The fact that the 3d geometric figure of the LC perfectly corresponds to the numbers of the binomial expansion of the tetraktys reveals a deep and fundamental connection, or union, of numbers and geometry. It's hard to overestimate the significance of this discovery, as the pursuit of such a union has driven the historical development of traditional mathematics, from ancient times. I've written much about this on my website at lrcphysics.com.

Here, though, I want to show how the math of the LC, in the form of the binomial expansion of the Tetraktys, should permit us to calculate the values of the LRC's RSt and thus the 1d, 2d and 3d properties of the S|T combos, which should, if valid, yield the magnitudes of the observed properties of the physical entities, which correspond to these various combos.

The key is understanding the expansion, or progression of the scale, or size, of the oscillating LC, since such an expansion corresponds to the sum of all its multi-dimensional scalar motion. To understand this, we must recognize that each physical dimension has two "directions." In the case of the unit LC, we can see that multiplying each dimension by the corresponding coefficient of that dimension, yields the correct number of dimensional units for a given dimension:

Recall that there is 1 0d point, 3 1d lines, 3 2d planes and 1 3d volume in the LC, and that these numbers are identical to the 1 3 3 1 sequence of numbers of the 4th line of the binomial expansion of the Tetraktys (BET). So, we can write this as:

1

^{0}x 1 point, 1

^{1}x 3 lines, 1

^{2}x 3 planes and 1

^{3}x 1 volume, or

1

^{0}(1mp) + 1

^{1}(3dp) + 1

^{2}(3qp) + 1

^{3}(1op) = 1(1) + 1(3x2) + 1(3x4) + 1(1x8) = 1+6+12+8 = 27 poles, where mp = monopole, dp = dipole, qp = quadrupole and op = octopole.

This is the multi-pole sum of the unit LC. So, to sum multiple units of the LC, we simply sum the coefficients of each dimension times the number of poles in that dimension:

n

^{0}(1mp) + n

^{1}(3dp) + n

^{2}(3qp) + n

^{3}(1op), or

n

^{0}(1) + n

^{1}(6) + n

^{2}(12) + n

^{3}(8).

On this basis, the progression of the LC, in terms of poles is, 27, 125, 343, 729, or

3

^{3,}5

^{3}, 7

^{3}, …(2n+1)

^{3}, n = 1, 2, 3, …∞

With this much understood, we can proceed to calculate the multi-dimensional magnitudes of the S|T combos. We have long since shown that summing the 1d numbers of the fundamental S|T combo, 1/2 + 1/1 + 2/1 = 4|4 num into the numbers of the triple combo, calculates the neutrino number, enabling us to calculate all the combos corresponding to the quarks and leptons of the first family of the LST's standard model of particle physics. For example, in the material sector:

Electron = 3(6|6) = 18|18;

Anti-Up = (2(6|6) + 4|4 = 12|12 + 4|4 = 16|16;

Down = ((2(4|4) +6|6 = 8|8 + 6|6 = 14|14;

Neutrino = 3(4|4) = 12|12;

Anti-Down = ((2(4|4) +6|6 = 8|8 + 6|6 = 14|14;

Up = (2(6|6) + 4|4 = 12|12 + 4|4 = 16|16;

Positron = 3(6|6) = 18|18;

with the inverse of these corresponding to the cosmic sector of the RST universe.

Combining these into the proton, neutron and helium atom entities, all the 1d charges work out perfectly, corresponding to the electrical properties of the observed entities, even in the beta decay process. However, these 1d numbers are not suitable to calculate the 2d and 3d magnitudes of the combos. We need to be able to calculate with 2d and 3d numbers to do this.

Of course, the vector mathematics of the LST are totally unsuited for this purpose. We must have a new scalar mathematics to calculate the multi-dimensional scalar magnitudes of the S|T combos, and I have shown why this is so on the LRC website. It's quite a story, but it's necessary to understand it, if one is to comprehend what we mean, when we refer to multi-dimensional scalar numbers and magnitudes.

Once it's understood how the scalar motion of an oscillating volume can form the basis for a multi-dimensional scalar mathematics, capable of describing the multi-dimensional magnitudes of scalar motion in various combinations, just as an oscillating vector motion can be used as the basis for multi-dimensional vector mathematics, describing magnitudes of vector motion, a whole new world of possibilities beckons us forward.

Hopefully, it's a valid world that can be shown to correspond to magnitudes of observed properties of matter, but, of course, there's no guarantee. It may turn out to be an exercise in futility, but I think it's worthwhile to explore it and see where it leads.

More next time.