The Larson Research Center

Because the M₁ motion of the Chart of Motion (CM) is based on the number 1, it’s not really motion in the same sense as M>1 motion, but it’s still motion. We’ve been referring to its magnitudes as “potential” motion, but this designation is not really adequate. Moreover, a better name for the CM would probably be “Chart of Magnitudes,” or maybe “Chart of Numerical Magnitudes,” rather than Chart of Motion, removing the sense conveyed by the word motion that “something” is necessarily moving.

However, while I think we will eventually be able to see how to name these things better, for now we will stick with what we have, even if it requires some extra explaining to make the meaning of the chart clear. In the RST based physical theory, everything in the universe of motion proceeds from M₁ motion. Or, to put it another way, all magnitudes in the universe of motion are derived from the RST’s unit progression. Without M₁ motion, there could be no other motion, and thus no universe of motion. Essentially, M₁ motion may be understood as the unit rate of two clocks, a time clock and its inverse, a space clock, at unit ratio, and, like all motion, its magnitudes are magnitudes of rate of change. In the M₁ case, a unit rate of change, but a rate of change nonetheless.

The implication of the CM is that magnitudes of space are simply the space aspect of magnitudes of motion. The primary magnitudes of the CM are magnitudes of motion, formed from two, reciprocal, magnitudes, one of space and one of time. Magnitude itself, in general, is thought of as having three aspects, quantity, dimension, and direction. However, in studying the RST, we identify another aspect of magnitude, the aspect of polarity, which we designate “direction.” The difference between the direction of magnitude, and the “direction” of magnitude, is that “direction” is referenced to a unit rate of change between the two aspects of motion, while direction is referenced to a relative position occupied by an object, but they are clearly related, and their differences must be carefully understood to avoid confusion.

One important way we can do this is to introduce the idea of duality, as a fundamental aspect of magnitude. Instead of having the four aspects of quantity, dimension, direction, and polarity, we will simply have three aspects of magnitude, quantity, dimension, and duality. Numeric duality would correspond to polarity, while geometric duality would correspond to direction. In this way, for every primal magnitude, there is dual magnitude, or complementary magnitude, regardless of the type of motion we are talking about.

In the case of M₂ magnitude, for instance, for every primal direction of the magnitude, there is a dual, or complementary, magnitude with a direction such that, when it is summed with the primal magnitude, the two magnitudes offset and the resultant magnitude of their sum is zero; The system is in a state of equilibrium; Also, for M₃ magnitude, for every primal interval, there is a dual, or complementary, interval that, when summed with the primal interval, “zeros” it; Likewise, in the case of M₄ magnitude, for every primal scalar, there is a dual, or complementary, scalar that, when summed with the primal scalar, balances it.

In this way, the magnitudes of the CM, whether magnitudes of motion, space, or time, all have their duals, depending upon the type of magnitude considered: all M₂ distances have two directions, all M₃ intervals have two phases, all M₄ scalars have two polarities, and all M_n motion has two aspects, space and time. With this clarification, we can see that the numbers of the CM can be modified to incorporate the RSM’s operational interpretation (OI) of number, so that in one chart, all three types of magnitudes (motion, space, and time) are included. Taking this approach immediately illuminates the true nature of the M₁ magnitudes, but at the cost of introducing additional complexity with regards to the nature of numbers.

Nevertheless, I believe the effort to modify the CM in this manner will be worthwhile, so I will make an initial attempt to do it in this post, but with the understanding that the results will be tentative and subject to change, as our understanding of the CM grows.

When we expand the dimensions of M₁ motion, from M₁⁰ to M₁ⁿ, we are expanding the dimensions of a unit ratio, or expanding the dimensions of the number 1 (1/1). In the legacy system of mathematics (LSM), 1ⁿ = 1 and that’s pretty much the end of the story, unless the numbers are given physical interpretations, by assigning physical dimensions to them. This has always been at the root of separation between numbers and magnitudes. Recall how the Greeks, especially Euclid, were careful to keep the two concepts of number and magnitude separate. One linear measure of magnitude was not the same as 1 square measure of magnitude, or 1 cubic measure of magnitude, and it’s not hard to see their point.

Nevertheless, the fact that (n/n) = 1 can take different forms, from 1/1 to infinity/infinity, is surely significant, and leads us to ask, “What’s the difference between (x/x)ⁿ and (y/y)ⁿ? Is there a limit to the values that dimension n can take? Obviously, not as far as numbers are concerned, but what about magnitudes? Is there a limit to the number of dimensions of unit magnitudes, because only three (four) dimensions of space magnitudes are observed? This is a very relevant question in physics today, since the only way that has ever been found to formulate a unified physical theory that includes gravity, requires more than three (four) spatial dimensions.

When the numbers in the CM are in their true form, as OI rational numbers, we gain a greater insight into what the chart is telling us. In fact, we can generalize the numbers of the chart, by expressing them in terms of unit, OI, magnitudes, as follows:

_M₁____M₂___M₃____M₄
(n/n)ⁿ (n/n)ⁿ (n/n)ⁿ (n/n)ⁿ,

In the first row of the chart, the dimension is 0,

(n/n)⁰, (n/n)⁰, (n/n)⁰, (n/n)⁰,

or,

1⁰/1⁰, 2⁰/1⁰, 3⁰/1⁰, 4⁰/1⁰,

and, since all these magnitudes are equal to 1/1, or a point magnitude, we can make the denominator of each of these numbers equal to their numerators, without changing the point magnitudes:

1⁰/1⁰, 2⁰/2⁰, 3⁰/3⁰, 4⁰/4⁰,

which are clearly unit motion magnitudes. Now, since we have seen that magnitude in general has a duality aspect, then we have captured the duality of these motion magnitudes nicely, simply by making this change. Moreover, our knowledge of numbers immediately gives us a hint that, whereas M₁ magnitude has no degree of freedom, M₂ magnitude has two (2/1 & 2/2), M₃ magnitude has three (3/1, 3/2, 3/3), and M₄ magnitude has four degrees of freedom (4/1, 4/2, 4/3, 4/4), which at dimension 0 in the CM doesn’t have any significance, because any number raised to the zero power is equal to one, but when the number’s dimension is raised to a non-zero value, the respective degrees of freedom may prove more significant.

Of course, each of these numbers has its dual as well, so we have the same number of degrees of freedom in the dual aspects, which would be the reciprocal magnitudes of these numbers, or (1/2, 2/2), (1/3, 2/3, 3/3), and (1/4, 2/4, 3/4, 4/4), respectively. However, while the duality of the motion magnitudes is expressed in their reciprocal degrees of freedom, the duality of space, or time, magnitudes is expressed in directions, phases, and polarities, as we’ve already discussed. Hence, we also need to capture the duality that specifically applies to the space and time magnitudes, and the way we do that is via the tetraktys, or the binomial expansion (binomial for duality, not for M₂ motion.)

Because M₁ magnitude has no degree of freedom, it is the easiest in some ways, but mysterious at the same time, as we shall see later on. Expanding M₁ magnitude from D = 0 to D = 3, we get:

1. (1/1)⁰ = 1
2. (1/1)¹ = 1
3. (1/1)² = 1
4. (1/1)³ = 1

which is where the concept of number seems to depart from the concept of magnitude, because each of these magnitudes is different, but the difference is not reflected in the number 1. More on this later. Now, expanding the M₂ magnitude, we get:

1. (2/2)⁰ = 1 term (0 directions, or point)
2. (2/2)⁰ + (2/1)¹ = 2 terms (2 directions, or 1D line)
3. (2/2)⁰ + [(2/1)¹+ (2/1)¹] + (2/1)² = 4 terms (4 directions, or 2D area)
4. (2/2)⁰ + [(2/1)¹+ (2/1)¹ + (2/1)¹] + [(2/1)² + (2/1)² + (2/1)²] + (2/1)³ = 8 terms (8 directions, or 3D volume)

Next, the M₃ magnitude expansion is

1. (3/3)⁰ = 1 term (0 phases, or point)
2. (3/3)⁰ + (3/1)¹ = 2 terms (2 phases, or 1D interval)
3. (3/3)⁰ + [(3/1)¹+ (3/1)¹] + (3/1)² ] = 4 terms (4 phases, or 2D Interval)
4. (3/3)⁰ + [(3/1)¹+ (3/1)¹ + (3/1)¹] + [(3/1)² + (3/1)² + (3/1)²] + (3/1)³ = 8 terms (8 phases, or 3D interval)

Finally, the M₄ magnitude expansion is

1. (4/4)⁰ = 1 term (0 polarities, or unipole)
2. (4/4)⁰ + (4/1)¹ = 2 terms (2 polarities, or 1D dipole)
3. (4/4)⁰ + [(4/1)¹+ (4/1)¹] + (4/1)² ] = 4 terms (4 polarities, or 2D quadrupole)
4. (4/4)⁰ + [(4/1)¹+ (4/1)¹ + (4/1)¹] + [(4/1)² + (4/1)² + (4/1)²] + (4/1)³ = 8 terms (8 polarities, or 3D octopole)

What this means, then, is that, in the first four dimensions of the CM, there are three sets of 8 dualities. There are 8 directions of M₂ spatial distances and 8 directions of M₂ temporal distances; There are 8 phases of M₃ spatial intervals and 8 phases of M₃ temporal phases, and there are 8 polarities of M₄ spatial poles and 8 polarities of M₄ temporal poles.

But what about the D = 4 level and beyond of the CM? For instance, at the D = 4 expansion level, what are the dualities of the 2⁴ = 16 magnitude of M₂? This has been a hard nut to crack, not just for us at the LRC, but for everyone in the LST community as well. String theorists hang their hat on the ten dimensions of the 1st generation tetraktys, and the 26 dimensions of the 2nd generation (1+2+3+4 = 10 & 5+6+7+8 = 26), but because LST physicists have treated D = 4, as the three space dimensions and 1 time dimension of spacetime, and can not yet benefit from the clarification of the RST and its fundamental revolution, with regards to the reciprocal nature of space and time, they take nine of the ten dimensions of the 1st tetraktys as space dimensions (length) and use the principles of M₂ magnitudes to generate theories that assume 6 of these nine space dimensions are hidden by what they call dimensional “compactification.”

This is clearly a mistake. The CM and the dualities inherent in it, enable us to see a much more simple and straightforward approach, and one that conforms to the Bott periodicity theorem which was first proven by Raul Bott. This theorem tells us that there are no new phenomena beyond three dimensions (four counting zero), that after three (four) dimensions, everything just repeats in a cycle of period 8. Although no one in the LST community really understands why this is so, the mathematics clearly shows the repetition.

Ironically, we’ve found that the secret is in a dimensional “compactification” of sorts after all, but in a much more elegant and inspiring form of nested dimensions. Here’s how it works: the pseudoscalar of the preceding tetraktys becomes the scalar of the succeeding tetraktys. It’s so simple that a child can understand it. The point at the center of Larson’s cube, which is a representation of the set of magnitudes defined by the four dimensions of the CM, is actually the cube of the preceding set, if there is one. Of course, for the first four dimensions, there is no preceding set. Figure 1 below illustrates the nesting of three tetraktyses.

Figure 1.  Nested, N-Dimensional, Magnitudes 

So, the magnitudes of the first four dimensions, or tetraktys, form the center point of the second tetraktys, which form the center point of the next tetraktys, and so on, ad infinitum, as far as numbers go. Certainly, there would be a limit to actual physical motion combinations set by environmental constraints such as temperature and pressure. To see how this works, let’s apply the duality expansion to the second tetraktys. First, the M₁ magnitudes:

1. (8/8)⁰ = 1
2. (8/8)¹ = 1
3. (8/8)² = 1
4. (8/8)³ = 1

Next, the M₂ magnitudes (2⁴ - 2⁷):

1. (16/16)⁰ = 1 term (0 directions, or point)
2. (16/16)⁰ + (16/8)¹ = 2 terms (2 directions, or 1D line)
3. (16/16)⁰ + [(16/8)¹+ (16/8)¹] + (16/8)² = 4 terms (4 directions, or 2D area)
4. (16/16)⁰ + [(16/8)¹+ (16/8)¹ + (16/8)¹] + [(16/8)² + (16/8)² + (16/8)²] + (16/8)³ = 8 terms (8 directions, or 3D volume)

Now, the M₃ magnitudes (3⁴ - 3⁷):

1. (81/81)⁰ = 1 term (0 phases, or point)
2. (81/81)⁰ + (81/27)¹ = 2 terms (2 phases, or 1D interval)
3. (81/81)⁰ + [(81/27)¹+ (81/27)¹] + (81/27)² ] = 4 terms (4 phases, or 2D interval)
4. (81/81)⁰ + [(81/27)¹+ (81/27)¹ + (81/27)¹] + [(81/27)² + (81/27)² + (81/27)²] + (81/27)³ = 8 terms (8 phases, or 3D interval)

Finally, the M₄ magnitudes (4⁴ - 4⁷):

1. (256/256)⁰ = 1 term (0 polarities, or unipole)
2. (256/256)⁰ + (256/64)¹ = 2 terms (2 polarities, or 1D dipole)
3. (256/256)⁰ + [(256/64)¹+ (256/64)¹] + (256/64)² ] = 4 terms (4 polarities, or 2D quadrupole)
4. (256/256)⁰ + [(256/64)¹+ (256/64)¹ + (256/64)¹] + [(256/64)² + (256/64)² + (256/64)²] + (256/64)³ = 8 terms (8 polarities, or 3D octopole)

Thus, the four types of magnitudes in the CM, written as powers of 4 through 7, the four dimensions of the second tetraktys, now makes just as much sense, as those written as powers of 0 through 3, the four dimensions of the first tetraktys, even though 4, and higher, dimensions of space magnitude don’t make sense geometrically. The reason is easy to see: there are no new phenomena beyond 3 (4) dimensions. All physical magnitudes are limited to 3 (4) dimensions, but they are not limited to the same four dimensions. The first set of magnitudes in four dimensions is based on 1; the second set on 1 x 8; the third set on 8 x 8, the third set on 8 x 8 x 8, etc. In other words, on successive powers of 8,

1. 8⁰ = 1
2. 8¹ = 8
3. 8² = 64…

simply because duality, in three dimensions, produces 2³ = 8 magnitudes, at each level, as shown in figure 1. Amazing! The universe of motion is made from 8-bit bytes!