One of the important topics of discussion in ISUS, since the days of Larson, was the “factor 3” he used, in the calculations of the gravitational and Planck’s constant. His calculation was challenged over and over again, because of this “factor,” which he seems to pull out of no where. He felt it was certainly connected with dimensions, but he wasn’t sure just how. He writes in Chapter 14 of Basic Properties of Matter:
The ratio of the natural unit of mass in the cgs system to the arbitrary unit, the gram, was evaluated in Volume I as 2.236055 x 10-8. It was also noted in that earlier volume that the factor 3 (evidently representing the number of effective dimensions) enters into the relation between the gravitational constant and the natural unit of mass. The gravitational constant is then
3 x 2.236055 x 10-8 = 6.708165 x 10-8 (with a small adjustment that will be considered shortly).
In calculating Planck’s constant, he refers to the same “factor 3.” He writes in Chapter 33 of The Structure of the Physical Universe:
If we call the energy of the photon E and the ionizing energy k, the maximum kinetic energy of the ejected electron is E - k. Energy, E, is t/s, but in the time-space region where velocity is below the unit value, the effective value of s in primary processes is unity, hence E = t. The unit value of s has a similar effect on frequency (velocity) s/t, reducing it to 1/t. The conversion factor which relates frequency to energy is therefore t divided by 1/t or t². Since the interchange in the photoelectric effect is across the boundary between the time-space region and the time region it is also necessary to introduce the dimensional factor 3 and the regional ratio, 156.44. We then have
E = 3 / 156.44 t²v (139)
In cgs units this becomes
E = (3 / 156.44) × ((0.1521×10-15)²/6.670×108) v = 6.648×10-27 v ergs (140)
The coefficient of this equation is Planck’s Constant, commonly designated by the letter h.
Later, Satz rederives Planck’s constant, without recourse to the factor of three. In “A New Derivation of Planck’s Constant,” he writes:
Larson¹ was the first to attempt to derive Planck’s constant from the Reciprocal System. Because of the change in the calculated natural values of mass and energy in the second edition of his work², the original derivation has been invalidated. The factor of three that was used is dimensionally incorrect since the photon is a one-dimensional vibration. And the use of the cgs gravitational constant in such an equation is wrong since the result cannot be converted to a different system of units such as the Sl (mks) system. The remainder of Larson’s original equation (including the use of the interregional ratio and the square of the natural unit of time) will be shown to be correct.
This paper assumes that the factor of three is definitely connected to dimensions, even though Larson evidently wasn’t convinced. On this basis, Satz eliminates it, because, as he indicates above, in Larson’s development the photon of radiation is one-dimensional, but he probably got his clue from Sammer, as the following letter to Larson, dated 1986, shows:
Jan Sammer 560 Riverside Drive Apartment 3Q New York, NY 10027
September 3, 1986
Dear Dewey:Re: revisions to chapter 4 of Basic Properties of Matter, “Compressibility.”
As I understand from Ron, the change involves the natural unit of pressure, which has increased by a factor of three, because the mass unit was increased by this same factor. The change in the mass unit affects enrgy, momentum, pressure, and force. But this increase is compensated for by the fact that pressure is one-dimensional, rather then three-dimensional. Thus the final figures should be close to the ones you calculated on the basis of the old value of the mass unit.
Regards,
Jan
However, the factor 3 that appears in the gravitational constant would not be affected by these one-dimensional considerations since mass is three-dimensional. Indeed, Larson’s uncertainty about the precise meaning of the dimensional character of the factor 3 can be seen in his discussion of one-dimensional electrical phenomena too, where the factor again shows up. He writes in Chapter 16 of The Basic Properties of Matter:
The same considerations apply to the size of the unit of this quantity. Since the charge is not defined independently of the equation, the fact that there is only one force involved means that the expression QQ’ is actually Q¹/2Q’¹/2. It follows that, unless some structural factor (as previously defined) enters into the Coulomb relation, the value of the natural unit of Q derived from that relation should be the second power of the natural unit of t/s2. In carrying out the calculation we find that a factor of 3 does enter into the equation. This probably has the same origin as the factors of the same size that apply to a number of the basic equations examined in Volume I. It no doubt has a dimensional significance, although a full explanation is not yet available.
The natural unit of t/s2, as determined in Volume I, is 7.316889 x 10-6 sec/cm2. On the basis of the findings outlined in the foregoing paragraphs, the value of the natural unit of charge is
Q = (3 x 7.316889 x 10-6)2 = 4.81832 x 10-10 esu.
There is a small difference (a factor of 1.0032) between this value and that previously calculated from the Faraday constant. Like the similar deviation between the values for the gravitational constant, this difference in the values of the unit of charge is within the range of the secondary mass effects, and will probably be accounted for when a systematic study of the secondary mass relations is undertaken.
But as we discussed in the previous post, the difference between the algebraic calculations and the geometric ones, has to be taken into account. At the LRC, the photon (bosons) and the matter particles (fermions) are combinations of three-dimensional entities, not one-dimensional vibrations that can then be rotated two-dimensionally and one-dimensionally, as in Larson’s development. In the LRC development, the initial vibration is a 3D vibration, so the factor 3 can’t be eliminated on the grounds of reduced dimensions in the case of photons, pressure, and charge.
It turns out though that the dimensional meaning of the factor 3 may have more to do with the inherent characteristics of Larson’s cube than those of physical entities and forces. The reason for this conclusion is purely mathematical, however. Recall that, as we have been able to assign geometric dimensions to natural numbers, by modeling the the 3D progression with Larson’s cube, where each dimensional component of the scalar progression pertains to a natural series, the algebraic series cannot correspond to the geometric series; that is, the numbers which progress algebraically, are not the same as the numbers which progress geometrically, or scalarly, because a scalar progression must occupy none or all of the possible dimensions to be scalar, by definition.
Thus, the RST’s time progression is scalar, because time has no direction in space. In other words, it has zero dimensions, while its space progression is scalar, or what we call pseudoscalar, because it expands in every direction, defined by all three spatial dimensions of Euclidean space. Hence, the RST’s scalar space|time expansion, from any given moment of time or space, algebraically produces Larson’s cube in one unit of time. The three dimensional components of this expansion are shown in table 1 below.
Table 1. The Algebraic Number Series of the RST Expansion
Referring to Table 1, we can see that in one unit of time, 10, the 1D expansion expands to 2 linear units in each of 3 directions (one unit left, right, up, down, forwards, backwards), while the 2D component expands to 4 square units in each of 3 directions (three orthogonal planes, each with 4 square units), and the 3D expansion increases to 8 cubic units, one in each of 8 diagonal directions. In the subsequent units of time, the numbers increase as shown in the table.
However, these algebraic numbers cannot be produced by a physical scalar expansion, because such an expansion expands in every direction simultaneously, producing an expanding ball, not an expanding cube. This brings us to consideration of the efforts to square the circle, as discussed in the previous post below. Recall that the inner circle of figure 1, with radius equal to 1/2 of the number of the 1D series, has a π*r2 area, which, at one unit of time, is equal to an area of π, since 12 = 1. The area of this circle is 1/2 the area of the outer circle, after one unit of expansion, which has a larger radius equal to the square root of 2, making the area of the larger circle 2π. Figure 1 is reposted below for convenience:
Figure 1. The Geometric Expansion vs. Algebraic Expansion
In other words, the geometric expansion is a fraction of the algebraic expansion. However, the two are related integer values, related by π, but π in terms of expansion area per unit of time, not in terms of rotation per unit of time. This is made clear by arranging the geometric calculations of four units of progression in table 2, below.
Table 2. The Geometric Number Series of the RST Expansion
As can be seen by comparing the two tables, besides the π factor, the coefficients of the two tables are remarkably different. What’s really remarkable, however, is that the geometric series reproduces the numbers of the periodic table of elements! In the progression of the area of the inner circle, Aic, we see the 0D expansion of the algebraic series squared, which corresponds to the four, 1/4, periods of the periodic table. In the progression of the area of the outer circle, Aoc, we see the reproduction of the 1/2 periods of the periodic table, corresponding to the number of elements between the noble elements, where the geometric numbers are the product of the 0D algebraic numbers and the 1D algebraic numbers of table 1.
When the progression of the surface areas of the inner and outer spheres (Ais & Aos) are calculated for the same four units of expansion, the progression of Ais surface area corresponds to the number of elements in each successive period of the periodic table (recall that the first period actually ends with deuterium, having three precursor entities before it (including hydrogen). The strange series of numbers in the last row is the series corresponding to the surface area of the outer sphere, Aos, but what do these numbers correspond to?
We don’t know yet, but notice the startling fact that they are a factor of three larger than the Ais numbers! What we have here is a purely mathematical, multi-dimensional, scalar, progression matching nature’s pattern of elements arranged by relative mass. The fact that combining these with their inverses, in the manner that we are studying at the LRC, in the form of SUDR and TUDR combinations, provides a number of “slots,” corresponding to these numbers, is simply breathtaking.
For instance, the 4π “slots” can be filled with four 1/4 magnitudes, the 16π slots with sixteen 1/16 magnitudes and so on. As regular readers of this site know, our major goal is to calculate the atomic spectra, using the new system of theory. One major milestone that we anticipated in this effort was reproducing the marvelous work of Larson in explaining the periodic table of elements, as a 4n2 pattern of physical magnitudes, rather than the 2n2 pattern of quantum mechanics. As can be seen from the right most column of table 2, we are closing in on this milestone, but we hardly expected to find the mysterious factor of 3 in the process.