In contemplating the postulated scalar, or magnitude only, motion of the universe of motion, we are assuming that it exists in three dimensions, in discrete units, with two reciprocal aspects, space and time. One of the first questions that arises then concerns how to express this motion, or even illustrate it, appropriately. One of the most common approaches is to use the spots on an expanding balloon analogy, which Larson himself used, but this approach has the drawback that it necessarily includes the geometry of the balloon (a good example of this can be found in Gopi's presentation here.)

Another approach that I first presented at an ISUS conference in 2003, uses a cellular (i.e. discrete) automata algorithm, identified by Wolfram as rule 254.

Figure 1. Rule 254

This computer algorithm produce two, inverse, arithmetic progressions, that can be graphically generated in 2D. The advantage of this approach, which I dubbed the progression algorithm (PA), is that it is a scalar, or magnitude only, expansion; that is, it contains no geometrical information. Moreover, the PA can be used to analyze the scalar motion concepts of the universe of motion very effectively. 

For instance, while the equation of rule 254 captures the RST's unit motion (s/t =1/1), the equations of rule 252

Figure 2. Rule 252

and rule 238

Figure 3. Rule 238

capture Larson's concept of one unit of time speed-displacement (s/t = 1/2) and one unit of space speed-displacement (t/s =1/2), respectively. Strictly speaking, these three PAs are just counting numbers (1, 2, 3...) counting the reciprocal units of space and time in the order of progression. However, to be useful, the space-time units themselves must be expressed in terms of their physical dimensions, which, as discussed in the previous post below, can be only one of four choices: 0, 1, 2 or 3, if the algebra of these scalar magnitudes is to be ordered, commutative and associative.

That the physical dimensions of one of the two reciprocal aspects of the scalar expansion has to be 3 is clear, given that we have postulated that the scalar motion of the universe of motion is three-dimensional. The motion equation then requires that the physical dimensions of the reciprocal unit be zero, since

v³ = s³/t⁰.

With these units of time speed-displacement and inverse units of space speed-displacement, as the basic building blocks of motion, it's possible to build a universe of motion, as Larson demonstrated. Though he provided precious few diagrams and equations in his development of the consequences of the RST, one that is especially enlightening appeared in his book New Light on Space and Time. Actually it was a series of charts that he used as a schematic of the units and combinations of units, making up the universe of motion. The first, Chart A, shows the two inverse units as postive and negative units of vibration:

Figure 4. Larson's Chart A

The 254 PA corresponds to the "Uniform Motion" in the chart, while the 252 PA corresponds to "LV^1-" and the 238 PA corresponds to "LV¹⁺". The letters LV in these labels are an acronym for linear vibration, the number 1 in the exponent indicates that the entity is a 1D unit, with the sign indicating that the speed of the vibration is relative to unity, positive being above unity and negative being below unity.

The reduction from the 3D motion of "Uniform Motion" (254 PA) to the 1D motion of "LV^1-" (252  PA) and "LV¹⁺" (238 PA) illustrates Larson's conclusion that the three dimensions of the postulated motion are independent dimensions, permitting the deduction that the "directional" reversals, creating the 252 PA and 238 PA from the 254 PA, occur in one of the three dimensions only, thereby explaining the propagation of the entities, as it then can be logically concluded that the LVs are "free" to continue with the 254 PA expansion in one of the remaining two dimensions that are not affected by the "directional" reversals.

As can be seen from Chart A, Larson logically develops two physical sectors of the universe of motion, one based on the high-speed 238 LV (called the material sector), and another based on the low-speed 252 LV (cosmic sector). In Chart D, we see the fully developed schematic of the material sector:

Figure 5. Larson's Chart D

Comparing Chart A with Chart D, we can see the course of Larson's development schematically. First comes the 238 LV, then the LV rotates, leading to the sub-atomic particles and atomic elements, which are composed of combinations of multi-dimensional rotations of the 238 LV. The fully developed logic, including much of the cosmic sector, is shown in the schematic of Chart E:

Figure 6. Larson's Chart E

The question, which has arisen from the beginning, however, is this: Can the logic of Larson's development be expressed with the rigor of mathematics? His answer to this question was that a new mathematical development is not necessary, because the development of the logical consequences clarifies the physical concepts expressed by the modern equations of current physics.

A change in [the definition of motion that is] the base of the [new] system naturally necessitates many modifications of the details of physical theory. However, the amount of change that is required is not nearly as great as might appear on first consideration, because the new development calls for very little change in the mathematics of present-day theory. The changes are mainly in the interpretation of the mathematics, in our understanding of what the mathematics mean.

This is especially true, when one realizes that the new system doesn't supplant the current system, but rather subsumes it; that is, the base of the current system, vectorial motion, is the motion of objects relative to one another. Of course, this motion is also part of the new system, and, as long as the vectorial motion is not a significant fraction of the speed of light, the new system really has little to add to the current system in its realm of vectorial motion. On the other hand, however, the vectorial motion of the current system cannot be defined without objects. Energy, radiation and matter must be put into the current system in order to apply it to the study of natural phenomena. This is not true under the new system of physical theory.

All forms of energy, radiation, and matter emerge from the RST strictly as a consequence of its fundamental postulates. All the theoretical constituents of the universe of motion and their mutual interactions are either motions, combinations of motions, or relations between motions, including the familiar, one-dimensional, translational, rotational, and vibrational motions, the mechanical (vectorial) motions of matter.
 
What the RST brings to the table, so-to-speak, is a new kind of motion, hitherto unrecognized as motion, the fundamental scalar motion of the theoretical universe, manifest in the observation of the continuously forward march of time and space. Clearly, the recognition of this new type of motion doesn't change the physical laws that science has already discovered that apply to relatively low speeds of objects, and it doesn't change the usefulness of the existing mathematical language used to express and exploit these laws, but is it really reasonable to think that a new mathematical language, suitable for expressing and exploiting the laws of the new scalar motion, would not actually be helpful?

Given the mysterious nature of the relationship between mathematics and physics, what Wigner characterized as the "unreasonable effectiveness of mathematics in physics," Larson’s view that a new, formal, language is not required for his revolutionary new system, that the mathematical changes required “are mainly in the interpretation of the [existing] mathematics, in our understanding of what the mathematics mean,” seems curiously out of step with the current fascination and emphasis on mathematics. As it turns out, however, this view was extremely insightful, perhaps even prescient.

More on this next time.