Outside, we have something called "step measure", where it is like walking in steps, each step is the same distance as the last, and the number of steps you take is the total distance you have traveled. The sequence goes like this:
1, 2, 3, 4, 5... where '5' is the total number of steps you have taken, and also the total distance traveled.
Inside unit space (the spatial aspect of motion is fixed at unity), we are walking in time, not in space. Our material existence gives us physical senses that only allow us to directly measure SPACE, not time. To address this measure, Larson introduced the concept of "equivalent space", which is how much of a spatial change we perceive when moving in time, as long as the spatial aspect of motion is fixed at 1.0.
Picture equivalent space as a sphere with a radius=1.0. The most we can move in space is ONE step, from 1 to the center, zero, and we can't go any further. But, we CAN take a temporal walk... but this results in each step getting smaller than the last. Since we are starting with ONE, a first step, the second step is 1/2, the third, 1/3, the fourth 1/4. Unlike the step motion outside, the total number of steps taken is NOT the same as the total distance traveled. On our 4th step (1/4) we aren't standing 0.25 away from where we started, but 1/2+1/6+1/12 = 0.75, 3/4ths of the distance.
One should also notice that each step is a fraction of the REMAINING DISTANCE, not the total distance. So each step is smaller than the last:
step=1 (zero reference)
step=2, (1-0)/2 = 1/2
step=3 (1-1/2)/3 = 1/6
step=4 (1-1/2-1/6)/4 = 1/12
step=5 (1-1/2-1/6-1/12)/5 = 1/20
step=6 1/30
step=7 1/42
step=8 1/56, ...
It takes an infinite number of steps (in time) to actually reach the center of equivalent space.
Gopi pointed out in an email that each step can also be viewed as a running interval (in time), where you simply multiple the current step by the last:
1/1*1/2 = 1/2
1/2*1/3 = 1/6
1/3*1/4 = 1/12
1/4*1/5 = 1/20
...
Gopi wrote:
Let's take a closer look at that. In rectilinear space, the shortest distance between any to points is a straight line--just one way to get there. In the polar space of the time region, the shortest distance between any to points is a unit arc, which can be either clockwise or counter-clockwise, so you have TWO "shortest distance" paths to follow in a polar space, which results in pairing of motion, twice the unit distance, such as wavelengths (-180 to +180) or concepts like bi-rotation. Therefore, when we look at a "temporal walk", we will see the steps paired up. The simplest combination is the pairing of adjacencies, which gives us our net temporal displacements:I think what the progression is, if I understood you correctly, is that in space, the intervals go as 1, 1/2, 1/2*1/3 = 1/6, 1/3*1/4 = 1/12, 1/4*1/5 = 1/20 etc. Now look at the denominators (time), you have 2, 6, 12, 20, 30 etc. Take the sum of the 2 steps: 2, 2+6, 6+12, 20+12, and you get the Bohr radii formula. (2,8,18,32...) This is basically 2(n2).
t=1 (0) + t=2 (1/2) = 2
t=2 (1/2) + t=3 (1/6) = 8
t=3 (1/6) + t=4 (1/12) = 18
t=4 (1/12) + t=5 (1/20) = 32
Or, to put an equation to it:
t(t-1) + t(t+1) = n2 - n + n2 + n = 2n2
Larson uses this sequence to compute the electric displacements in his Periodic Table. Each sequence would apply to ONE double-rotating system, and since the atom is composed of TWO double-rotating systems, we will see the periodicity doubling-up, 2,2, 8,8, 18,18, 32,32.
Gopi wrote:
Which is quite an interesting observation. The electron, being a rotating unit of space, CAN be captured in the equivalent space of the atom, since the relation of space to space is not motion. Spectral lines, such as the Balmer series, are determined by the emission of photons captured by electrons. (In RS2, electrons CAN capture and emit photons; captured photons imposing a rotational vibration upon the electron, creating electric charge).Now take its contributions in space (Invert back), and you get 1/(2n2) and hence your Balmer stuff.
So what we end up observing is an atomic nucleus, composed of 1D and 2D temporal rotation, with a cloud of charged electrons distributed around it; their positions dependent upon the numerous speed ranges in equivalent space created by the atomic rotations.
It also infers that the "orbital electrons", though responsible for spectral lines, have nothing to do with chemical interactions!
Gopi wrote:
If you fill out the sequence of folds based on the temporal intervals, you get:Another thought hit me when I was pondering this. I think in "Nature of Scalar Rotation" Nehru had identified the "folds" structure of the time region. I just tried this with the series of "steps" instead of the intervals.
First step is UNITY, equivalent of no step : 1.
Next step is 'inward', -1/2.
Next step is 1/3 of this step, but outward: 1/6.
And the next: -1/24. And so on.
We get the series: 1 - 1/2 + 1/6 - 1/24 + 1/120.... inf. = e(-1)
This is the basis of exponentially decreasing functions!! All we need is identifying the value of the scaling factor for these steps and we get different rates for the exponential function: e(-x).
+0
-0
+2
-6
+12
-20
+30
-42
Now take a delta on the positive and negative sides:
2 - 0 = 2 (+)
6 - 0 = 6 (-)
12 - 2 = 10 (+)
20 - 6 = 14 (-)
...
Which gives us the L subshell sequence of quantum numbers, 2s, 6p, 10d, 14f..., but with some conceptual rationale behind it. We are observing speed zones again, which appear to be the byproduct of a polyhedral projection of motion into space. If you think about it, since everything in the RS is quantized, the projection of motion will not be spherical, but polyhedral -- faces, edges and vertices. Look at the terms in the time intervals as geometric faces:
2: Plane
6: Cube
12: Dodecahedron
20: Icosahedron
If this is cross-referenced with Dr. Moon's article on the polyhedral structure of protons, I'll bet we have a solution to the atomic quantum levels and a bit of a different view of the atom.