The Larson Research Center

When I first showed the LRC’s RST-based “Wheel of Motion” to our new LRC associate, Larry, the mathematician, he said, “Well that’s nice, but how come you don’t make it 3D?”

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.

As we make the course correction discussed in the entry below, it’s useful to note that we can now view Einstein’s mass -> energy equation in terms of the SUDR/TUDR combos.

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 = hv, in the light of the LC..

The iconoclastic discovery of faster-than-light neutrinos, if confirmed, again places the Reciprocal System of Physical Theory (RST) in the lime-light. Or at least it would, if the new system weren’t ignored by the Legacy System of Physical Theory (LST) community.

Here at the Larson Research Center (LRC), we are getting woefully behind major new developments, as they come in ever faster. Nevertheless, it’s interesting to see how the RST easily accommodates the new superluminal neutrino findings. In our preon model of the LST standard model of particle physics, we can see that there are two of every fermion particle and anti-particle, except neutinos.

For instance, a left-handed and a right-handed version of the negative electron, and the two corresponding versions of the positive electron (positron), occupy the left and right end positions in the LRC schematic, as shown in figure 1 below:

Figure 1. LRC Chart of Standard Model Fermions Based on SUDR|TUDR Triplet Combos

From this chart, we can see that the central positions of the neutrino and anti-neutrino are unique in that, like the fulcrum of a lever, they are in-between the left and right sides of the chart. This means that, if we rotate the chart horizontally, the positions of all the particles are swapped, except those of the neutrinos, which maintain their central positions along the axis of rotation.

However, notice that such a rotation will invert the postions of the constituent preons (S|T units) in every particle of the chart, including those of the neutrino and anti-neutrino, so that the S|T units on the left and right sides of each triangle are swapped and the left - right orientation of the bottom S|T units in the triangles is reversed.

Thus, the two sides of the chart, the front and backsides if you will, represent two, reciprocal, sets of S|T combos, which correspond to the two sectors of the RST universe of motion.  Hence, there are actually two neutrinos and two anti-neutrinos in the system, which Larson would have designated the m-neutrino and the c-neutrino, together with their corresponding anti-particles, inhabiting their respective material and cosmic sectors of the universe of motion.

Therefore, we see that the constituent S|T combos of each sector of the universe of motion exhibit a “handedness,” or chirality, as it is called, which is an observed property of the respective fermions. Of course, the LST standard model also includes a description of interactions between these constituent particles. These interactions are described in terms of mediating particles called bosons, which are the photons, the W and Z bosons and the gluons, together with the missing Higgs boson.

While we have been concentrating our study mostly on the fermions at the LRC, it should be noted that particle interactions in the universe of motion must be described in terms of motion, rather than in the terms of the autonomous “forces,” which the LST community has concocted to suit their purposes, and that this necessarily includes the interaction between the material sector (constructed of material fermions) and the cosmic sector (constructed of cosmic fermions). 

Such material sector|cosmic sector interactions are rare, occuring only at the boundaries of the two sectors, where vector motion in space approaches the speed of light, which is the unit speed of the scalar universal space/time expansion.

In the material sector, the 3D oscillations of the SUDRs (space-like motion of fundamental constituents of material fermions) necessarily consist of less-than-speed of light motion, from the expansion of space, or the Material Sector, point of view, while the 3D oscillations of the TUDRs (time-like motion of fundamental constitutents of the cosmic fermions) necessarily consist of less-than-the-speed of light motion, from the expansion of time, or Cosmic Sector, point of view.

With this much understood, the observed faster-than-light neutrinos, if confirmed, would be explained in the RST community, in general, as a sector boundary phenomena, where the time component of c-neutrinos, moving through material sector matter, are “sling-shotted” along their way by the time component of space-like oscillations of the m-fermions. The SUDRs of these fermions have two units of time expansion, for every unit of space expansion, making them less-than-the-speed of light entities, from the space expansion point of view, but effectively increasing the speed of c-neutrinos by their uniform time expansion.

This interaction amounts to the c-neutrinos hitching a ride on the time expansion of m-fermions (i.e. the c-neutrino takes advantage of the m-fermion’s space oscillations, when passing through them,) which is tantamount to effectively increasing their speed, from the expansion of space (material sector) point of view.

Relative to the unit expansion, then, the c-neutrinos are moving faster, like a man running forward on the top of a moving train, from the material point of view, even though they would be moving slower, like a manning running backward on the top of a moving train, from the cosmic point of view.

The bottom line is that Einstein’s equations remain valid, within each sector, but motion in time (space), viewed from across the sector boundary, appears to violate the universal speed limit, just as the inverse of a rational number appears to be greater or smaller, depending upon one’s point of view. Put another way, if the left end of an East-West aligned teeter-totter is lower than its right end, as viewed from its South side, the reverse will be true, when it is viewed from its North side.

The trouble with LST physics, however, is that they do not understand that there are two, reciprocal, sectors of the universe: There is one where matter is constructed of fermions with more less-than-the-speed-of-light material SUDRs, the oscillations of which are space-like, and one where cosmic matter is constructed with more less-than-the-speed-of-light cosmic TUDRs, the oscillations of which are time-like.

But they’re getting there…

The 3D model of the S|T unit is a composite of two 3D oscillations, the space oscillation, the SUDR, and the time oscillation, the TUDR. Since they are the inverse of each other, the SUDR is 27 times smaller than the TUDR. We can therefore use the SUDR volume, surface or radius to express the relationship numerically. For example, let’s use the volume:

1(27)/2(27), 1(27)/1(27), 2(27)/1(27), or

27/54, 27/27, 54/27

As explained in the previous entry, only the unit measure of 3 volumes out of the lower 27, actually cube to an integer. 9. 18 and 27, which enables us to use them as 1/3, 2/3 and 3/3 of the whole, in 3D terms. Interestingly enough, however, in 1D terms, the numbers are different. Only three SUDR radii fit in a TUDR radius (i.e. 3 * (1/31/2) = 31/2), so the volume sum in which the SUDR radius is 1/3, 2/3 and 3/3 of the TUDR radius is 1, 8 and 27, respectively. I will defer the discussion of the ramifications of this fact until later. Right now I want to complete the other half of the numeric system, where the SUDR is larger than the TUDR, what we can refer to as the upper half of the number system, which terminates at 54/27.

Figure 1 below shows the chart of the upper half that corresponds to the chart of the lower half provided in the previous entry.

Figure 1. Chart of 27 TUDR Volumes Summing to Two TUDR Volumes

The concepts being introduced here are somewhat subtle. First, we assume that our universe is nothing but motion, with two reciprocal aspects, space and time, existing in discrete units. Since nothing is perfect, we must introduce a deviation into the perfect, one for one, expansion of space and time. The only deviation possible is a “direction” reversal in one of the two aspects, which produces a unit displacement in the otherwise perfect expansion.

We label the unit space displacement, SUDR, for space unit displacement ratio, and the unit time displacement, TUDR, for time unit displacement ratio. In simple terms the SUDR is a 3D space oscillation, while the TUDR is the inverse of this, a 3D time oscillation.

These two fundamental units progress, or increase, in their respective non-oscillating aspects. For the SUDR, this means that for each half of its oscillation, time continues to expand, and for the TUDR, space continues to expand, during each half of its oscillation. If we chart these motions we see that they are perpendicularly oriented, because they are reciprocals. Hence, it is possible that the time expansion of a space oscillation (SUDR) could bring it into contact with the space expansion of a time oscillation (TUDR).

A combination of these two fundamental entities constitutes a new ratio of space and time, a SUDR|TUDR (S|T) unit. A little consideration will show that an S|T unit is an oscillating combination that propagates in both space and time, and therefore can come into contact with other instances of S|T units, forming compound units. Consequently, we need to calculate the properties of this fundamental scalar motion combo in order to explore the possible combinations that might occur and then compare them to observations.

From simple geometric and mathematical properties, we deduce that the radius of the smallest TUDR is the square root of 3, and that the radius of the largest SUDR is the inverse of this, which means that, while it is three times smaller than the radius of the TUDR, its volume is exactly 27 times smaller than the volume of the TUDR. 

Of course, these sizes are all relative until we choose a unit for the scale, which undoubtedly must be based on the speed of light.