The Larson Research Center

The Structure of the Physical Universe

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Material Sector Microcosm

Mathematical Formulation

(xn/2xn)<————— (xn/yn) —————>(2yn/yn)

Given the toy model of reciprocal unit displacement from the unit progression ratio (UPR), the space unit displacement ratio (SUDR) and the time unit displacement ratio (TUDR), which can be combined to form the observable entities of the standard model of particle physics, which is the supreme accomplishment of the legacy system of theory (LST) community, the next task is to find a mathematical formalism that calculates the properties of these scalar entities in terms of their scalar magnitudes, dimensions and “directions.”

We begin, of course, with the unit progression ratio, the fundamental motion of the reciprocal system of theory (RST). This space/time ratio is 1/1, and since it is an eternal progression, selecting any reference (0/0) point in it immediately expands numerically as

1) 1/1, 2/2, 3/3 …n/n,

forming an expanding volume from the selected point, as space/time progresses.

However, since these units are three-dimensional, and since three dimensions necessarily contain lower order two-dimensional, one-dimensional and zero-dimensional components, and each of these dimensions expand both mathematically and geometrically as two, reciprocal “directions” outward from unity (i.e. as an increase in both the numerator and denominator of the unit ratio), it is obvious that the progression of simple counting numbers in 1) above does not sufficiently express the scalar properties of the expansion, in terms of magnitude, dimension and “direction.”

For this purpose we employ Larson’s cube (LC), the stack of 2x2x2 unit cubes discussed in the Magnitude section of the SPUD. The LC is the geometric expression of the binomial expansion of the tetraktys, the 3D line (fourth line) of the expansion is written

2) 20, 21, 22, 23,

which, when combined with the numbers in the fourth line of the pattern of Pascal’s triangle, 1331,

3) 1(20)+3(21)+3(22)+1(23), 

mathematically encapsulates the geometric properties of the LC, with its single point (20) at the origin of the 2x2x2 stack of eight 1-unit cubes; Its 3, 1D (21) axes (x, y, z); Its 3, 2D (22) planes (xy, xz, yz), and its single volume (23) (xyz).

Obviously, the 3D geometric/mathematical expansion of the LC is not isomorphic to the 3D physical expansion of space/time, since the physical expansion consists of continuous magnitudes, not discrete magnitudes. Nevertheless, if this issue is sidelined for the time being, it’s easy to see that selecting one axis and one plane in the stack, the numbers in 2) above can be counted as in 1) above to form a 3D numerical progression, in terms of magnitude, dimension and “direction:”

4)

((1x2)0+(1x2)1+(1x2)2+(1x2)3)/((1x2)0+(1x2)1+(1x2)2+(1x2)3) (2, inverse, 2x2x2 stacks of 8, unit cubes) 

((2x2)0+(2x2)1+(2x2)2+(2x2)3)/((2x2)0+(2x2)1+(2x2)2+(2x2)3) (2, inverse, 4x4x4 stacks of 64, unit cubes)

((3x2)0+(3x2)1+(3x2)2+(3x2)3)/((3x2)0+(3x2)1+(3x2)2+(3x2)3) (2, inverse, 6x6x6 stacks of 64, unit cubes)

.

.

.

((nx2)0+(nx2)1+(nx2)2+(nx2)3)/((nx2)0+(nx2)1+(nx2)2+(nx2)3) (2, inverse, nxnxn stacks of (2n)3, cubes)

Given this quantification of the 3D UPR, the next step is to break the symmetry. This can be represented graphically, using progression algorithms adopted from Wolfram’s celluar automata rule 254, as explained here. 

First, the unit progression is perfectly symmetrical.

Figure 1. Graph of Unit Space/Time Progression Ratio

As the graph of figure 1 shows, the unit progression of scalar motion consists of one unit increase of space for each unit increase of time. Since space and time are reciprocals, the progression is manifest as an increasing number of space units, accompanied by an equal increase of time units. In other words, it is a graphical equivalent of the numerical progression shown in 1) above, the unit progression ratio.

Of course, the graph is only a two-dimensional representation of a one-dimensional numerical progression ratio, while the actual unit progression ratio is three-dimensional. Nevertheless, it is well suited to illustrate the concept of the UPR, especially its symmetry. At the same time, though, its symmetry is misleading, since it is the symmetry of opposites, not of inverses.

Units of space and time in the RST are inverses, not reflections, or opposites. Positive and negative poles are opposites, but not inverses. The group of positive and negative integers are opposities:

5) -n, …-3, -2, -1, 0, 1, 2, 3, …n,

but they are not inverses. The group of rationals shown in 4) are numerical inverses. The difference is that while the magnitudes and dimensions of the opposite units are equal, their “direction” is not. In the case of the inverse units, the magnitudes and “directions” are not equal, only the dimensions of the numbers remain unchanged.

Thus, for the inverses, the “direction” of the left side is decreasing from unity to zero, while the “direction” of the right side is increasing from unity to infinity.