In the previous entry, I explained how we use the 3D space/time oscillations of our theory, or the fundamental SUDRs and TUDRs, to form S|T units that are “preons,” triplets of which are used to build quarks and leptons and their anti-particles, as described in the LST standard model of particle physics.
Only the first family of the three families of the standard model have been modeled with the S|T preons, since it has seemed more important to concentrate on the periodic table of elements, and the atomic spectra, than to spend time considering the other two families of the standard model. However, this may have been a mistake.
In correspondence with Dave (see comments), who seems to have an uncanny ability to discover new relationships between known physical constants, I have learned that there is an important number that some researchers in the LST community refer to as “that damned number,” because, as unlikely as it might seem, it keeps popping up in the study of particle masses (see discussion here.)
Coincidentally, the same number, 2/9, plays an important part in Larson’s RSt, as he explains here. In the LST work that Dave has been contributing to, where he makes some amazing clarifications, the number 2/9 relates the charged lepton masses of the standard model with each other, as described by Carl Brannen in several papers, but probably most simply, starting with this one.
In the meantime, I’ve also discovered “that damned number” in the S|T preons. It turns out that, when we include the inner and outer balls defined by Larson’s Cube (LC), we are constrained to the inverse of the square root of 2/9 in the volume formula for the SUDR and TUDR.
I say constrained, because, regardless which measure we choose for the unit, or metric, of the LC, the geometry defines two, inverse, volumes, containing this number: The unit LC defines the volume of the outer ball, which always has a radius of the square root of 2, and the inverse of this ball, which has a radius of the inverse of the square root of 2.
Of course, this conforms to our new number system and bodes well, given the Le Cornec ionization energy (IE) analysis of the atomic spectra of the periodic table, which is based on the ratio of the square root of Hydrogen’s IE, to the square root of the IE’s of the other elements.
All this is very motivational, to say the least, and has lead me to suspect that the LC can be used to unravel these fundamental relationships, in our RST based theory.
But now Dave has shown me that a conversion constant can be derived from Carl Brannen’s and/or Hans de Vries’ studies of the ratios of lepton masses, based on the Koide formula, which also incorporates squares and square roots. It’s a very simple formula that divides the sum of the masses of the charged leptons (electron, muon and tau), by the square of the sum of their square roots.
What Carl did was to use the curious fact that the formula can be related to spin matrices, and his equivalent density matrices, to find mass ratios that can then be algebraically “back calculated” from the known energy values of the leptons to find conversion constants. For instance, Dave showed me that the conversion constant calculated using the electron mass-energy goes like this:
(1 + sqrt2*cos(2*pi*1/3 + 2/9))^2 = 0.001628115,
where the 1/3 term is used for the electron, because it is the first generation charged lepton. For the muon, or the second generation charged lepton, we would use 2/3, and for the tau we would use 3/3, in the formula.
Now with these values, we can “back calculate” the conversion constant for the mass of a given charged lepton. For instance, for the electron:
0.001628115093 * X = .511 MeV
X = .511 MeV / 0.001628115093
X = 313.8598753 MeV
In fact, it turns out that this calculation from the formula yields the same 313.8 Mev conversion constant for all three masses. At this point, Dave decided to call this common constant the “unit lepton mass,” something that would probably not be appreciated as much by the LST community, as by the RST community, which recognizes the crucial importance of the unit value as lying between reciprocals.
Larson, for example, went to some length to point out to the LST community that E = mc2 does not mean that mass and energy should be regarded as the same, but it means that a quantity of mass is the equivalent of a quantity of energy: the one can be converted into the other by the conversion constant, c2, but they are two quantities of different physical dimensions and therefore two different entities altogether, which have something that is more fundamental in common - namely, motion.
In this case, the “back calculated” mass constant enables Dave to fit the results of the lepton mass formula into a reciprocal relation, via the squaring operation, which is the last step of the formula; that is, if the penultimate output of the formula happened to be 1, for some reason, squaring 1 and multiplying the result by the constant would not change the constant: clearly, the result would be the constant itself.
However, squaring the actual penultimate results of the formula yields values above and below the imaginary unit value of the formula, producing the reciprocal relation around the constant. Dave explained it to me this way:
0.0403499082^2 = 0.001628115 <— squaring result with the 1/3 term (electron) << 1
0.5802119201^2 = 0.336645872 <— squaring result with the 2/3 term (muon) < 1
1.000000000^2 = 1000000000 <— squaring result with unknown term (imaginary) = 1
.379438172^2 = 5.661726014 <— squaring result with the 3/3 term (tau) > 1
which is an amazing insight, I think. It tells us that the lepton mass formula, which requires “that damned number,” 2/9, no doubt incorporates a reciprocal relation of some kind. Not that the lepton masses themselves are somehow reciprocal, but that the reciprocity of factors determining them may play an unrecognized, but crucial role.
There’s more to explain here than I can include in one post entry, so, as a start, I will focus on the physical meaning of the mysterious number, 2/9, which, in the LST-based findings, has no physical meaning, but which appears in the volume formula of the oscillating S|T units of our RST-based physical theory here at the LRC.
The calculation of the two, reciprocal, unit displacement volumes, the SUDR and the TUDR, is straight forward enough. We begin with the LC, which is a 2x2x2 stack of 8 unit cubes. The outer ball that just contains the LC necessarily has a radius of 21/2. To find the inverse of this ball, which necessarily has a radius of 1/21/2, we only need to construct a second LC inside the inner ball, which is just contained inside the first LC. This second, smaller, LC is also a 2x2x2 stack of 8 cubes, but each cube necessarily has length, width and height of 1/21/2, instead of 1.
Figure 1. 2D View of Nested LCs and Associated Outer and Inner Balls. The radius of the outer ball (dashed blue circle) of the unit LC is 21/2, while its inverse (dashed red circle) is 1/21/2. The volume of the outer ball is eight times the volume of its inverse, which is π(2/9)1/2 . The ratio of the difference between the volume of the inner ball and the volume of the outer ball, and the difference between the volume of the inner ball and the volume of the inverse of the outer ball is 81/2.
The volume formula is 4/3 π * r3. Thus, the volume of the smaller, SUDR, ball, the inverse of the outer, TUDR, ball, is 4/3 * π * 1/21/2 = 1/4.51/2 * π. But the inverse of 4.5 is 2/9, so the volume of the SUDR turns out to be: π(2/9)1/2. The volume of the TUDR, based on its radius of 21/2, is 8 times this value, or 8π(2/9)1/2.
Hence, the number, 8, or 23, is the 3D number system’s equivalent of the 0D number system’s number 4, or 22; that is, whereas in the 0D scalar number system (what we could call the digital number system), the ratio of the two, inverse, unit displacements is
(2/1)/(1/2) = 4,
in the 3D pseudoscalar number system (what we could call the analog number system), the corresponding ratio of the two, inverse, unit displacements is
((21/2/1)3/(1/21/2)3 = 8.
However, whereas in the digital number system, the difference between the unit 1 and the unit 2 is 2-1 = 1 and the difference between the unit 1 and the inverse of the unit 2, 1/2, is
1-1/2 = .5,
and their ratio is
1/.5 = 2, or 21,
in the analog number system the corresponding ratio of differential volumes is
(8π(2/9)1/2)-(4/3 * pi)/(4/3 * pi)-(π(2/9)1/2) = 81/2, or (21/2)3.
Therefore, there is a symmetry in the pseudoscalar number system that is not found in the scalar number system.
In other words, in the operational interpretation of number, which allows us the quotient AND the difference interpretations of numbers, (giving us fractions of a unit as inverses on the one hand, and negative whole numbers as inverses on the other hand), the digital number system’s two fundamental ratios are 22 and 21, respectively, introducing a dimensional asymmetry between the two operational interpretations, but in the corresponding fundamental ratios of the analog number system, they turn out to be 23 and (21/2)3, wherein the dimensional symmetry is preserved between the two operational interpretations of the 3D numbers.
Right now, the significance of this is mostly academic, and probably should have been explained in the New Math blog of the LRC, but, given the teasing information that Dave keeps providing us and the beauty of the simplicity of these findings, one is reminded of the old saying: where there is smoke, there is fire.
More later.
Update: To clarify: The equation of the ratio of differential volumes should be written:
((8*pi*sqrt(2/9))-((4/3)*pi))/((pi*sqrt(2/9)-((4/3)*pi))) = sqrt(8),
where the unit volume is ((4/3)*pi*13) and the order of subtraction in the numerator is reversed from that in the original equation.