In the last entry I made a big mistake, using the radius of the square root of 2 and its inverse to calculate the minimum volume of the TUDR and the maximum volume of the SUDR. Whether it was because I was fixated on the “that damn number,” 2/9, or because I confused the 2D view of Larson’s Cube (LC) with the actual 3D stack of cubes, I don’t know, but for some reason I spaced it out and, though I caught it, my readers quickly did too.
I was highly concerned, however, fearing that what seemed to be working out so well with the square root of 2 radius, would no longer be so promising, with the square root of 3 radius. I dreaded re-doing the calculations that I had worked out for the periodic table, but since I couldn’t see how those calculations related to preon model of particles, I had some hope that the new calculations would fit both.
Yet, I found that, while I could fit the square root of 2 based numbers into the pattern of the periodic table quite nicely, but not into the preon model of particles, the square root of 3 numbers did fit the preon model quite nicely!
I suspect that this is a clue that the numerical pattern of the periodic table is based on 2D properties of scalar motion combos, while the preon model is based on the 3D properties of their constituent motions. This makes sense, because the 3D oscillations contain all the lower dimensional oscillations as well, and both our atomic model and photon model are 3D oscillations. A triple combination of 3D S|T units can have a 2D configuration topologically, a triangle, if the S and T units are not coincident, while the double combination can only have a 1D configuration, a line, under these assumptions.
However, I don’t know if it can be maintained that the two aspects of motion ought to be separated to any degree, by space or time, since the separation, if any, would have to be a separation of motion (i.e. space and time), and motion not participating in the oscillations at that.
But regardless, we actually have three different sets of radii to work with: First, we have the center radius, which is the product of the two inverse radii that is always equal to 1, designated radius r’, the radius of the inner sphere contained inside the LC (technically a ball).
On the right of the unit sphere, we have the radius of the outer sphere containing the LC, designated r”. On the left, we have the inverse of the outer sphere, obtained by constructing a second, smaller, LC, just contained by the unit sphere. This third radius is designated r.
In the 2D case, the third dimension does not exist, so the spheres are flattened to circles, with magnitude in the plane of the x and y axes only, but in the 3D case, the two inverse radii are actually diagonals of a cube and therefore have an x, y and z component. The magnitude of the largest, with its radius designated r”, corresponds to the radius of the TUDR. It is the square root of 3, while the radius of the SUDR, designated r, is the inverse of the square root of 3. The three sets of lengths are shown in the figure below, depicted as different number systems:
Figure 1. The 1D, 2D, and 3D sets of lengths (radii) contained in the SUDR and TUDR
The idea is that the 2D and 3D number systems, based on units of the square root of 2 and the square root of 3, respectively, are isomorphic to the familiar 1D number system, all things considered. That is to say, the ratios 1n/2n, 2n/2n, and 2n/1n, where n equals the appropriate dimensional unit, constitute a valid number system in each case.
However, the 2D system subsumes the 1D system and the 3D system subsumes them both. Hence, the 3D system is a composite of 0D, 1D, 2D and 3D numbers. Surprisingly, however, the ratios of the various numbers contained in the reciprocal components of the composite form a pattern of integers that corresponds to physical patterns based on 0, 1, 2 and 3 dimensions, as well.
The most fundamental aspect of the system is the reciprocity of the larger numbers on the right, the smallest 0D number of which is r”, and the smaller numbers on the left, the largest 0D number of which is r. Recall that the continuous magnitudes of the circles and spheres are constrained by the discrete magnitudes of the squares and cubes of the LC and vice-versa. Each circle, or sphere, has a set of four squares, or 8 cubes of sequential sizes between it and the next circle or sphere.
As the discrete, right, lines of the LC expands, the 0D numbers (unit lengths) progress, 1, 2, 3, 4, …, the 1D numbers (side lengths of stack) progress, 2, 4, 6, 8, …, the 2D numbers (faces of stack) progress, 4, 16, 36, 64, while 3D numbers (# of cubes in stack) progress, 8, 64, 216, 512, ….
At the same time, the outer circle, sphere, containing the squares, cubes, expands, delineated by the increasing 0D diagonal radii, 21/2, 81/2, 181/2, 321/2, the increasing 2D surface, 8, 32, 72, 128, …, and the increasing 3D volume, 481/2, 30721/2, 349921/2, 1966081/2, ….
As figure 1 indicates, each of these n-dimensional numbers in the 3D expansion has it’s inverse. The question now is, “What does all this mean, physically, if anything?”
To tell the truth, there is so much popping out that I seem to have a tiger by the tail. There is a growing set of rational numbers emerging from all this that, like a set of Legos, can be used to build many wonderful things, but I want to share one, new, and unexpected, clue, which in particular intrigues me. Consider the following equation,
Td/Sd = Tv/Sv * Sa/Ta,
2(r”)/2(r) = 481/2/.257 * ((4/3)/12)
3 = 27 * .111…
3 = 3
Where T and S designate TUDR and SUDR respectively, and d, v and a are 1D diameter, 2D area, and 3d voluume designators, respectively. Of course, it turns out that the dimensions are the same as those in the famous equation:
E = mc2
which, in space/time terms, is written:
t/s = t3/s3 * s2/t2
Maybe this is just a happy coincidence. Maybe it’s nothing new, but please feel free to inform me, if so, because I missed it before, somehow.