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2D Rotation as Representation of 3D Oscillation

Posted on Tuesday, March 5, 2013 at 03:19AM by Registered CommenterDoug | CommentsPost a Comment | References21 References

In the discussion thread of the last entry, a reader pointed out that Ron Satz has published a paper on his website showing that the calculation of the atomic spectra can be accomplished using RST-based theory. It was announced last May as a “News Flash,” but we missed it:

News Flash (05/06/2012):  Dr. Satz’s new paper, “Theory of Atomic Spectra and Ionization Energies,” is now available.  Both simple and complex spectra are explained with ease using the Reciprocal System.  So is the splitting of spectra in magnetic and electric fields.  Ionization energies of the elements are calculated and compared with observation.   There are no jumping electrons in the Reciprocal System; Quantum Mechanics has been vanquished.

This is apparently good news, but we have some concerns based on a preliminary study of the paper. A complete LRC review of the paper is pending.

In the meantime, continuing with the study of the LRC concepts of 3D oscillation, the last entry in this blog showed how the change in volume constituting the spherical oscillation can be compared to rotation and when it is, the enigma of 4π rotation, which has stumped the LST community for many decades now, becomes clear and easy to understand: It takes the equivalent of one 360 degree revolution to fill the volume and a second equivalent 360 degree revolution to empty it again, as the volume returns to its original state after 720 degrees of apparent rotation.

Again, there is no actual rotation involved. It’s just that by mapping the change in volume to a corresponding rotation, as the oscillation proceeds, we can exploit the well understood mathematical concepts of rotation and its relationship to waves. 

The fact that the 8 1-unit cubes of Larson’s cube (LC), and the ball of radius 1 it contains, are equally partitioned so that each cube has the same proportion of spherical volume, enables us to map these 1/8 volumes of the 3D oscillation to the 2D unit-circle rotation based on units of π radians.

This is really fortuitous, since, while π/2 radians equals 90 degrees of rotation, or 1/4 of a complete rotation cycle, two, 1/8 sub-volumes of the inner ball of the LC constitute 1/4 of its total volume.

This fact makes 90 degrees of 2D rotation equivalent to a 1/4 change of 3D volume.

Consequently, we can represent the 3D volume oscillation in terms of a 2D rotation, where one 2π rotation in the meridian plane of a ball represents an increase of its volume, from 0 to 1 unit, and a subsequent 2π rotation of the meridian plane of a ball represents a decrease of its volume, from 1 unit to 0, as explained in more detail below:

Figure 1. The Volume of a Spherical Wedge

As it turns out, the volume of the spherical wedge in figure 1 is to the volume of its sphere (actually ball) as the angle of the wedge is to 2π:

\frac{V_w}{V_s} = \frac{\alpha}{2\pi},

or, as the angle of the wedge in degrees is to 360 degrees.

So, in our case, the ratio 1/8 = 45/360.

This is not only interesting with respect to the wave equation, but also to the finding of Dave Lackey that the relative quark and lepton masses are related to definite angles of rotation. That these might be wedge angles seems highly likely to me (see here.)

The two figures below show the 2D rotation representation of the 3D oscillation graphically:

Figure 1. First 2π Rotation

Figure 2. Second 2π Rotation

The reason this is so significant is that it provides a physical orgin of 4π quantum spin for the first time anywhere and also provides the basis for including the wave equation in our discrete system of theory. 

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