Hi Nehru,
I have been deriving a computer model from my "Ancients" experience, and it has been turning up some things that might be relevant to our goals. First, I have some questions for you:
1) Motion in the time region is rotationally distributed in the time-space region, correct?
2) Is there any reason why that rotationally-distributed motion cannot be polyhedral, rather than spherical? Since we are dealing with discrete units, to me it would seem plausible that the rotational distribution would be discrete (faceted), rather than continuous.
3) What constitutes a "dimension" in the time region? Is it the iX, jY, kZ axes, the axis of rotation (as in your tetrahedral diagram), or something else? (I am wondering if the 4th power expression in the time region has to do with dimension, or just the number of primary rotations involved)?
I found some info on the internet on how to transform quaternions into homogeneous coordinates, so now I have a better understanding of them. I constructed a quaternion model of the TR rotations using HC, based on your information, starting with a basic double-rotation. Using pi as Rnat, the equivalent-space projection of the double-rotation showed up as a 4-faceted sphere -- a tetrahedron (4 discrete units, with the vertices at the circumradius). The 2 double-rotations transform to an octahedron (8 facets). The tetrahedron did not preserve parity, and wanted to either bond or dissolve, whereas the octahedron DID preserve parity, and was harmonically stable. ("Parity" is the concept of meshed gears -- adjacent, opposite rotations are stable, whereas adjacent, similar rotations are unstable and repel).
This leads me to believe that your idea of two double-rotations being the rotational base is correct (separation is pi radians, +/- i , which solves my hypersphere problems).
Increasing the rotations transformed into the Platonic solids, becoming increasingly "smooth" and sphere-like as the number of facets increased, moving from the octahedron, to the dodecahedron, to the icosahedron, to a 28-faceted solid that I don't know the name of. I couldn't help but notice that this follows the pattern of scalar reversals you have referred to as "folds". Opposing faces of the solids are the same rotational axis, so the number of "rotations" expressed are the facets/2: octahedron: 4 (2 per double-rotation, so we could say 2, 2 rather than 4), dodecahedron, 12/2=6, icosahedron, 20/2=10, 28-facet, 28/2=14... the same pattern of "folds", 2, 2, 6, 10, 14, 18, 22... corresponding to the shell structure, but including a crystalline geometry for the atom.
I am going to use "turn" to refer to the unbounded measurement of rotation in the TR, to distinguish it from bounded "angle" in the TSR.
Here is where I am questioning "dimension". Per our conversations, the two double-rotations (the octahedral projection) forms the sub-atomic particles, so we see them as 4-rotations, being 4-dimensional. But, once you reach helium, the rotational projection appears to change to the dodecahedron, which has 6 rotational axes plus the original 4 rotational axes (being a turn, rather than an angle). (I think you were trying to express this in your 'tetrahedron' model (TR #8) that was 6 dimensional, that I was unable to follow).
I can't help but notice that there is a 1:1 correspondence between the number of facets on these polyhedral projections, and the number of electrons allowed per Bohr "shell". Though the facets, themselves, being merely a projection of turn into time-space would only account for atomic number -- not the energy level interactions. The thought also occurs that since each of these facets represents an internal rotation -- a temporal displacement, that each facet could, in fact, "capture" an uncharged electron, thus giving the APPEARANCE in time-space of an electron "cloud" arranged in shells about a nucleus. (The electron, of course, would be captured inside unit boundary, but it would be projected into the polyhedral face when viewed outside the TR.)
I know you've suggested that electron capture may account for these energy level interactions, and from what I've found, that appears correct. I have some more detailed info on this, but I want to work up some figures and tables, to try to make it clearer.
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I am beginning to suspect whether the property called 'mass' exists inside the Time Region, or it appears only in the Time-Space Region? I have been endeavoring to show that rotational motion is primary in the Time Region, like linear motion is in the Time-Space Region.
You may be on to something here. To me, 'mass' and 'gravity' appear backwards, given their space-time units. We see "speeds", s/t, as particulate (local), and "energy", t/s, as waves (non-local). But then you have mass, a local, particulate phenomenon, as t^3/s^3, with the dimensions of energy, and gravity as non-local, with dimensions of speed, s^3/t^3. This may indicate that we are on the other side of the unit boundary for the mass/gravity concepts, than for other phenomena. As such, "mass" would be a time-space measurement, rather than a time region measurement.
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A principal characteristic of the "spin", or the intrinsic angular momentum, is the resistance to change the direction of the spin axis, called the 'moment of inertia'. This rotational property is analogous to inertia, or mass, which is the resistance to the change of direction of the linear speed.
Let me see if I follow you... momentum in the TSR is t^2/s^2. In the TR, s=1/t, so momentum in the TR would be t^2/(1/t)^2 = t^4?
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If the basic motion constituting the atom is a two-dimensional rotation (existing in the Time Region), its primary manifestation (in the Time Region) should be ROTATIONAL INERTIA---a resistance to change the 'angle' (direction) of the axis of rotation in space (or equivalent space, as the case may be).
I follow you up to the last line. Wouldn't it resist "change of angle" (called "shift" in Projective Geometry) in TIME, rather than space/eq space? Is inertia preserved across frame-inversion? If so, would that mean that a temporal "direction" is being translated to a vectorial direction in time-space?
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Since on crossing the Time Region and emerging into the Time-Space Region the primacy shifts from rotational to linear, the rotational inertia manifests as mass.
I think I understand... since it resists any change of rotational direction in time, it will also resist any change of linear direction in space (it matters not which direction).
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Further there would be no preferred direction in the Time-Space Region, even though within the Time Region axial direction of the "spin" has existence and relevance. This explains the observed SPIN DEPENDENCE of the so-called nuclear (strong) force: that the force between two nucleons of parallel spin is stronger than the force between two nucleons of antiparallel spin.
I just read about this in Nick Thomas's book on Counterspace. They refer to the concept as "shift" between two points (0 shift = parallel points -- strong force, if pi/2 shift, then orthogonal points -- weak force). It is apparently related by a cosine function. Is this the case with spin dependence?
I'll get you a follow-up shortly on the shells and energy levels I derived from the polyhedral double-rotation model. It is amazing how much it looks like the Bohr atom (equiv space view of time region rotation).
Bruce