The Larson Research Center

Things are moving quite fast in the LRC development of the RST. I hope and pray that the Peter Principle doesn’t overcome us, before we can explain the atomic spectra, which is our immediate research goal. Mathematically and geometrically, we are focused on two things and their relationship: These two things are the geometrical set of right lines and circles constructed with Larson’s Cube (LC), and the algebraic set of numbers in the tetraktys, generated by the binomial expansion.

Given Sir Hamilton’s complaint that the science of algebra pales in comparison to the science of geometry, which we have referred to often, it is gratifying that we have discovered that the tetraktys is a mathematical map of the LC. That is to say, the numbers of the tetraktys correspond to the lengths of the right lines in the LC’s 2x2x2 stack of unit cubes:

1. 20 = 1 = 0D unit expansion of LC (point)
2. 21 = 2 = 1D unit expansion of LC (Line)
3. 22 = 4 = 2D unit expansion of LC (Area)
4. 23 = 8 = 3D unit expansion of LC (Volume)

This correspondence of the numbers of the tetraktys with the geometry of Larson’s cube is highly significant, since the LC not only contains the discrete magnitudes of the geometric cubes, but also the continuous magnitudes of the geometric balls.

Indeed, the LC contains the new number line, in the form of nested right lines and circles, which we have been investigating in light of the 4n2 numerical patterns of the wheel of motion, especially in light of the Le Cornec findings.

In the previous post, we discussed the two operational interpretations of the rational numbers, the quotient interpretation and the difference interpretation and how there are two units involved: One unit, 1/2, is the inverse of the other, 2/1, which are the units of the LC, and its inverse, but also the corresponding units of the SUDR and TUDR, respectively.

Since the SUDR is the 3D oscillation of space/time, while the TUDR is the 3D oscillation of time/space, this means that the magnitude, or the speed, of the TUDR is four times greater than the magnitude, or speed, of the SUDR. This was troublesome actually, because the number of preons in the S|T triplets depends upon a 1:1 relative weight between the two.

However, we recognize that, from the perspective of unit speed, the two entities are equal, because each is a unit displacement from unit speed, albeit in opposite “directions.” This equality works out beautifully for identifying the various entities of the standard model as combinations of preons, but not so much for the energy properties of the wheel of motion.

In the latter case, the number four, the quotient relation of the relative number of S|T units in the preons appears to be more important, because we are dealing with the relative energies of the SUDRs and TUDRs in our investigation of the atomic spectra.

But now that we have the quantitative number four (i.e. T|S = (2/1)/(1/2) = 4) , what we need is to understand the n2 part of the equation. What physical property corresponds to the n and why is it squared? We have sought an answer to this question in the mathematics and geometry of the LC and tetraktys for years, but have only been teased with intriguing hints. 

In the quantum mechanics of the LST, the n in their 2n2 equation corresponds to the energy shells of the nuclear atom, and the shells host the orbits of the two electrons allowed by the Pauli Exclusion principle in each orbit, which all fits so nicely into the classical idea of angular and orbital momentum and the four quantum numbers of QM. However, truth be known, you can’t look too closely, or some serious flaws appear in the model.

In the RST based model we are building, consisting of combinations of 3D space oscillations (SUDRs) and 3D time oscillations (TUDRs), the number of electrons is associated with the number of protons, but the electrons are not modeled as residing in concentric shells orbiting a nucleus, but oscillating in connection with the associated proton, which, again, leaves us with the question, “If the n term in the 4n2 periods of the wheel does not correspond to shells, as does the principle quantum number, N, of the QM atomic model, what does it correspond to?”

Whatever the answer is, it has to have a square relation, not a cubic one, which is puzzling given that the the volumes of the atoms would seem to determine their order, not their cross sections. Yet, the 22, 42, 62, 82 of the periods correspond to the increasing areas of the expanding LC, not the increasing volumes. 

Well now it appears that the square relation might reside in the relation of the inverses, at least at the unit level. To see this, we merely need to recognize that the inverse of the tetraktys is the double of the binomial, just as the inverse of the LC is it’s double. Remember, this follows from the equation of inversive geometry, where

r’2 = r * r”

When r is 1, then r’ is the square root of 2 and r” is 2, which is the next set of right lines in the expanding LC. In other words, the 2x2x2 stack of 8 unit cubes expands to a 4x4x4 stack of 64 unit cubes in the discrete expansion of two units of time, which just contains the continuous outer circle expansion of the LC. So, if we relate a 2D slice of the LC to the number line, the nested right lines and circles correspond to the numbers on the 0D time line (radii of the circles), the 1D diameters, the 2D areas and the corresponding 3D volumes (that are implied from the 2D slice.)

However, recall that we found that the number 1, the radius of the inner ball of the LC, is troublesome, since 1n is always equal to 1, regardless of the magnitude of n, so our new number line drops down one level, so-to-speak, to the next smaller ball, with radius r equal to the inverse of the square root of two, which is the inverse of the radius of its associated outer ball, with radius equal to the square root of 2. This is shown in figure 1 below.

Figure 1. The Right Lines and Circles of Larson’s Cube Fitted to the New Number Line

The relation of inversive geometry is still the same, but now r = 1/(21/2) (green radius), r’ = 1 (red radius) and r” = 21/2 (blue radius) and the equation, r’2 = r * r” still holds true: 12 = 1/21/2 * 21/2.

Generating a new unit LC on this basis gives us a tetraktys of

1. 20 * (1/21/2) = 1/21/2 
2. 21 * (1/21/2) = 21/2  
3. 22 * (1/21/2) = 81/2  
4. 23 * (1/21/2) = 321/2 

Therefore, instead of the corresponding inverse tetraktys being doubled to a quadnomial expansion, 40 = 1, 41 = 4, 42 = 16, 43 = 64, it is doubled to the square root of 2, since 2 * (1/21/2) = 21/2, giving us:

1. 20 * (21/2) = 21/2 
2. 21 * (21/2) = 81/2  
3. 22 * (21/2) = 321/2  
4. 23 * (21/2) = 1281/2 

for the inverse tetraktys. 

Hence, whereas the ratio of the unit expansion of the tetraktys to the unit expansion of its inverse tetraktys, at each level of the tetraktys, using the unit of the traditional number line, is,

1. 20:40 = 1:1 = 1:1 (point to point ratio (duration ratio))
2. 21:41 = 2:22 = 1:2 (line to line ratio)
3. 22:42 = 4:42 = 1:4 (area to area ratio)
4. 23:43 = 8:82 = 1:8 (volume to volume ratio),

now, the ratio of the expansion of the tetraktys to the expansion of its inverse, which corresponds to the unit of the new number line, is a constant ratio, 1:2, at all four levels:

1. 1/21/2:21/2 = 1:2 (point to point ratio (duration ratio))
2. 21/2:81/2 = 1:2 (line to line ratio)
3. 81/2:321/2 = 1:2 (area to area ratio)
4. 321/2:1281/2 = 1:2 (volume to volume ratio)

This is very interesting, using the new number line like this, because, since the TUDR tetraktys is the inverse of the SUDR tetraktys, their product is 1/2, at each level, while the inverse of this ratio is, of course 2/1.

Could it be that we are on to something here? More later.

Update: I don’t know why I wrote “product” in that last statement, when it obviously should read “quotient.” However, the product too is interesting, since it gives us:

1. S0 * T0 = 12  
2. S1 * T1 = 22 
3. S2 * T2 = 42 
4. S3 * T3 = 82   

Notice that the exponents of these factors are not summed in the product, because they indicate the geometric dimensions of S and T, not the number of factors in a number. The number of factors is contained in the tetraktys itself, but when we multiply these by the unit, 1/21/2, or the inverse unit, 21/2, it’s as if we are counting these as the sides of the square (4), and the edges of the cube (8).

The exception is the 0D components, because they are mathematical inverses, while the others are not. None of this may matter in the final analysis, since the product of space and time normally doesn’t make sense conceptually. 

In the next post, we will discuss the analog magnitudes of the tetraktys and LC.