The Larson Research Center

In 1929, Charles Janet published his paper, “Deliberations on the Structure of the Nuclear Atom,” in which he proposed an interesting alternative to today’s Periodic Table of Elements. Although it has been almost completely ignored, his table was loosely based on the same 2n² periodic relationship used in quantum mechanics, but, as shown in figure 1 below, with an interesting twist. The first period contains four elements!

Figure 1. The Periodic Table of Charles Janet Published in 1929.

This table, appearing on Albert Tarantola’s website, is based on Janet’s work, which Tarantola refers to as the “first ordering” of the elements. It follows the accepted electron configuration sequence in ascending atomic order, so even though the two columns of akali metals and akaline earth metals are moved over to the right under hydrogen and helium, the accepted order of electron shell filling is maintained, as shown in table 1 below.

Table 1. Z-Ordered Electron Configuration of Janet Table (Read left to right, top to bottom)

This is interesting in the context of the LRC’s RST-based research, because the periods of Larson’s periodic table (here) are based on his 4n2 relationship, which requires four elements in the first period. Of course, it’s always been assumed that the three missing elements in the first period were incomplete elements such as the proton, neutron and electron (or neutrino, or massless neutron, etc.)

However, the placing of helium, lithium and beryllium into the first period is very interesting, although it “dethrones the noble gases from their key positions in the table, which is especially disappointing in view of the LRC’s Wheel of Motion form of the table.

Interestingly, as Tarantola and others have pointed out, another form of the table can be constructed from a ratio of square roots of atomic ionization potentials, as first shown by Le Cornec in 2002, using actual ionization data from the existing literature. Although Tarantola dismisses the significance of Le Cornec’s work, because he offers no theory to explain it, he does acknowledges that it shows a reversal in the order of the “s” and “p” energy levels, which, unlike Le Cornec, who asserts that this indicates a major failure of quantum mechanics, he simply attributes to the fact that the atom is a “complex object.” Tarantola calls this table the “second ordering” and leaves it at that.

For us, though, the fact that the ionization pattern shows a different order, inconsistent with the standard atomic model, is extremely interesting. One of the reasons why appears in the mapping of the s, p, d and f energy levels to the tetraktys. If we place these sets of 2 energy levels into the tetraktys, with subscripts indicating each, we get table 2 below.

Table 2. Duality of Electron Configurations Mapped to the Duality of the Tetraktys

As will be readily noticed, the energy levels map perfectly to the tetraktys, in some respects. There is one of the four dual “s” sets at the top, with the remaining three “s” sets positioned in the same repeating fashion, at the beginning of each row, as the 2⁰ values in the tetraktys. The total number of sets is consistent as well, with four dual “s” sets, three dual “p” sets, two dual “d” sets and 1 dual “f” set, each set having two, six, ten and fourteen members, respectively (the difference between the successive 2n² periods), arranged into four groups, corresponding to the four 4n² periods of Larson’s table and the Wheel of Motion.

In fact, the only missing characteristic seems to be the accepted electron configuration order. However, Le Conec’s ionization’s order doesn’t comply either. As shown in figure 2 below, Tarantola’s table of second ordering takes lithium and beryllium out of the first period and puts them back into the second period, as indicated by Le Cornec’s ionization results.

Figure 2. Tarantola’s Modification of the Periodic Table (Second Ordering) Given Le Cornec’s Results

But, if we take the fact that Janet’s table, based on a 2n² period, concurs with Larson’s table, based on a 4n² period, and the fact that Le Cornec’s results contradict quantum theory, based on a ratio of square roots, we have to take notice, because, as it turns out, the oscillating pseudoscalars of the S|T units, contain both the 2n² and the 4n² terms, as the two factors in a series of ratios of square root products that fit the four periods of the table!

Again, the only thing missing is the order of the energy levels. If we map the four periods of the square root equations to the tetraktys, we get a perfect match, as shown in table 3 below.

Table 3. Duality of S|T 2n² x 4n² Periodicity Mapped to Tetraktys

Though, at first, this seems counter intuitive, it works because the equations of the table are based on the equation of inverse geometry (r’² = r * r’’), where the inverse of a given circle’s, or sphere’s, radius, works out to be twice, or half, its radius. Students of the RST will recognize the significance of this immediately, since 1/2 and 2/1 are the basic units of speed-displacement in the system.

Details to follow soon.

One of the most important and immediate objectives of our research here at the LRC is to address the “glaring lacuna” in Larson’s work, the inability to calculate the atomic spectra. Larson tried to work it out, but temporarily gave it up, when it appeared so complex and obtuse a subject that it became apparent that it would drain his resources and slow his progress in developing his RST-based theory, what we call his Reciprocal System theory (RSt).

It was Nehru who gave it the label with which we still refer to it today. While noting this great contrast with the success of the LST’s wave mechanics in explaining the “vast wealth of spectroscopic data,” he goes on to elaborate on the much heralded success of the LST theory:

The several quantum numbers, n, l, m, etc. come out in a natural way in the theory. Even the “selection rules” that govern the transitions from one energy state to another could be derived. The fine and the hyperfine structures of the spectra, the breadth and intensity of the lines, the effects of electric and magnetic forces on the spectra could all be derived with great accuracy. In addition, it predicts many non-classical phenomena, such as the tunneling through potential barriers or the phenomena connected with the phase, which found experimental verification.

Of course, all of this began with Heisenberg’s great discovery of the non-commutative multiplication in the first mathematical structure of quantum mechanics. As Bohr wrote Rutherford:

Heisenberg is a young German of gifts and achievement. In fact, because of his last work, prospects have at one stroke been realized, which, although only vaguely grasped, have for a long time been the center of our wishes. We now see the possibility of developing a quantitative theory of atomic structure.

In developing an RST-based theory, however, the second postulate limits us to an algebra of magnitudes that is ordered (absolute), commutative and associative (meaning its geometry is Euclidean). Nehru’s approach, an attempt to clarify the physical concepts of quantum mechanics, while keeping the mathematics, is therefore problematic, unless we drop the second fundamental postulate of the RST.

Fortunately, as described in the previous post, the new mathematics developed at the LRC, based on reciprocal numbers as analogs of the scalar/pseudoscalar ratios, maintains these essential algebraic properties. Using this algebra, we have been able to build a simple toy model of the motions and combinations of motions that is consistent with the second postulate and that contains the bosons and fermions (both quarks and leptons) of the standard model of particle physics and the baryons of the periodic table based on Larson’s 4n² concept, as opposed to the 2n² concept of quantum mechanics.

The challenge, however, has, again, been to identify the quantitative relations between these bosons, quarks and leptons, as observed in the experimental energy levels of the baryons. In short, we need the RST breakthrough that corresponds to Heisenberg’s quantum mechanical breakthrough.

Well, I’m happy to announce today that the needed breakthrough has arrived; at least it has as far as the toy model is concerned. Recall that our RST-based model, unlike the LST model and Nehru’s model, is not based on rotation, but rather the vibrations of the pseudoscalars, which we call S|T units, the SUDRs and TUDRs, which are inverse pseudoscalar/scalar oscillations. Figure 1 shows how these combine to form the entities of the standard model.

Figure 1. Scalar Motion Combinations

The middle colors, green, red and blue are used to indicate the relative balance of red SUDRs and blue TUDRs in a given combination. A green circle in the middle of a combo indicates an even number of each. A red color indicates more SUDRs than TUDRs, while a blue color indicates the reverse. We assume an initial value of one SUDR and one TUDR in the green balanced combo, and an excess of one kind of unit in the unbalanced combos.

This works out perfectly in terms of combining protons, neutrons and electrons, but the question has been how to fit the photons into these combos. In the quantum mechanical model, the electron is regarded as orbiting the nucleus (Bohr’s model), even though this was modified (the cloud model) to fit Heisenberg’s mathematical structure, wherein no definite orbital path could be identified.

In the S|T model, the electron doesn’t rotate around the nucleus, but becomes an integral part of the atomic structure, as shown in figure 2 below:

Electron

Figure 2. Combining Quarks and Leptons into Baryons 

Even though the standard model is empirical, the mathematical relations of its entities are described by Heisenberg’s fundamental mathematical structure developed in two theories: The first theory is quantum electro dynamics (QED), and the second theory is quantum chromo dynamics (QCD). Both of these theories are based on rotations and suffer from the algebraic pathology of higher dimensional numbers, as explained in previous posts. The underlying mathematical structure is described as based upon the principles of symmetry in U(1)xSU(2)xSU(3) groups, but these groups are formed from numbers derived from rotations.

In our theory, there are no rotations, so the challenge has been how to show the different energy levels of the atomic spectra in the new model, which has no orbits or electron shells. The breakthrough has come by realizing that the combinations of motions in the S|T combos, while numerically balanced and accurate in the baryons and their combos are nevertheless what we might call internally polarized. This was understood by simply tabulating the SUDR|TUDR counts at the nodes of the combos. For instance, tabulating the S|T nodes in the combination of a proton, neutron and electron, as shown in figure 3a below, results in a polarized triangle, representing the deuterium atom, as shown in figure 3b.

Figure 3. The Polarization of Non-Ionized Deuterium

This is a remarkable and fortuitous result, since it permits the binding of a boson to the atom, as shown in figure 4 below, without changing anything, but the energy of the combo:

Figure 4. Combining Photons with Baryons

On this basis, we may now begin to develop a quantitative relation between photons and atoms in our model – the atomic spectra! Indeed, we might say that, perhaps, the prospects at one stroke have been realized, which have, for a long time, been the center of our wishes!

The importance of the tetraktys in mathematics and physics is well known, as the Clifford algebras are intimately associated with the study of geometry and physical theory. Here at the LRC, we know that this is due to the fact that the union of mathematics and physics requires understanding that there are three properties of numbers and motion: magnitude, dimension and “direction,” where each non-zero dimension has magnitude in two “directions,” which we can think of as up and down, left and right, forward and backward.

The fact that rational numbers and scalar motion have inverses (two “directions”) means that the tetraktys should also have its inverse, and, in fact, it does. We can show this by writing the numbers of the tetraktys as ratios:

2⁰/2⁰ = 1/1
2⁰/2⁰ 2¹/2⁰ = 1/1, 2/1
2⁰/2⁰ 2¹/2⁰ 2²/2⁰ = 1/1, 2/1, 4/1
2⁰/2⁰ 2¹/2⁰ 2²/2⁰ 2³/2⁰ = 1/1, 2/1, 4/1, 8/1

then its inverse is obviously,

2⁰/2⁰ = 1/1
2⁰/2⁰ 2⁰/2¹ = 1/1, 1/2
2⁰/2⁰ 2⁰/2¹ 2⁰/2² = 1/1, 1/2, 1/4
2⁰/2⁰ 2⁰/2¹ 2⁰/2² 2⁰/2³ = 1/1, 1/2, 1/4, 1/8

This may not be of interest to the LST community, because they don’t recognize scalar motion, let alone multi-dimensional scalar motion, and even if they did no doubt they would be stopped cold by the notion of super-luminal speeds.

But in the RST community, aware of the cosmic sector and its inverse speeds, the inverse tetraktys is of great interest, since it is a generalization of the multi-dimensional numbers, and their algebras, of the cosmic sector, just as the tetraktys is the generalization of those things in the material sector.

To truly understand the theoretical universe of the RST, both the tetraktys and the inverse tetraktys are necessary. Of course, this means that we need multi-dimensional numbers and units of motion that are scalar, or pseudoscalar, but, unfortunately, these don’t exist in the LST community’s system of mathematics. They have devised an imaginary number (the so-called square root of -1) to raise the dimension of scalar numbers, because they needed to stick with the 1D motion of objects, not realizing that motion without objects is also possible.

However, as we know, this has sickened their multi-dimensional algebras, forcing them to cope with trying to understand physics with nothing but scalars and vectors, or the 0D numbers of scalars and the 1D numbers of complex numbers. Because of this, most physicists and engineers would think we’ve gone batty if we start referring to multi-dimensional scalars - to them, it’s a contradiction in terms.

Nevertheless, that’s really what pseudoscalars are. So now we want to go back and replace the imaginary square root of -1, upon which the superstructure of modern mathematics has been built, with the square root of 2, which is not an imaginary number, but a very real one (no pun intended here - lol).

The way I hope we can do that is, first, by recognizing that the 3D pseudoscalar has all the properties we need to coax matter out of scalar motion: Like the 3D gravitational property of matter, with its two “directions,” in and out, the 3D pseudoscalar has 3D volume with two “directions,” expansion and contraction. Likewise, the 2D magnetic property of matter, with its two “directions,” north and south, is analogous to the 3D pseudoscalar’s 2D surface, having an inner and outer side, which cannot be separated. Finally, the 1D electrical property of matter, with its two “directions,” positive and negative, is reflected in the 3D pseudoscalar’s 1D circumference, which also has two “directions,” clockwise and counter-clockwise.

Add to this the fact that the inverse pseudoscalar is already studied in the field of inversive geometry, and we have at least two good reasons to suspect that this line of investigation is worth pursuing.

As explained recently in the New Math blog (See here), the square root of 2 turns out to be quite useful for defining numbers, in terms of the geometry of the three spheres, which correspond to -1, 0, +1, when -1 is understood to be 1, 0 is understood to be the square root of 2, and +1 is understood to be 2, the inverse of 1, and we replace these integers with the rational numbers, 1/2, 1/1, 2/1, using Larson’s concept of speed-displacement.

This may sound crazy at first, but it’s only necessary to understand that 0 doesn’t really have to be understood in the way it’s normally portrayed on the real number line, or in Cartesian coordinates. That is to say, the way we understand it in our LRC research is not like this:

-n …, -3, -2, -1, - 1/2, -1/3, … -1/n, 0, 1/n…1/4, 1/3, 1/2, 1, 2, 3, … n

where the unit distance is seen as infinitely divisible, but rather as,

1/n, …1/4, 1/3, 1/2, 1/1, 2/1, 3/1, 4/1, …n/1

where each unit is not infinitely divisible, but is discrete and has its discrete inverse unit, as in group theory. In this way, 0 does not have a place on the number line per se. Instead, it is implied by 1/1 = 0, when the binary operation of the group is changed from multiplication to addition, and the meaning of the “/” symbol becomes a “displacement” operation between the numerator and denominator, not an inverse multiplication operator.

With this much understood, we want to map the set of pseudoscalars, corresponding to the first two units of the expansion (t₁-t₀ = 1 and t₂-t₀ = 2), to the tetraktys and the inverse tetraktys, in the same way we map the rational numbers to it, using the intermediate pseudoscalar, with the unit ratio of the square root of 2, as the 0D tetraktys value, the ratio of the 1D circumferences as its 1D value, the ratio of the 2D surfaces as its 2D value, and the ratio of the 3D volumes as its 3D value.

We can use circles with differing patterns to indicate which aspect of the spheres is being symbolized, a pattern of dots for the 0D pseudoscalar, a pattern of lines for the 1D one, a pattern of squares for the 2D one, and a solid fill for the 3D one, as shown below:

Figure 1. Symbolic Notation for Radii, Circumferences, Surfaces and Volumes of Pseudoscalar Spheres with Radii r =1, r’ = (2)^1/2, r’’ = 2.

First, we construct the pseudoscalar tetraktys, then the inverse pseudoscalar tetraktys:

Figure 2. The Tetraktys and Inverse Tetraktys of Pseudoscalar Sphere Ratios

As it turns out, each of these pseudoscalar ratios is a power of the square root of 2. The unit ratio, which is just the ratio of the radius r’ to itself, or the square root of 2, divided by itself, is equal to 1. Letting R equal this unit ratio, we can write it as R⁰ = (2^1/2)⁰ = 1, because any number raised to the zero power is equal to 1. The ratio of the 1D circumferences of r’ and r, 2πr’/2πr, is equal to the square root of 2, or (2^1/2)¹, which we can write as R¹. The ratio of the 2D surfaces, 4πr’²/4πr², is equal to 2, or (2^1/2)², which we can write as R². Finally, the ratio of the volumes, (4/3)πr’³/(4/3)πr³, is equal to 2*2^1/2, which we can write as R³, since 2 * 2^1/2 = (2^1/2)³.

Substituting these values of R for the graphics shown in figure 2, we can rewrite the pseudoscalar tetraktys, and its inverse, symbolically, as shown in figure 3 below:

Figure 3. The Symbolic Pseudoscalar Tetraktys and Its Symbolic Inverse

This is a remarkable result, because it appears to constitute the basis for a multi-dimensional number system with no algebraic pathology, since each associated pseudoscalar algebra would presumably be as completely ordered, commutative and associative, as the well known scalar algebra.

Clearly, the R⁰ numbers are ordered, since the square root of 2 is a real number. The R¹ numbers are ordered, because we can’t have an R¹ number smaller than 1. The R² numbers and the R³ numbers are also ordered for the same reason, because no areas and volumes smaller than the unit areas and volumes exist in the system, by definition.

Indeed, I think the proof that all these numbers are ordered, commutative and associative is trivial, so I won’t bother to show it here.

Of course, I could be wrong – and very embarrassed – again!

It’s been so long since I’ve posted anything on research, I’m afraid people will think I’ve abandoned the work. However, the truth is, I’ve had to turn my attention to practical matters, leading to a neglect of the theoretical.

But I have some unpublished articles that have lain around for sometime, because I haven’t been able to get up enough momentum to finish them, or think them out completely. I think I will go ahead and post them anyway, though, just to get something out there to think about. Maybe that will help me get going again, even if it might be embarassing. Here’s the first one:

In our virtual lab in Second Life, we’ve been playing with SUDRs and TUDRs and their combinations, S|T units. Here is a short video of three S|T units combined as a neutrino triplet from our toy model of standard model entities:

The animation of the SUDR (red pseudoscalar) is driven by the changing diameter generated by the difference between the sine and -sine function, while the TUDR (blue pseudoscalar) is driven by the changing diameter generated by the difference between the cosine and -cosine functions.

In essence, this means that the expansion/contraction of the pseudoscalars is a function of two, counter-rotating, rotations, as shown in figure 1 below:

Figure 1. Two Counter Sine Functions (left), and Two Counter Cosine Functions (right), Define Inverse Diameters of Oscillating Pseudoscalars (not synchronized)

Consequently, with these two functions, we can analyze the pseudoscalar oscillations, and their combinations as S|T units. The first thought was to plot the changing 1D, 2D and 3D pseudoscalars in terms of π, which produced some interesting wave forms, but then the idea ocurred to us to take a point on the surface of the pseudoscalars as a zero reference. This means that the origin “moves” with respect to the reference point and gives us a way to compare the n-dimensional magnitudes as a function of time (space); that is, the 20 point increases from 0 to 1 * 10 , and back to 0, while the 21 function changes from 0 to 6 * 11, during the same time, while the 22 function changes from 0 to 12 * 12, and the 23 function changes from 0 to 8 * 13 and back, during one cycle.

In this way, everything is positive, and never negative, just as the magnitude of the diameter is always positive and never negative.  While this is interesting, the big challenge is to capture the inverse relationship. In what way is the TUDR oscillation the inverse of the SUDR? From the standpoint of the expanding/contracting diameter, there is no difference between the two oscillations of figure 1. The oscillation on the left is the +/- cosine projected on the horizontal diameter, while the oscillation on the right is the +/- sine projected on the vertical diameter, but the geometric inverse of the unit diameter is twice its size.

If this were not bad enough, how do we represent the temporal diameter with a spatial diameter? The answer, I believe, is to follow the math. As far as the math is concerned, the inverse of 1/2, is 2/1, and this is simply a doubling of the numerator, from 1 to 2, and a halving of the denominator, from 2 to 1, the mediato/duplacio math of the ancients.

Another way to express the same result is to keep the size of the diameter the same, but to quadruple the frequency of the TUDR, with respect to the SUDR. Figure 2 incorporates this idea.

       Figure 2. Normalized SUDR and TUDR Oscillations

Of course, this is tantamount to assuming that the relative frequencies of the oscillating pseudoscalars is a valid comparison, but I don’t see any other way of comparing them. If this works, we can leverage the knowledge of hetrodyning and harmonics. Something we’ve already explored to some extent.

Philip Gibbs, in his FQXI essay, “This Time – What a Strange Turn of Events!” writes: 

Minkowski used the symmetry in the Lorentz transformation to bring together space and time making them merely different dimensions of spacetime. Yet time is somehow different in our mind. This difference is characterised by an arrow of time that defies the symmetry. In our conscious experience our past is clear and fixed but our future is uncertain. From the laws of thermodynamics we learn that this difference is due to entropy which always increases as time passes. Entropy is a measure of information and by the rules of quantum mechanics information of (sic) conserved. There is a paradox, but information can be as clear to us as the letters on this page, or as hidden and disordered as the states of the molecules in the air around it. As time passes, the disorder increases and entropy measures this change.

Time can distinguish itself from space in this way because the spacetime metric has a Lorentz signature that assign a different sign in the time dimension versus the three space directions. Thus in locally flat Minkowski spacetime distances are measured by the invariant quantity

ds2 = dx2 + dy2 + dz2 – c2dt2

Part of the mystery of time is to understand where this signature comes from. Why three plus signs for space and only one minus sign for time? Even with this separation of dimensions there should remain symmetry under time reversal t -> -t, but the arrow of time breaks this symmetry. What is the origin of this arrow? From what bow did it take flight?

When we understand that the progression of time is only one aspect of the space/time progression, and that the progression of space is its reciprocal, we can understand the broken symmetry. It’s broken when the uniform motion is quantized by the continuous reversals in the space, or the time, aspect of the progression, as shown in the graphic of the previous post below, which is shown again in figure 1.

Figure 1.Two Fixed Reference Systems Created by Pseudoscalar Oscillations.

In figure 1, we see the arrow of time is created when the s/t pseudoscalar oscillations nullify the space progression, and the arrow of space is created when the t/s pseudoscalar oscillations nullify the time progression. Of course the two systems are separated by the unit space/time progression, which is c-speed from the 0 point of both systems.

From the perspective of either system, the zero speed (or frequency) of the inverse system is four times its own zero speed (or frequency); that is, 1/2 * 4 = 2/1. Another way to say the same thing is that, if we take the frequency of one system as the fundamental, the frequency of the inverse system is two octaves above that frequency, regardless of which one we select as the fundamental (i.e. 1/2 + 1/2 = 1 and 1 + 1 = 2).

If fermions are triple combinations of s/t and t/s pseudoscalars, whose net frequency is at the fundamental, or whose net motion is at the spatial zero, then a natural question to ask is, “What effect does vectorial motion have on their time (space) flow?” Einstein’s theory shows that time slows down relative to inertial systems in motion. We can illustrate this effect as shown in figure 2.

Figure 2. Vector Motion Slows Down Time.

Of course, the difference between the space/time of figure 1 and the spacetime of figure 2 is that, in figure 1, both space and time are progressing, whereas, in figure 2, only time is progressing, and while the change in space of figure 2 is a vectorial motion of an object, a one-dimensional change of x, y, z, locations, tied to events that are separated by spacetime, the events in one inertial frame happen slower (height of green arrow), relative to the events in another inertial frame, depending upon their relative speed (length of the purple and blue arrows).

In figure 1, we see that it’s the oscillation in the space progression, effectively nullifying it, that creates the inertial frames of figure 2. So, a more accurate representation would show the oscillation of an inertial frame, in both the s/t and t/s cases, as shown below in figure 3.

 Figure 3. The S/T and T/S Pseudoscalar Oscillations in a World Line Chart

The space/time progression of the oscillating s/t pseudoscalar is illustrated in the vertical bar of figure 3, where the space aspect of the continuous expansion is oscillating, while the time aspect continues its uniform increase. In the horizontal bar, the oscillation of the t/s pseudoscalar is illustrated, as, in this instance, the time aspect oscillates, while the space aspect continues its uniform increase.

In either case, the orthogonal paths of the oscillations show that the indicated system is at the zero point of their respective fixed reference system, created by the oscillations. Now, let’s give the same vector motion to the pseudoscalars as that shown it figure 2. Notice, depending on the vectorial speed, that the vertical bar will slant toward the horizontal, just as the green arrow does in figure 2, and the horizontal bar will slant toward the vertical.

However, there is a difference in how the x, y, z, spatial dimensions are to be understood in the two figures. In figure 2, the change in locations is defined by 1D motion, whereas, in figure 1, every point in the graph is a 3D change in the size of the locations; that is, vectorial motion causes the bar to slant to the horizontal, but it’s a physical impossibility to represent vectorial motion in three directions at once.

Therefore, we have to understand the oscillations of figure 3, not as 3D psuedoscalar oscillations of figure 1, but as 1D pseudoscalar oscillations. On this basis, it would take a composite of three charts like that in figure 3 to illustrate all the vectorial motion possibilities (this has important implications later.) But, to illustrate the relation of vector and scalar motion, we can imagine a 1D pseudoscalar oscillation, affected by high-speed 1D vectorial motion, as shown in figure 4 below.

Figure 4. 1D Pseudoscalar Vector Motion

In figure 4, the red space/time arrow increases diagonally, to the upper right, as the s/t pseudoscalar expands, in space and time equally. Subsequently, it increases diagonally to the upper left, as the s/t pseudoscalar decreases in space, as time continues to increase.

Inversely, the blue time/space arrow increases diagonally, to the lower right, as the t/s pseudoscalar contracts, in time, but continues to increase in space, while it subsquently increases diagonally to the upper right, as the t/s pseudoscalar expands in time, while space continues to increase.

Hence, we can see the perfect symmetry of the space/time | time/space relationship. But now, when we add vectorial motion to these pseudoscalars, the vertical, s/t, pseudoscalar rotates right, to the unit speed diagonal, while the horizontal, t/s, pseudoscalar rotates left, to the unit speed diagonal, represented by the green boxes with green arrows.

The most interesting thing to note at the unit boundary is the directional changes of the red and blue arrows. As the s/t pseudoscalar’s speed increases to c-speed, the time component of its expansion arrow disappears. It increases in space only, while on the contraction part of the cycle, the space component of the contraction arrow disappears, indicating that only the time component is increasing in this half of the cycle.

The full implications of this development are not well understood as yet, but it seems clear at this point that any increase in vectorial speed of the s/t pseudoscalar, beyond the unit level, crossing the c-speed boundary so-to-speak, is tantamount to a decrease in the vectorial speed of a t/s pseudoscalar. Moreover, we can see that, what would appear to be an increase of s/t vectorial speed, is actually a decrease in t/s vectorial speed, which completely transforms the red diagonal arrows of the s/t pseudoscalar into blue diagonal arrows of the t/s pseudoscalar and vice-versa!

Thus, the arrow of time, the arrow that defines the entropy on the s/t pseudoscalar side of unit speed, reverses direction, as the arrow of space, on the t/s side of unity, defining a reverse entropy! (note to John: is this not tantamount to the direction of matter’s time arrow (s/t) being opposed to the direction of energy’s time arrow (t/s), described by you?)!

To underscore that the directions of the two scalar arrows (i.e. the two “time” arrows) progress in opposite “directions,” a final graphic serves to more clearly illustrate it geometrically, in figure 5 below.     

Figure 5. The “Directions” of the Two Arrows of “Time” in Our RST-based Physical Theory are Opposed.

While our RST-based theory differs from Larson’s, in that his theory doesn’t incorporate the principles of the tetraktys in its development of the consequences of the RST, yet it continues to amaze me how the trail he blazed continues to be our guiding light.

Not only was he the first to solve the problem of the asymmetry of the “arrow of time,” and in the process uncover the t/s side of the universe, but he went on to show how high-speed vector motion, with the dimensions of scalar motion, and the unit c-speed datum, produces a cosmology of such beauty and grandeur that the contemplation of it is in itself almost a religious experience.

Of course, there is much, much more to learn about it. We have scarcely begun.