in normal 3D space so it doesn’t transform with the usual rotation matrices from classical mechanics. Classical objects rotate using a representation of the group SO(3), whereas spin vectors rotate with a representation of the group SU(2).

Commentor B: “

A 2pi rotation changes the sign of the state vector, |\sigma\rangle\mapsto -|\sigma\rangle, but any state vector of the form C|\sigma\rangle where C is a complex number represents the same state as |\sigma\rangle.I don’t think this has anything to do with the fact that we’re using SU(2) instead of SO(3). SU(2) is used instead of SO(3) to make sure that we always have U(R’R)=U(R’)U(R) where U(R) is the rotation operator (acting on state vectors) that corresponds to the rotation matrix R (acting on vectors in \mathbb R^3). If we use SO(3), we’d sometimes get U(R’R)=-U(R’)U(R), depending on what paths in SO(3) that we use to get from the identity to R, R’ and RR’.I think that any representation of SO(3) on a 2-dimensional complex vector space will always have the property that a 2pi rotation changes the sign of the vector on which it acts. One could ask why we would choose to use a 2-dimensional representation. The answer to that is that we didn’t choose. The number of dimensions needed to represent the spin state of a spin S particle is always 2S+1.

Commentor C:

Why is it 4pi (it’s not 2pi) in spin space? Please give a picturesque explanation.

Commentor B:

There’s no way to really draw a picture of this. 2 complex dimensions are equivalent to 4 real dimensions and we can’t draw more than 3. Roger Penrose suggests the following picture, but I don’t think it’s helpful at all: Put one end of a long belt on the table, under something heavy, and put the other end between two pages in a book. Now rotate this book by 4 pi around the axis defined by the belt. Without rotating the book any further, you can “undo” the 4 pi rotation by looping the belt around the book. This doesn’t work with a 2 pi rotation.

Commentor D:

I read also that a waitress with a tray works nicely. If she rotates the tray once, her arm is twisted. To regain the original state, she has to rotate the tray by 720 degrees. I think you have to try it to see what I mean. I believe it was Feynman who gave me this picture, to give credit where it’s due.

Commentor E:

It is cute, but not perfect. Like a recently deceased kitten. Cheers.

Commentor F:

It’s nice to see a 4pi rotation is significant somehow, but this doesn’t really make it clear what this has to do with spin 1/2 particles. What part of an electron is analagous to the belt?

Here’s one way to understand it. Let the belt represent the path of the electron through spacetime. The electron may twist and turn as it travels, but let’s say that at certain points, it has the same orientation it started with. When this happens, the electron must be in a state related to the original one by some constant factor.

What can this constant be? Well, we need to assign it in such a way that the state of the electron varies continuously along its path, ie, no sudden jumps. Let’s say the electron has done two complete rotations since it started. Can the constant we assign at this point be something besides one?

The belt trick says no. To see why, let’s suppose we could make it something else, say A. Then we’ve assigned a state to the electron at every point along the path in such a way that it starts at some state |\psi> and ends at some state A|\psi>. Now, using the belt trick, we can continuously deform this path to one where the electron doesn’t rotate at all. But we’re holding the ends of the belt fixed, so it still has to start at |\psi> and end at A|\psi>. But this should be true no matter how short the path is, and when we take the path to be very small, this means the state must change discontinuously, unless A=1. On the other hand, a single rotation cannot be so deformed, so the constant doesn’t have to be 1. However, it’s restricted by the fact that doing this twice must return you to the original state, so it must be either 1 or -1. For an electron, it turns out the constant is -1.

Here’s a more technical explanation. The belt is supposed to represent is a path through SO(3), the group of rotations in 3 dimensions. Namely, each point along the belt corresponds to a rotation: the rotation that brings that slice of the belt into its current orientation. So we have a one parameter family of elements in SO(3), ie, a path in SO(3). Then by deforming the belt while holding its two ends fixed, we get a deformation of the path, ie, a homotopy of paths. The fact that a twice twisted belt can be deformed to a straight belt (while a once twisted belt can not) is then just the demonstration that a path in SO(3) corresponding to a 4pi rotation is homotopic to the constant path (while a 2pi rotation is not), ie, that the fundamental group of SO(3) is Z2. Thus the rotation group is not simply connected, and so it has representations that are not single valued. For one reason or another, nature chooses to use one of these representations for certain particles, such as the electron.

Commentor C:

Thank you! It is a visualized explanation. But I think it is similar to a picture of Berry phase, isn’t it?

Commentor G:

Another model is when two complete revolutions are only projections of single complete revolutions which are structured from two nested rotations:

You can imagine it like path in 3D space for pendulum with same inner and outer frequencies (axes of rotations are perpendicular). Resulting trajectory is on half-sphere (Viviani window curvature). One of the trajectory projection is circle with half radius and this is what we are able to measure like magnetic moment for particle with ½ spin. When structured rotors complete 2pi revolutions their composite projection complete 4pi.

That was the end of the discussion. I don’t know why the last comment remained unanswered. It’s not clear to me what this person is saying, but it seemed to be out of the ordinary context of rotation groups, which is what legacy physics (LST) is based on today.

Even though Schumm says physicists haven’t a clue as to the physical origin of quantum spin, this undefined concept is an indispensable part of legacy physics, as can be seen in this online discussion of spin 0 particles:

http://physics.stackexchange.com/questions/31119/what-does-spin-0-mean-exactly

Alas, however, you will not find a definition of “particle” in any of these discussions of quantum spin, since that concept itself is undefined. If you dig deep enough, you will find that the word “particle” is qualified, as “point-like.” That is, it’s not strictly a point, since a point is defined as something with no physical extent!

Consequently, the theoretical concepts of legacy physics are necessarily a mess, but they work well, to a certain extent, for calculations of observed physical properties. This places the new system of physical theory at a disadvantage, but then we don’t have the research resources the LST has.

Still, it’s nice to have a simple, straight forward explanation of the physical origin of quantum spin, as shown in the previous entry. The real satisfaction, though, comes from the fundamental explanation of magnitude, dimension and “direction,” which underlies the RST concept.

]]>News Flash (05/06/2012): Dr. Satz’s new paper, “Theory of Atomic Spectra and Ionization Energies,” is now available. Both simple and complex spectra are explained with ease using the Reciprocal System. So is the splitting of spectra in magnetic and electric fields. Ionization energies of the elements are calculated and compared with observation. There are no jumping electrons in the Reciprocal System; Quantum Mechanics has been vanquished.

This is apparently good news, but we have some concerns based on a preliminary study of the paper. A complete LRC review of the paper is pending.

In the meantime, continuing with the study of the LRC concepts of 3D oscillation, the last entry in this blog showed how the change in volume constituting the spherical oscillation can be compared to rotation and when it is, the enigma of 4π rotation, which has stumped the LST community for many decades now, becomes clear and easy to understand: It takes the equivalent of one 360 degree revolution to fill the volume and a second equivalent 360 degree revolution to empty it again, as the volume returns to its original state after 720 degrees of apparent rotation.

Again, there is no actual rotation involved. It’s just that by mapping the change in volume to a corresponding rotation, as the oscillation proceeds, we can exploit the well understood mathematical concepts of rotation and its relationship to waves.

The fact that the 8 1-unit cubes of Larson’s cube (LC), and the ball of radius 1 it contains, are equally partitioned so that each cube has the same proportion of spherical volume, enables us to map these 1/8 volumes of the 3D oscillation to the 2D unit-circle rotation based on units of π radians.

This is really fortuitous, since, while π/2 radians equals 90 degrees of rotation, or 1/4 of a complete rotation cycle, two, 1/8 sub-volumes of the inner ball of the LC constitute 1/4 of its total volume.

This fact makes 90 degrees of 2D rotation equivalent to a 1/4 change of 3D volume.

Consequently, we can represent the 3D volume oscillation in terms of a 2D rotation, where one 2π rotation in the meridian plane of a ball represents an increase of its volume, from 0 to 1 unit, and a subsequent 2π rotation of the meridian plane of a ball represents a decrease of its volume, from 1 unit to 0, as explained in more detail below:

**Figure 1**. The Volume of a Spherical Wedge

As it turns out, the volume of the spherical wedge in figure 1 is to the volume of its sphere (actually ball) as the angle of the wedge is to 2π:

,

or, as the angle of the wedge in degrees is to 360 degrees.

So, in our case, the ratio 1/8 = 45/360.

This is not only interesting with respect to the wave equation, but also to the finding of Dave Lackey that the relative quark and lepton masses are related to definite angles of rotation. That these might be wedge angles seems highly likely to me (see here.)

The two figures below show the 2D rotation representation of the 3D oscillation graphically:

**Figure 1.** First 2π Rotation

**Figure 2.** Second 2π Rotation

The reason this is so significant is that it provides a physical orgin of 4π quantum spin for the first time anywhere and also provides the basis for including the wave equation in our discrete system of theory.

]]>Our immediate research goal here at the LRC is to arrive at the point where we can calculate the atomic spectra, using our RST-based theory (our RSt). The LST community’s ability to do this (well, in principle, anyway) and our inability to do it is embarassing. Nehru called it the “glaring lacuna” of Larson’s RSt (see here.)

While pursuing this goal, we’ve discovered many wonderful things, but none of them has enabled us to do the calculations yet. Perhaps, we may be finding ourselves agreeing with Nehru that the use of the wave equation may be inevitable after all, if we are going to reach our goal.

However, there are some intriguing conceptual developments that are closely related to the atomic spectra, which show the fundamental difference between the LST and the RST that has to be taken into account, if we are to use the wave equation in our calculations. The LST defines motion as a change in an object’s location, the distance between locations over the time it takes to travel that distance.

This idea of translational motion is extended to rotation, but the “distance” traveled is measured in terms of a time rate of change in the quantity of radians, or unit angles. We can count between the cardinal points on a circle, in terms of π:

t0…………t1…………t2………….t3…………..t4

0………(π/2)……….(2π/2)……….(3π/2)………(4π/2)

But when this convention was employed in quantum physics to describe quantum spin, it was found necessary to modify it to:

0……..(π/2)………(2π/2)………(3π/2)……..(4π/2)……..(5π/2)………(6π/2)………(7π/2)………(8π/2),

because one revolution of quantum spin requires twice as much rotation as one revolution of non-quantum spin, something not understood to this day in the LST community.

Since the LST concept of quantum spin, as incomprehensible as it is, was the key to the breakthrough of quantum physics in the LST community and its successful calculation of atomic spectra energy levels, it’s important to understand how it relates to the RST theory of atomic spectra energy levels, which has no concept of spinning electrons, whirling around a nucleus of protons and neutrons.

Moreover, in our LRC RSt, unlike in Larson’s RSt, and in Nehru’s modification of Larson’s RSt, there is no concept of rotation! Nehru sought to employ complex numbers to represent what he called “one-dimensional spin” in the time (inverse) region of the universe of motion, because time is 3D in this region and space is scalar. He also extended this approach to the use of quaternions to represent what he called “two-dimensional spin.” The 1D spin applies to what Nehru called the “Atomic Zone,” while the 2D spin applies to what he called the “Nuclear Zone.”

These time region studies see a 1D “electronic” potential between rotation in the complex plane and the 3D unit progression, and a 2D “nuclear” potential between the rotation of the 1D rotation itself in an orthogonal dimension and the universal progression. The former, is formulated in terms of 2D complex numbers, while the latter is formulated in terms of 4D quaternions. This follows pretty much the LST approach to calculating these potentials with Lie algebras, but replaces the LST concepts of the atom, such as electron clouds and nuclei, with RSt concepts of n-dimensional scalar rotations, forming the atoms, which oppose the universal expansion.

However, since the LRC RSt concept requires an initial 3D oscillation of the space (time) aspect of the progression, with compounds of the resulting entities forming the bosons and fermions of the universe of motion, we must find the corresponding “electronic” and “nuclear” potentials in these 3D entities, without recourse to n-dimensional “scalar rotations.”

Apparently, the way to proceed in our case is to extend the idea of Larson’s 1D scalar speed-displacements in a natural way, which we have tried our best to do, by first understanding the mathematics of scalar motion, which is manifestly different than that of vector motion. Three-dimensional scalar motion does not involve the changing location of an object, but the changing size of an oscillating volume.

This periodic change in volume occurs from postulated changes in the outward “direction” of the universal space/time expansion, at a given point in that expansion. We can express the unit 3D expansion in algebraic terms, by expanding Larson’s cube (LC) over time:

t0…………t1…………t2………….t3…………..t4

10…………23………..43………….63………….83

But the 3D ocillation, reversing at each unit, introduces a differential between it and the progression, which is determined by a given dimension of the oscillation. For its 1D component, for example, the 3D scalar expansion is:

11…………21………….41………….61………….81,

creating the unit speed-displacement in one dimension:

10…………out:21/(out:1x21)………..out:41/(in:2x21)………….out:61/(out:3x21)………….out:81/(in:4x21),

or a 1D unit speed-displacement: 1:2.

However, for the 2D component of the 3D scalar expansion, we get:

10…………22………..42………….62………….82,

creating a unit speed-displacement in two dimensions:

10…………22/(1x22)………..42/(2x22)………….62/(3x22)………….82/(4x22),

or a 2D unit speed-displacement:12:22.

For the 3D component of the scalar expansion, we get:

10…………23………..43………….63………….83,

creating a unit speed-displacement in three dimensions:

10…………23/(1x23)………..43/(2x23)………….63/(3x23)………….83/(4x23),

or a 3D unit speed-displacement:13:23.

This shows we’re in the game and what’s more, the discovery of Miles Mathis that π = 4, in kinematic equations, places the 3D potential squarely in our Wheel of Motion, as each group of elements follows a 4n2 , or a πn2 relation.

But we need to quantify the physical expansion/contraction using continuous magnitudes, since nature doesn’t expand by cubes. One way to do this is to divide the volume of the unit ball into 8 sub-unit balls that have a volume corresponding to the 8 1-unit cubes in the 23 stack of unit cubes of the LC.

The volume of the unit ball is just 4π/3, since r = 1. Thus, (4π/3)/8 is the sub-unit volume we need, and, as it turns out, the radius of this 1/8 volume is just the cube root of its ratio to the unit volume, or the cube root of 1/8, which is 1/2.

V1 = (4π/3), V2 = V1/8

V2 = (4π/3)r3

r3 = (V2/V1) = 1/8

r = (1/8)1/3 = .5

Now, if we map these sub-unit volumes (SV) to angles of rotation of the unit circle, we can construct a rotational analog of the volume oscillation. Our first four points of reference on the unit circle for the first four volumes are:

SV0…..SV1…..SV2…..SV3……SV4.

and our first four angles of rotation are:

0………(π/2)……….(2π/2)……….(3π/2)……….(4π/2),

Happily, 1/8th of the unit volume is equal to (π/2)/3, so our 3D scalar to 2D rotation map is:

0…..(π/2)(2/3)……(2π/2)(2/3)…..(3π/2)(2/3)…..(4π/2)(2/3)

which fills from 0 to unit volume in 1 unit of time, the expansion, and then the contraction deflates, from unit volume back to the starting point (0), taking another unit of time:

(4π/2)………(5π/2)……….(6π/2)……….(7π/2)……….(8π/2),

(4π/2)(2/3)…..(3π/2)(2/3)……(2π/2)(2/3)……..(π/2)(2/3)……….0

This is a significant achievement of the RST. Bruce Schaumm, on page 187 of his book, *Deep Down Things*, linked above, writes:

We don’t really have a clue about the physical origin of [quantum] spin. To describe spin as “intrinsic angular momentum” is like your best buddy describing how your car’s differential works by explaining that it “employs a mechanical linkage;” the only useful information contained in this statement is that its author probably knows next to nothing about how a differential actually works.

Well, it’s not all that hard to explain in the universe of motion:

The reason it seems that it takes two 2π revolutions to complete one cycle of quantum spin is that the “spin” is not actually 1D rotation, in 2D space, but the 3D oscillation of a volume in 3D space, as depicted below:

In the equivalent of one revolution of 2π, the 3D oscillation has fully expanded, which is one-half of its cycle. To return to the starting point at zero, requires a second unit of time, and the equivalent of a second 2π revolution. With this map of 3D oscillation to 2D rotation, we ought to be able to adapt the wave equation to move forward in the goal to calculate the atomic spectra.

**UPDATE:** As (π/2)/3 = .523598… = V1/8, two of these volumes, or V = (π/2)(2/3), are the equivalent of one π/2 rotation (90o.)

**2nd Update: **The above update is in error, since (π/2)/3 = .523598… radians = V1/8 = 30 degrees, not 45 degrees. Hence, the calculations shown are incorrect. The correct calculations are shown below:

If we map these 8 sub-unit volumes (SV) to 4, 90, degree segments of 2π rotation of the unit circle, we can construct a rotational analog of the volume oscillation. Our first 4 points of 90 degree reference on the unit circle for the first 4 sub volumes are:

SV0…..SV1…..SV2…..SV3……SV4

and our first 4 segments of 90 degree equivalent rotation are:

0………(π/2)……….(2(π/2))……….(3(π/2))……….(4(π/2)),

Mapping the 1/4 volume to π/2 radian of rotation, the required delta in volume per 90 degrees of rotation, gives us .66667 volume units per 90 degrees of rotation:

V/x = π/2

x/V = 2/π

x = V(2/π)

x = 2.6666,

V/x = 4.18879/2.66667 = .66667 volume units, per 90 degrees of rotation.

Hence, the value of the 8 SVs in the original calculation does not correspond to the four, 90 degree rotations, as indicated in the graphic. In terms of radians, each of the 8 SVs is equivalent to 30 degrees of rotation, not 90.

Therefore, each delta in volume, corresponding to 90 degrees of rotation has nothing to do with the eight sub-volumes in the LC.

Instead,

0……..(π/2)………..(2(π/2))……….(3(π/2))……….(4(π/2)) =

0…..(.66667)…..(2(.66667))…..(3(.66667))…..(4(.66667)),

which fills the volume from 0 to unit volume in 1 unit of time, the expansion, and then the contraction deflates, from unit volume back to the starting point (0), taking one more unit of time, for a total of 360 degrees of equivalent rotation, per cycle, but taking two cycles, or 720 degrees of equivalent rotation, to return to the starting point (0):

(4(π/2))…………..(5(π/2))…………(6(π/2))…………(7(π/2))………(8(π/2)) =

(4(.66667))…..(3(.66667))……(2(.66667))……..(.66667)……………0

I’ll correct the graphic later.

]]>The LST community covers the enigma up with “Poincaré stresses,” but truth be told, it was the reason the LHC was built: They want to resolve the issue, not just cover it up. The RST community is still striving to resolve it, as well. K.V.K. Nehru challenged Larson’s concept of “simple harmonic motion,” which Larson described as “…a motion in which there is a continuous and uniform change from outward to inward and vice versa.”

Nehru objected to the validity of this conclusion, based on the fact that scalar “directions,” inward and outward, are discrete. There is no scalar “direction” that is partly outward or partly inward. He writes:

Since there is nothing like more outward (inward) or less outward (inward) the question arises as to the meaning of the statement “a continuous and uniform change from outward to inward”? Outward and inward, as applied to scalar motion, are discrete directions: the scalar motion could be either outward or inward. There are no intermediate possibilities.

In the LRC RST-based theory (RSt), the periodic “direction” reversals are 3D, thus avoiding the saw-tooth vs. sine-wave dilemma that plagued Larson and that drove him to positing his concept of simple harmonic motion. The reversal from a 3D expansion to a 3D contraction, and vice-versa, clearly has the gradual change, to which Nehru objected, built right into it: As the expanding volume grows toward unit size, its outward rate of spherical expansion slows, even while the radius’ rate of expansion remains constant. At the point of reversal, the decrease of the volume in the inward “direction” is again gradual at first, even though the radius’ change of “direction” is instantaneous.

At the zero point (3D origin), however, this is not the case, unless we recognize the nature of the point described in the FQXI paper: In that case, the gradual change in “direction” of the spatial sphere, at one end, is matched by the gradual change in “direction” of the temporal sphere, at the other end, and, thus, it is perfectly analogous to the concept of the interchange of inverses that is inherent in rotation and also in simple harmonic motion.

Nevertheless, while 3D oscillation solves the enigma of the point, it introduces another one, an enigma that is uniquely ours: If the 3D space (time) unit oscillates by changing into its inverse, isn’t that tantamount to the numerator changing into the denominator, in the case of the SUDR, and vice-versa, in the case of the TUDR?

This question has gnawed at me ever since I wrote the FQXI paper. The tentative conclusion that I have been forced to come to is that it’s a matter of accounting. If 8 units of space are converted into 8 units of time, during an expansion to 64 units of space and 64 units of time, then the net balance is 64 - 8 = 56 units of space and 64 + 8 = 72 units of time, an 8 unit deficit of space and an 8 unit surplus of time.

During the next step, when 8 units of time are transformed into 8 units of space, the space deficit is made up from the time surplus. This is not unlike the swinging pendulum, when the potential energy is max, it’s all on one side of zero, and this surplus is transferred back to the reciprocal side, from which it came, before the cycle repeats itself.

If it works for mass, momentum and vector motion, why not for space, time and scalar motion? Maybe Ted’s quantum wave equation would be applicable after all.

:)

To say the least, I needed something interesting to write about, before the year ends. I did post an article about the LST preon theory published in the 2012 November issue of Scientific American, but I wanted something new to write about our RST-based theory of preons (RSt). To go on record in this momentous year 2012, with something significant, something to advance the theory, just seems important.

Well, in the general discussion the other day, the fact that the LRC’s RSt, unlike Larson’s RSt and unlike Peret’s RSt, is developed using mathematics, albeit a new mathematics, as well as logic, came up and I decided that I had better go back and reread it to refresh my memory, in order to be ready to answer questions about it.

Wow, was I surprised! Here, before leaving the LRC for six months, Larry the mathematician and I had been working out the mathematics of the SUDR and its inverse the TUDR in pursuit of the properties of the preons and thus the standard model particles. I had discovered my silly mistake of using the square root of two and its inverse in this endeavor, instead of the square root of three.

Big difference. Instead of dealing with neat values with a factor of 8, we were facing a somewhat awkward factor of 27! It takes 27 volumes of a SUDR to equal the value of one TUDR! I didn’t want to believe it. I tried and tried to figure a way out of it, but finally had to resign myself to the fact that we would have to work with it.

Then, to my utter surprise, I found it confirmed in the development of the new mathematics that I had written up four or five years ago. My doubts originated from the assumption of inverse geometry, where we used the equation r’2 =r x r”, to quantize the SUDR and TUDR and found that the SUDR volume is 1/27 the volume of the TUDR. Yet, writing years before learning anything about this equation, I had come up with the exact same thing, only in terms of poles!

I had forgotten all about it, but counting the poles of the dimensions of the tetraktys,

1) 20 = 1

2) 20 + 21 = 3

3) 20 + 21 + 22 = 9

4) 20 + 21 + 22 + 23 = 27

the 3D line, the fourth dimension counting from 0, contains 27 poles: 1 monopole, 3 dipoles, 4 quadrupoles and 1 octopole (1 + (3x2) + (3x4) + 1(8) = 27.)

Now, how do calculations of geometric ratios, using the equations for circumference, area, and volume, all requiring the use of pi, and employing the equation of inversive geometry, in other words, the use of equations of continuous magnitudes, turn out to be ** identical** to calculations based on fundamental numbers, dimensions and polarities (“directions”), all based on discrete quantities?

A surprise it may be, but it is the truth. Check it out for yourself, dear reader.

It turns out that the connection is the sacred number three, the union of duality, which, impressively enough, is the very definition of the Reciprocal System, the fundamental of which are embodied in Larson’s Cube.

The graphic above is outdated, but appropriate, because when I developed the new mathematics, I had yet to even think about the inner and outer circles contained by the LC and also containing it. This is all I had at the time.

I think this is a worthy announcement, a discovery to match the year 2012, and just in the nick of time too!

:)

]]>

I’ve always hated that question, because in a 3D universe, the periodic table of elements ought to be 3D, but the wheel of motion, like the periodic table, is only a 2D representation of the periods of its elements. It has always seemed to me that showing the true periodic nature of the elements in the wheel format, which eliminates the confusing gaps in the table format, was accomplishment enough, especially since the wheel format clearly shows the true magnitudes of the RST-based 4n2 numbers, rather than quantum mechanics (QM)-based 2n2 half-period numbers of the table.

Well, it turns out that several years ago, studying the numbers of the tetraktys, we discovered how they are actually the algebraic equivalent of the geometry of Larson’s Cube. Since then, we’ve been trying to understand the SUDR and TUDR, and their combinations as preons, which we call S|T units that combine to form the so-called “fundamental particles” of our preon version of the standard model, and connect them to the wheel. These entities combine to form the seqence of elements in the wheel, following a 4+16+36+64 = 120 pattern of mathematical “slots” for the 118 elements plus the proton and the neutron.

It’s not been easy, but we have made some progress, with many starts and stops along the way. One important connection links the progression of the LC (and thus the tetraktys) with the wheel of motion. We’ve found that we can encode the universal space/time expansion in terms of the expanding LC, which can be written as the expanding 3D level of the tetraktys

TEn = ne0 (+) 2(ne0)1 (+) 4(ne0)2 (+) 8(ne0)3

where TEn is the expanded LC equivalent of the tetraktys, n is the unit variable of its time expansion, and e0 is the scalar time unit (i.e. e0 = 1, the point, but it also represents one line, one square and one cube, after one unit of time expansion), corresponding to one edge of the LC (measured from the interior center point of the LC to the face center, lying along any one of the three axes of the stack of 8 one-unit cubes, making up the LC.)

So, in 1 unit of time, the value of TEn is

TE1 = 1*10 (+) 2*11 (+) 4*12 (+) 8*13 = 1 (+) 2 (+) 4 (+) 8

The (+) operator indicates the joining together of the four components of the LC, the respective values of the scalar point unit, the linear line unit, the square area unit, and the cubic volume unit of the LC, into one, unified, entity, isomorphic to the 3D tetraktys (20, 21, 22, 23).

This is similar to the concept of joining of the scalar, vector, bivector and trivector into a multivector in Hestenes’ Geometric Algebra. Only, in our case, we join the 0D scalar and the three pseudoscalars of the tetraktys into the LC.

Hence, at t=2,

TE2 = 2e0 (+) 2(2e0)1 (+) 4(2e0)2 (+) 8(2e0)3 = 2 + 4 + 16 + 64

Similarly,

TE3 = 3 + 6 + 36 + 216,

and

TE4 = 4 + 8 + 64 + 512.

Now, clearly, any of the six 2x2 = 4 face units of the 2x2x2 = 8 stack of cube units of TE1 corresponds to the four slots of the innermost wheel of the Wheel of Motion, where the RST-based period is 4n2 = 4, when n = 1 (The four slots for the Proton, Neutron, Protium, Deuterium (H).)

Similarly, any one of the six 4x4 = 16 face units of the 4x4x4 = 64 stack of cubic units of TE2 corresponds to the sixteen slots of the second, outer wheel, when 4n2 = 16, n = 2.

The third wheel corresponds to one of the six 6x6 = 36 face units of the 6x6x6 = 216 cubic units of TE3, where the RST-based period is 4n2 = 36, n = 3, and the fourth wheel corresponds to one of the six 8x8 = 64 face units of the 8x8x8 = 512 stack of cubic units of TE4, where the RST-based period is 4n2 = 64, n = 4.

Of course, the question is, how do we know that this is anything other than a mathematical coincidence? Where is the physical connection to these numbers? Well, the answer is that we are still trying to clarify that connection, but it’s certainly interesting to note that the ratios of the associated balls of the LC follow this same numerical pattern that we see in the expanded right lines.

That is to say, the ratios of the respective radii, surfaces and volumes of the two balls associated with TE1, TE2, TE3 and TE4 (the outer ball and its inverse), follow the same numerical pattern as do the edges, lines, squares and cubes in the expanding LC, because, as it turns out, the ratio of the radius of an initial volume to that of the nth volume of equal magnitude added to it, is equal to the cube root of n. Everything follows from there.

If we take the outer ball of the LC, which geometry and algebra agree must have a radius equal to the square root of 3, and its inverse ball, which must have a radius equal to the inverse of the square root of 3, and follow their expansion as the LC expands, we find that it takes the sum of the volume of 8 balls in each case to expand the ratio of the radius of the initial ball to the radius of the nth ball from 1 to 2, (81/3 = 2).

It then takes 64 of these volumes to expand the ratio of the two radii to 4, (641/3 = 4). It takes 216 volumes to expand the ratio to 6, (2161/3 = 6), and it takes 512 volumes to expand the ratio to 8, (5121/3 = 8).

Thus, we see that there is not only a 0D, 1D, 2D and 3D mathematical (discrete unit) connection between the algebra of the tetraktys, the geometry of the LC and the Wheel of Motion, but there is also found a corresponding physical (i.e. continuous unit) connection between them.

Moreover, we see that it is the area aspect of a given volume that yields the 4n2 periods of the wheel, which means that the associated sums of the 3D volumes are actually incremented to form the 2D elemental slots in each period; That is, summing the volumes leads to the RST’s 4n2 relation, because the six faces of each expanded stack of one-unit cubes are degenerate. Thus, only one need be selected to select associated two-unit volumes. Consequently, since the smallest face possible has a magnitude of four squares, when the successive expanded LCs are divided by 4, we get the required relation:

4/4 = 12; 16/4 = 22; 36/4 = 32; 64/4 = 42.

And because this is so, we can just as easily represent the periods in the wheel as a factor of cubes:

8n3 = 8, n=1; 8n3 = 64, n= 2, 8n3 = 216, n=3; 8n3 = 512, n=4,

which is just the cubic progression of the LC.

So, to answer Larry, from a graphics perspective, it’s just much easier to draw a 2D wheel of motion than it would be to draw a 3D version. However, from a mathematical perspective, the 2D aspect of the wheel cannot be separated from its 3D aspect, because, in reality, they are simply two aspects of the same thing.

The bottom line is, even though the Wheel of Motion is not drawn in 3D, it represents a 3D volume sequence nevertheless.

Now, we need to find the way to build the expanded LCs (TE1, TE2, TE3, TE4), using the S|T units that serve as the preons to our standard model of particles. Some ideas on that next.

]]>To see this, we need only change the way the equation is usually written. We change from

E = mc2,

to

E = c2 * m,

which, in space/time terms, may be written,

t/s = (s/t)2 * (t/s)3,

but this can also be written in terms of SUDR and TUDR ratios, as a ratio of their respective 2D area scalar speeds to their respective 0D radius scalar speeds, in this manner:

t/s = ((s/t)/(t/s))/((s2/t)/(t2/s))

In the reciprocal speeds of the first term, the 0D radii space units yield unit speed ratios:

s/t = 2(20)/2(20) = 2/2 = 1 and t/s = 2(20)/2(20) = 2/2 = 1,

and the physical dimensions are squared, when the inverse multiplication operation is carried out:

((s/t)/(t/s)) = ((s/t)*(s/t)) = s2/t2.

However, in the 3D term, the physical dimensions are raised up from 2D to 3D, by the inverse multiplication operation:

s3/t = 2(22)/2(20) = 8/2 and t3/s = 2(22)/2(20) = 8/2.

Hence:

((s2/t)/(t2/s)) = ((s2/t)*(s/t2)) = s3/t3,

which are the volume dimensions of inverse mass.

Consequently, when the final inverse multiplication operation is carried out, we get Einstein’s familiar equation:

t/s = (s/t)2 * (t/s)3.

The surprising thing about this, however, is that the inverse multiplication operation on the ratio of these two geometric magnitudes, equates them numerically with the next higher dimensional unit, what we might call the fundamental magnitudes of Larson’s Cube (LC), the 1D line width (21), the 2D area face (22) and the 3D volume cube (23); that is, the *necessary* mathematical operation of the fundamental reciprocal relation raises 2 to 4 (22 = 4) and 4 to 8 (22 * 2 = 23 = 8), and then, subsequently, the 2D area magnitude multiplied times the 3D volume magnitude is equal to the 1D line magnitude, just as it should be.

This will no doubt be really useful in working with the S|T units of the preons in terms of energy, starting with the neutrinos and working our way down to the electrons and positrons and finally combining the quarks into protons and neutrons and adding the electrons to the protons, to begin working our way up the periodic table of the elements.

But more than this, it appears that this marvelously clear insight into the meaning of the terms in Einstein’s famous equation might reduce the horribly arcane subject of modern physics to a toy model that can be taught, starting in elementary school.

Now, we need to take a look at the less famous, but more enigmatic equation of Max Planck, E = h*v, *in the light of the LC..

I just finished watching an interesting talk by Ed Witten entitled “Knots and Quantum Theory,” given at the prestigious Institute for Advanced Studies (IAS) at Princeton. Witten more or less describes his exploration of the connection that knot theory has to quantum theory, which serves to exemplify how researchers spend most of their time “floundering around in the dark,” and finding themselves “actually stuck” in their efforts to make sense out of the physical structure of the universe.

I have found the same thing in my little microcosm of theoretical physics research. Only in my case, instead of struggling to understand the monumental and vast complexity of the arcane subjects with which the intellectual giants at IAS work, I struggle to understand the straightforward first four numbers, one, two, three and four. Nevertheless, I too must admit that I spend most of my time floundering in the dark and actually stuck.

For example, for a long time I have tried to connect in my pea brain the LRC’s preon model of the so-called elementary particles of particle physics with the two balls of Larson’s cube (LC); that is, the inner and outer balls defined by the LC, the one contained by it and the one containing it.

These two balls are nested in an intriguing way that can be extended in both “directions” analogous to the way numbers can be extended in the positive and negative “directions.” I had noticed that the radius of the next smaller inner ball (in the “negative direction”), nested inside the radius of the initial inner ball, and is the inverse of the first outer ball, with the inner ball in between them representing unity (to distinguish these three balls, I will refer to them as inverse, unit and outer, from this point on) And since, presumably, this is the relationship of the unit space and unit time oscillations (SUDRs & TUDRs) used to construct the preons of our preon model, it seemed to be a natural conclusion that they could be used to quantify the SUDR and TUDR. However, so far, the attempt to proceed on this basis has only led to floundering and I have been stuck ever since.

At length, I’ve had to conclude that, in spite of tantalizing clues that this is the direction to go, I must be doing something wrong. I must not be thinking about the LC in the correct way and this has caused me to reflect on the curious observation that I made some time ago, that these balls are nothing real. The only real part of the construction is the LC itself. The balls can only exist as some sort of phantom representation of the discrete numbers of the LC. They appear as soon as the LC appears and disappear as soon as the LC disappears, even though their radii are both smaller and larger than the unit magnitude of the LC, regardless of how small or large that unit might be, ad infinitum.

Yet, what I have also known for some time is that the ratio of the inverse and outer balls is a rational number, which relates to the discrete numbers of the LC in a remarkable way, but I think I tried to use that fact in a way that that led me to equate the SUDR and TUDR with the two, inverse, balls directly, when I should have been using their ratio only.

This means that the reciprocal relationship of the SUDR and TUDR is still to be found in the original 1/2+1/1+2/1 = 4/4 equation, not in its equivalent, substituting the square root of 2 or the square root of 3 for the unit. However, both of those square roots are associated with the discrete equation in the sense that their ratio translates the discrete numbers of the LC to the continuous magnitudes of the two, inverse, balls.

For example, the numerical progression of the LC is 23 = 8, 43 = 64, 63 = 216, 83 = 512, where the base number is 2 because each dimension has two “directions,” and the exponent is 3 because there are no new phenomena beyond 3 dimensions (Bott’s periodicity theorem). But just as the LC’s volume expansion is a discrete number, the expansion of its two associated, inverse, balls is a continuous magnitude, given by the volume formula for a ball, V = 4/3 * π * r3.

Consequently, since the r of the 2D slice of the unit ball is always 1/2 of the cube root of the LC discrete number, the associated discrete progression of its radius is, 1/2*81/3 = 1, 1/2*641/3 = 2, 1/2*2161/3 = 3, 1/2*5121/3 = 4, and since the r of the 2D slice of the outer ball is always the square root of 2 times the corresponding r of the 2D slice of the unit ball, the discrete progression of its radius is 21/2 * 1 = 21/2, 21/2 * 2 = 81/2, 21/2 * 3 = 181/2, 21/2 * 4 = 321/2.

Now, the question is, what are the corresponding radii of the 2D slice of the inverse balls? The procedure that I have been trying to use depends on the assumption that this radius can be determined by geometric construction: Simply construct a new LC inside the unit ball and take the radius that fits inside it as the next lower radius in the progression. This procedure comes from the fact that each upper radius can be constructed similarly. So, how can it matter what size we choose as the unit size to relate the two inverses to?

However, as I said, this has led to floundering, even though it seems logical. I should note that it was compelling to me for at least two reasons. First, it’s easy to see that the radius of the first ball constructed in this manner is the inverse of the square root of 2, the radius of the outer ball. Second, it allows us to extend the radii in two “directions” from the unit ball, situated between these two, inverse, radii, just as the number 1/1 is situated between the two inverse numbers, 1/2 and 2/1.

In spite of these two sirens, however, I think it’s necessary to resist the temptation to go that way and instead to look at unity as the real part that must be increased. This means that we take the unit progression as 1/1, 2/2, 3/3, …n/n. While this might seem to be a trivial assumption given the fundamental postulates of the RST, the 1/2+1/1+2/1 = 4/4 equation was misleading, since it seemed to imply that the traditional number line, 1/1, 2/1, 3/1, …n/1, should be taken as representing the positive displacement values we need: To the left of 1/1, we have the increasing values of what Larson called time displacements, and to the right we have the increasing values of what he called space displacements, which are the inverse of the time displacements.

This is logical and straightforward, but perhaps it is wrong in the sense that it only succeeds in describing the inverse displacements from 2/2. After all, we can’t get any displacement from 1/1. If this is so, then the next two displacements we have to go to are at the 4/4 and the 6/6 units in the progression, and so on:

1) 2/2 = (1/2)/(1/2)

2) 4/4 = (1/4)/(1/4)

2) 6/6 = (1/6)/(1/6)

.

.

.

n) n/n = (1/n)/(1/n)

The advantage of this line of thinking over the previous is that it brings the mathematical development into conformity with the postulates, and it makes the relationship of the SUDR and TUDR to be inverse in the same sense that space and time are inverse: The quantity on the left is the inverse of the quantity on the right, by virtue of the division symbol; that is, they are not inverse numerically (i.e. both are positive), but they are inversely related in the equation, just as two opposed, but equal, 1D line segments are both positive magnitudes.

It’s also obvious that the 1 in the numerator is not the quantity 1 numerically, but it is the same number that is in the denominator, raised to the power of 3, representing 1 volume unit as a whole, expanded and contracted, as the number 1 represents one cycle of 2π radians in the LST equations. Thus, in terms of space and time dimensions, the discrete progression turned oscillation is different in each dimension. For three dimensions, the volume progression (i.e. volume frequency of oscillation, in units per cycle) is:

2*(23)/2*(20) = 8, 2*(43)/4*(20) = 32, 2*(63)/6*(20