Sometimes it might be misleading to call the Cosmic Sector the ‘inverse’ Sector. This is because the phenomena of that Sector are really the ANALOGS (or CONJUGATES) of the corresponding Material Sector phenomena. Thus, for example, the Cosmic Sector quantity corresponding to the Material Sector acceleration [s/t²] is [t/s²] and not the inverse, [t²/s]. Only in the case of those quantities with equal powers of the space-time dimensions—like, say, [t³/s³]—the conjugate turns out to be the inverse too.

This has some significance that we need to recognize explicitly. Consider the quantity speed [s/t]. The corresponding pertinent quantity in the counterspace of the Time Region (of the Material Sector) is the INVERSE speed [t/s], which manifests to us as energy, as we already know. Now if we consider the quantity pertaining to the Cosmic Sector, namely, the CONJUGATE speed [t/s], it manifests to us not merely as the material energy, but as PRANA—the self-organizing power of LIFE—apparent in all the living systems.

1. There is another implication of the maximum polar space (angle) being infinite in the Inside Region, coupled with the inversion that takes place at the unit boundary. Within the circle of radius 1 L_nat, representing the Inside Region, the increase in angle follows an inverse pattern. That is to say, instead of the series 1, 2, 3, 4,… , it would be:

   1, (1 + 1/2), (1 + 1/2 + 1/3), (1 + 1/2 + 1/3 + 1/4), …

   The physical effect of this diminishing nature of the angular increment in the Inside Region is to pack infinite angle in the finite Cartesian circle of 2π radians. Once again, an implication not foreseen by common sense.

2. There is also a dimensional implication. In the Outside Region multiplying two orthogonal lines (m * m) produces an area (m²). Further multiplying the area by an orthogonal line produces volume (m³). In the Inside Region, an angle (θ) sweeps an area (A = 0.5 * θ * r²), and a two-dimensional angle (θ²) sweeps a four-dimensional hyper-volume (H = 0.25 * θ² * r⁴).

All our human observations refer to the Outside Region. When we try to “understand” (visualize) phenomena pertaining to any other Region—which might be within the precincts of the Outside Region (like the Inside Region), or totally beyond it (like the Cosmic Sector)—we consider a projection of the phenomenon concerned onto our observational domain, namely, the Outside Region. A mathematical/logical investigation might lead to knowledge of the true properties of a system. But it may not always be possible to visualize them, or some of them. Further, when we do create the projections of such phenomena, these projections might not show characteristics that our common sense view is accustomed to. We shall keep this factor in mind.

We have noted in “Minimum and Maximum Limits” that while linear motion is primary in the Outside Region, rotational motion is primary in the Inside Region. Consequently we need to take that the reference frame that is pertinent to the Outside Region has to be a Linear Coordinate System (LCS) and that to the Inside Region a Rotational Coordinate System (RCS). In KVK Nehru's article, “Some Thoughts on Spin,” Reciprocity, XXVI (3), Winter 1997-8, pp. 15-18) we derived for a self-sufficient system of space (or time), the dimensionality—that is, the number of independent coordinates—has to be 3. Thus while the LCS is comprised of 3 linear coordinates, say, x, y, z, the RCS is comprised of 3 rotational coordinates, R1, R2, R3.

Whenever we talk of the expanding space, we do not have difficulty mentally imagining it. We naturally envisage a continuous increase of Cartesian distance in 3-space (that is, volume), starting from the zero or the “origin.” Furthermore, we do not have difficulty imagining an un-ending, infinite, expansion.

Larson's crucial discovery that space, time, and motion are quantized demands a re-examination of our common sense (Cartesian) view of the expansion of space. Since there would be a minimum quantity of space from the natural reference viewpoint, S_nat, we are not justified in envisioning the expansion of space as commencing from the Cartesian origin/zero—it rather commences from S_nat. In other words, the effective origin (of the expansion) is a spherical surface of radius S_nat—not the Cartesian zero point.

The next question is what happens when we cross this sphere of radius S_nat and go inside? The effective (physical) origin of the “expansion” is uniformly distributed on the entire inside of this spherical surface. Let us call this the “distributed origin.” There is still expansion but this expansion appears as a radial contraction to us (the Cartesian observers).

There are two important consequences of this fact, namely, that the “origin” of motion in the inside region is not zero but uniformly distributed over the inside surface of the sphere of radius S_nat.

Firstly, all radial directions, from any point on the `spherical origin' toward the center of the Cartesian frame are totally equivalent. They become scalar from the point of view of the Cartesian observer.