Meeting a Terrific Challenge

Discussion of Larson Research Center work.

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dbundy
Posts: 141
Joined: Mon Dec 17, 2012 9:14 pm

Re: Meeting a Terrific Challenge

Post by dbundy » Fri Jan 04, 2019 5:53 am

The LST community views the physical universe as a stage of space and time for matter and its interactions. As David Hestenes expresses it, the goal of their research program is to find the set of the fewest interactions among the set of the fewest particles of nature, This has resulted in the standard model of particle physics.

Larson's program of research is radically different. It assumes that everything in the physical universe stems from the reciprocal relation between space and time, which it assumes exists in discrete units of three dimensions. This has led us to deduce that, instead of adding time as a fourth dimension to the Pythagorean theorem equation, which is the basis for the LST community's concept of spacetime, we conceive of two separate Pythagorean equations, one for space and one for time, each with three dimensions. which are reciprocals.

The equation of space, √(12 + 12 + 12) = √3, is calculated from the LC in units of space, while the equation of time is the inverse of this, 1/√(12 + 12 + 12) = 1/√3, calculated from the LC in units of time. The respective volumes of space and time associated with each of these equations for their respective radii are inverse, and the space volume is calculated to be 27 times smaller than the time volume.

For unknown reasons, this value coincides with the value of the number of poles in the 3d space calculated from the tetraktys:

0d = 30 = 1
1d = 31 = 3
2d = 32 = 9
3d = 33 = 27

when the coefficients of the binomial expansion equation of the tetraktys (see Pascal's triangle) are taken into account, which, for three dimensions, are 1, 3, 3, and 1. Now, the question is, how do we calculate the number of poles for the inverse tetraktys, the tetraktys of time, we might say? Do we just invert the numbers:

1/0d = 1/30 = 1/1
1/1d = 1/31 = 1/3
1/2d = 1/32 = 1/9
1/3d = 1/33 = 1/27

This seems logical, but who has ever heard of inverse dimensions? Nevertheless, when we calculate the number of poles in our revised "Bott clock" we get:

0d = 34 = 81
1d = 35 = 243
2d = 36 = 729
3d = 37 = 2187

which numbers, at first glance, look nothing like the inverse dimensional numbers. However, notice that, if we take the 34 = 81 value of poles as the new unit, as indicated should be the case in our "clock," and make it the inverse of the first tetraktys, as also indicated in our "clock," we get the exact inverse of the number of poles in the first tetraktys:

1/0d = 81/81 = 1/1
1/1d = 81/243 = 1/3
1/2d = 81/729 = 1/9
1/3d = 81/2187 = 1/27

The question is now, of course, is this just a little mathematical sleight of hand on our part? Is the logic twisted to suit our objective? The reader will have to make that decision, but I find it hard to believe that we could continue to make these observations by virtue of mere coincidence.

dbundy
Posts: 141
Joined: Mon Dec 17, 2012 9:14 pm

Re: Meeting a Terrific Challenge

Post by dbundy » Mon Jan 07, 2019 3:08 pm

When we mess with the foundation of mathematics, we mess with the foundation of physics and the technology that is the foundation of Western civilization. God have mercy on my soul, because I can't stop now!

The posts on page 3 of this topic deal with the relationship between the tetraktys and the LC and its consequences. The major conclusion is that the tetraktys is as far as we can take the binomial expansion. There are no higher dimensions in physical phenomena, in spite of Clifford algebra.

Notwithstanding this apparent fact, we've seen how John Baez et. al, explain incredibly complex mathematics in an attempt to go beyond three dimensions (four counting 0). It is very interesting to see how they have mistaken the "directions" of dimensions, or at least mis-labeled them, so that it's difficult to not confuse non-specialists with their mathematical and geometrical concepts.

Nevertheless, the "Bott clock," which we have revised (see previous posts above), helps us to understand the new scalar concepts tremendously, I think, even though it still lacks the clear connection with the LC. I think I can add that too to the clock.

On page 3 of this topic I show how the number 8 of the 3d expansion follows the expansion of the LC: 23, 43. 63, ...2n3.

This connection of numbers and geometry is real easy to show, when n = 1, because the 3d expansion generates the 2x2x2 stack of 8, 1-unit cubes of the LC, which contains all the dimensional sub-spaces of the LC, and the numbers of Pascal's triangle, which are just the coefficients of the binomial expansion (1 3 3 1), at the 3d (fourth) level.

However, if there really are no higher dimensions than these three (four), then how do we modify the binomial expansion to follow the geometric expansion of the LC? The clue is found in the LC expansion, which is actually a cubic expansion, when period 8 is taken into account; that is, 8/8 = 13, 64/8 = 23, 216/8 = 33, ...2n3/23.

Fortunately, this means that n3 is a factor in the third dimension, implying that n2, n1 and n0 are also factors in their respective subspaces of the tetraktys:

20 x n0; 21 x n1; 22 x n2; 23 x n3 = 20; 21; 22; 23, which, when multiplied by their respective binomial coefficients, 1, 3, 3, and 1, produces the LC, when n = 1:

1 monopole, 3 dipoles, 3 quadrupoles, 1 octupole = 27 poles

When n = 2, we get 125 poles, and 3 gives us 343, 4 gives 729 and 5 generates 1331 (just a coincidence, I think). The interesting thing is that these products and sums also form a cubic progression, starting with n =1 ----> 33 = 27:

n = 2 ----> 53; n = 3 ----> 73; n = 4 ----> 93; n = 5 ----> 113. This is an important factor relative to the atomic spectra, but more on that later. Right now, I want to incorporate this into our Bott clock:

Image

Poles are an alternate way of thinking of magnitudes in two "direction" from a center point:

30 = (1) point:
o
31 poles = 2 end and 1 center point arranged to form a line :
<----o---->
32 poles = 6 end and 3 center points arranged to form an area:
<----o---->
<----o---->
<----o---->
33 poles = 18 end and 9 center points arranged to form a volume:
<----o----> <----o----> <----o---->
<----o----> <----o----> <----o---->
<----o----> <----o----> <----o---->

When n = 5:
<---<---o--->--->
When n = 7:
<---<---<---o--->--->--->

In the case of the clock, n = r, so that for each r = 1, 2, 3, … n, there is a dual set of tetraktii generated, each corresponding to its respective LC.

This is much easier to work with than thinking that the dimensions can be incremented higher and higher, as in the fourth dimension and beyond of Clifford algebras:

Image

dbundy
Posts: 141
Joined: Mon Dec 17, 2012 9:14 pm

Re: Meeting a Terrific Challenge

Post by dbundy » Mon Jan 21, 2019 9:35 am

In the previous post, we saw how the progression of physical dimensions is limited to three non-zero dimensions, and how these four (counting zero) relate to the geometric progression of the LC in two numeric progressions. The first is (2n)3, which we can label as the stack number. It counts the number of 3d cubes in the successive stacks of cubes: 8, 64, 216, ....(2n)3 of the progression.

The second geometric progression, which we can call the pole number, is (2n+1)3. It counts the number of poles in a given stack number. Now ignoring for the moment that nature does not expand in cubes (unfortunately), what we want to do is relate these two progressions to the atomic spectra and the elements of the periodic table, in terms of our new scalar motion equation, S|T = (1/2+1/1+2/1) = 4|4. in such a way as to derive a new atomic model of the atom.

The first step is to clarify the fact that our revised Bott clock is not a clock at all. Rather, it is an expansion (appropriately enough) of a radius, r, where r is a natural number (r = 1, 2, 3, ...∞). The expansion generates the two, reciprocal, pole numbers and the two, reciprocal, stack numbers, for a given radius, r. It also generates the corresponding scalar equation for the combination of the two, reciprocal values, since it turns out that the appropriate num of the corresponding scalar equation is just a product of r and the unit num of the scalar equation: or r(4|4).

Recall that the natural units of motion (num) are an unusual combination of space and time oscillations. They are actually space (time) | time (space) ratios, or magnitudes of speed, but hardly in the ordinary sense of that word. As oscillations, they are a measure of frequency, but since they propagate relative to matter, they may be characterized as a combination of a wavelength and a waveduration, where the former can be expressed in terms of rotation, or cycles per unit time, 1/t (change in space-volume over time), and likewise for the latter, as cycles per unit space, or 1/s (change of time-volume over space). Both expressions are "frequencies," but one is an expression of cycles per unit time, while the other is an expression of cycles per unit space, and the cycles are 3d cycles of expansion/contraction, equivalent to 4π, 2d rotations.

The image below shows this Bott expansion graphically, with the stack numbers, pole numbers and respective num contained within concentric circles of radius r:

Image

The fact that the concentric circles of the graphic above resembles the 4n2 periods of our Wheel of Motion version of the Periodic Table of Elements is no accident. Notice that the quotient of the num (natural units of motion) values of the scalar motion equation and the total number of cubes (2r)3, in each circle, corresponds to 2(2r2), disregarding that one side is the inverse of the other side:

8/4 = 2; 1/8 ÷1/4 =1/2
64/8 = 8; 1/64 ÷1/8 = 1/8
216/12 = 18; 1/216 ÷1/12 = 1/18
512/16 = 32; 1/512 ÷1/16 = 1/32

This result is very interesting. It seems to indicate that the dual, (2n)2, periodic pattern of the elements follows the pattern of the spectral tetraktys and the Bott expansion, where one-half of the elements in a given period are the inverse of the other half (at least in terms of our scalar equation), even though the number of nucleons and electrons increases linearly with each additional atom in the series.

Moreover, as it turns out, if we multiply the scalar equation by 2r-1, we get the Bott spread of the atomic spectra groups directly, when r runs from 1 to 4:

r=1: (2r-1)(1/2+1/1+2/1) = 1(4|4) ~ 4;
r=2: (2r-1)(1/2+1/1+2/1) = 3(4|4) ~ 12;
r=3: (2r-1)(1/2+1/1+2/1) = 5(4|4) ~ 20;
r=4: (2r-1)(1/2+1/1+2/1) = 7(4|4) ~ 28;

Again, this seems to indicate that there is a connection between these various aspects of our scalar magnitude combinations and the dimensions of the atomic spectra and the periodic table of elements, but exactly what it is remains unclear, as of yet.

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