The Larson Research Center

In the previous post below, I laid out what amounts to a 3D “coordinate” system of operationally interpreted (OI), rational, numbers. Of course, in turning our attention to the quantitative development of our RST-based theory, these numbers will not be referred to spatial locations in a vectorial reference system, where zero is at the origin of three, orthogonal, axes, as we are accustomed to using it in the legacy system of physics (LST).

Rather, these numbers will be referred to the net speed-displacements of the constituent units of S|T triplets (see “The Big Bet” and “Preon Models” and subsequent posts). Recall that these OI numbers are developed from the increasing degrees of freedom in the quantity, duality, and dimensional properties of OI numbers. To develop the physical magnitudes, which correspond to these numbers, requires the application of a set of physical principles of motion, just as to develop physical magnitudes that correspond to the numbers of the zero reference system also requires the application of a set of physical principles of motion.

In the latter case, however, continuous magnitudes of velocity, energy and momentum are developed from principles of mass, length and time, in what is referred to as classical physics. Beyond this, in the so-called quantum physics of the LST, discrete magnitudes of velocity, momentum and energy are developed from modified principles of mass, length and time, which include principles of uncertainty, of exclusion, and of complementarity, of non-commutativity, etc. These are principles that are not needed in the classical version of the mass, length and time system, but are indispensable in its quantum version.

Consequently, the continuous principles of physical magnitude are not replaced by the additional discrete principles of physical magnitude, in the LST, but the continuous principles are seen to arise out of the discrete principles, which apply only to the properties of particles existing below a certain scale. Nevertheless. because these particles can only be studied while moving relative to one another at very high speed, both sets of principles are required. In other words, in LST physics, both continuous and discrete magnitudes are needed to understand experimental observations correctly, and this necessitates the application of two separate sets of physical principles.

Of course, this approach has worked, to a certain extent, but it is ultimately problematic, because continuous and discrete magnitudes are ultimately different animals. Successfully modifying the physical principles of continuous magnitudes, in order to “quantize” them, making it possible to adapt these principles to serve our purpose in deriving the observed discrete magnitudes of nature, is admittedly ingenious, but, ultimately, it is doomed to failure, as we are discussing in the The Trouble with Physics blog.

However, it is important to recognize that basing the continuous and the discrete physical magnitudes of nature on one type of motion, what we call the M₂ type, or the vectorial type, of continuous motion, is no longer necessary, since the discovery of a new set of physical principles, one that applies strictly to discrete magnitudes of motion, now portends a way out of the current continuous-discrete quagmire. In particular, we should now understand that the discrete principles, which the LST has employed in its quantum system, the rotational form of M₂ motion, where magnitude, expressed as a continuously changing direction in one location, is substituted for a continuously changing location in one direction, is a contrived approach, used to generate discrete magnitudes, and that it is no longer necessary.

This changes everything. Now it should only be necessary to distinguish between the M₂ type of motion, whether of the linear or rotational form, and the M₄ type, or the scalar type, of motion, in calculating the discrete physical magnitudes of nature from the principles of discrete space and time only. The new principles should be completely capable of explaining how the continuous type of magnitudes arise from the discrete type of physical magnitudes, without the need to modify the continuous principles one iota. At least it seems clear to us, here at the LRC, that, in order to properly derive the discrete physical magnitudes of nature, it behooves us to seek to understand and to employ the new set of discrete physical principles that Larson has discovered, rather than to continue to rely on a modification of the continuous physical principles of the Newtonian system.

This means, among other things, that we cannot entertain a model of the nuclear atom, with a nucleus consisting of a tightly bound mix of quarks, surrounded by a cloud of orbiting electrons, occupying various levels of discrete energy, derived from complex treatments of angular momentum, the nature of which is mysterious. Hence, in the new system, Planck’s energy constant is not regarded as a unit of angular momentum, generated by the wavefunction of the massive electron. Indeed, the whole idea of kinetic and potential energy, in its Hamiltonian and/or Lagrangian form, as operators on the wavefunction, is rendered completely irrelevant, by our transition to an entirely new system of discrete physical principles, used to develop the discrete physical magnitudes of the new theory.

In short, we have developed a new set of n-dimensional, OI, rational numbers, based on 1, not 0, which we seek to use to express discrete physical magnitudes, and these magnitudes must be developed using a new set of physical principles of discrete units of motion, which does not include the idea of particles and atoms that consist of moving subparticles, bound together by autonomous, fundamental, forces. Instead, the new approach embraces the idea of combining discrete units of space|time, or scalar motion, from which the properties of mass, energy, and radiation emerge in such a way as to produce the observed forces of interaction.

Accordingly, in the previous post below, we show how the new “coordinate” system of n-dimensional OI numbers, with varying degrees of freedom, are derived. Notice how, in the 2D case, the pattern of numbers that we developed follows that of the periodic table of elements, as developed by Larson; that is, it follows the 4n² pattern:

1. (2|2)² = 4*1² = 4
2. (3|3)² = 4*2² = 16
3. (4|4)² = 4*3² = 36
4. (5|5)² = 4*4² = 64

Please note that, if the normal, quantitative, interpretation (QI) of rational number is used, the above equations are incorrect: That is to say, in the QI case, all four of these numbers are equal to 1 raised to the second power, but when the new, operational interpretation, of rational number is used, as shown in figures 1 - 4 of the previous post, then they are all valid equations, reflecting the total, discrete, number combinations that are possible in each case (except that (1|1)² = 0 was shown in the graphic instead of (5|5)² = 64.)

However, due to our natural inclination to think of orthogonality in terms of multiplication, we need to strive to understand that, in the discrete case, the mathematical meaning of dimension signifies the number of independent terms involved, not necessarily the geometric dimensions involving directions differing by 90 degrees of angle. Thus, while we represent the independence of the dimensional degrees of freedom in figures 1-4, as orthogonal lines, separated by 90 degree angles of direction, this does not imply a geometric relationship between the dimensions. We can just as easily represent them as independent, parallel, lines, for example.

We are simply saying that with a value of 1 unit of speed-displacement, the second power of this number, (2|2)², has two, dual, values, in each of the two dimensions, or a positive and negative value in each dimension, if you will. Thus, there are four possible magnitudes in the 2D number 2:

1. D₁ +1
   
2. D₁ -1
3. D₂ +1
4. D₂ -1

Similarly, the 2D number 3, (3|3)², as show in figure 3 of the previous post, has 4 values in each of 4 quadrants, or 16 numbers, because now we have two positive and two negative values in each of the two dimensions:

1. D₁ +1, +2, -1, -2
   
2. D₂ +1, +2, -1, -2
   

So, combining the 4 numbers of dimension 1, with the four numbers of dimension 2, will give us 4*4 = 16 unique values, but these have nothing to do with orthogonal geometric directions. They are just numbers. We can add them together, as weights, multiply them together, as areas, or use them, as binary inputs, in a logic circuit, but the numbers are always independent of the applications that employ them. They are only abstract symbols with a logical, 4n², pattern.

Therefore, to develop multi-dimensional physical magnitudes that we can express in terms of these multi-dimensional numbers, we simply need three, independent, physical variables, such as the speed-displacement of the three constituent S|T units of the S|T triplet combination of scalar motion that we’ve been discussing (see previous posts below). These units of discrete motion are independent in the sense that they constitute three adjacent physical locations, partially merged together as one, as illustrated in figure 1 below:

Figure 1. The Physical Representation of the S|T Triplet

In the figure above, we see a graphical illustration of three, independent, S|T units merging together to form a combination unit. Notice that, while the geometry of the combination is necessarily two-dimensional, because the center points of the three spheres must lie on a plane, the independent space|time magnitudes of displacement (represented by the three different colors) are three-dimensional and each dimension has the dual “directions” of the OI numbers; that is, the net magnitude of the space|time displacement of each sphere can be green, red or blue, corresponding to magnitudes that are balanced (neutral green), or unbalanced in one of two “directions,” red (negative) or blue (positive).

A balanced (green) S|T unit can be any value that has the same number of SUDRs as TUDRs, but the minimum number it can have is, of course, one of each. If we denote the constituent S|T units in the triplet as A,B,C, then the equation for the minimum green unit in the triplet is

S|T_A = (1|2) + (1|1) + (2|1) = 4|4 nm,

where nm is the symbol used for natural units of motion. The equation for the minimum blue unit is then

S|T_B = (1|2) + (1|2) + (4|2) = 6|6 nm,

and the equation for the minimum red unit is

S|T_C = (2|4) + (2|1) + (2|1) = 6|6 nm

The total number of natural units of motion is the sum of these, or

(4|4) + (6|6) + (6|6) = 16|16 nm

However, in terms of independent variables, we have 0 (green), +1 (blue), and -1 (red), and we can see that, whenever one of the S|T triplet’s three constituent units is green, the 2² = 4 possibilities of magnitude that the triplet can take are those of two, orthogonal, dimensions:

1. blue + red = +, -
   
2. blue + blue = +, +
   
3. red + blue = -, +
   
4. red + red = -,-
   

But now an important question arises: What is the difference between the (green + red + blue) combination, and the (green + blue + red) combination? Numerically, (0 + (-1) + 1) = 0 and (0 + 1 + (-1)) = 0, because quantitatively interpreted (QI) numbers are always zero-dimensional numbers (scalars). However, the numbers of the S|T units are comprised of three terms, not one, and while these three terms represent one space|time location, which is separate and distinct from adjacent space|time locations, the values of the space and time displacements, in a given location, vary.

So, if the three magnitudes were QI numbers, the sum of the (red + blue), or (blue + red), would be the same, in both cases, but the OI numbers do not work in the same way. There are three, adjacent, space|time locations in the S|T fermion triplet, each with its red (space) location, and each with its blue (time) location. However, because the configuration of this triplet is a triangle, the time location of S|T unit A is associated with the space location of S|T unit B, and its space location is associated with the time location of S|T unit C. The same arrangement holds for all three S|T units: Each is joined to the other two, through a space|time connection.

Thus, this situation seems to raise the same type of issue raised by multiplication of M₂ vectors, a and b. In taking the outer product of two vectors, the order of the operation makes a difference; that is, a^b is not the same as b^a. Yet, we are not multiplying n-dimensional vectors here, but, rather, we are summing n-dimensional scalars. Regardless, however, we see that we must still distinguish the sum of ((+) + (-)) from the sum of ((-) + (+)), just as vector algebra must distinguish the bivector of a^b from the otherwise identical bivector b^a, by making it -b^a, indicating a directional property of the bivector that Hestenes calls “orientation.” The orientation of an outer product, in Hestenes’ GA, indicates the direction of its construction, if you will, because there are two ways you can raise the dimension of a number, represented by sweeping the vector a along the vector b, or the vector b along the vector a.

However, in the RSM, we don’t have n-dimensional vector magnitudes, only n-dimensional scalar magnitudes, which sounds like a contradiction in terms, without an understanding of the difference made by the OI rational number. Yet, as explained in the previous post below, the new OI numbers do, in reality, have the three properties, quantity, duality, and dimension, the same three properties observed in physical magnitudes. They are not just symbols of quantities alone. Therefore, the sum (-A) + (+B) is different from the sum (+A) + (-B), because all three properties of the OI numbers must be taken into account. The easiest way to understand this is to consider the four quadrants of a plane formed from two, independent, reciprocal numbers (i.e. two, three-term, OI numbers), as shown in figure 2 below.

 

Figure 2. Plane of Two Independent OI Numbers

In figure 2 above, the magnitudes of quadrants 1 and 3 (disregarding the middle terms for now), in terms of polarity, or color, are straightforward, but quadrants 2 and 4 are ambiguous:

1. Q1 = +, + (blue + blue)
2. Q2 = -, + (red + blue)
3. Q3 = -, - (red + red)
4. Q4 = +, - (blue + red)

Yet, summing the OI numbers in each quadrant, we can see that, actually, there is a big difference between Q2 and Q4:

1. Q1 = +, + (blue + blue) = (4|2) + (2|1) = 6|3
   
2. Q2 = -, + (red + blue) = (1|2) + (2|1) = 3|3
   
3. Q3 = -, - (red + red) = (1|2) + (2|4) = 3|6
4. Q4 = +, - (blue + red) = (4|2) + (2|4) = 6|6

It turns out that Q4 is twice the value of Q2. Notice that this is not the same thing as the non-commutative property of bivectors, because (-A) + (+B) = (+B) + (-A). Thus, OI numbers are commutative, but they can be misinterpreted, if all three properties of the numbers are not considered.

An interesting aspect of all this is the light it throws on the concept of non-commutative mathematics. It’s clearly the result of raising the dimensions of numbers through rotation via the ad hoc invention of the imaginary number, used to form complex numbers from real numbers. If we do it this way, we still have to somehow preserve the duality property of the number, and we do this in GA (unwittingly), by invoking the direction of rotation as a numerical property; that is, the representation of the two sides of the plane, the positive and negative sides we might say, is accomplished by changing the sign of the bivector, under multiplication, indicating that the direction of rotation (the “sweeping of the vectors”) changes with a change in the order of the multiplication operation.

However, when we use OI numbers, this expedient is no longer necessary, because the duality of the numbers is preserved through the summing operation, since it is built into the numbers themselves, in a natural way. Actually, what we are doing is replacing the artificial plane of rotation, the complex plane, with its zero origin, with the plane of duality and its origin of unity. Fortunately, doing so is perfectly logical, from a mathematical point of view, and now we see that it is well motivated from a theoretical point of view, as well. All that remains is to justify it, by deriving actual physicical magnitudes, based on the same, reciprocal, principles of discrete values that the “coordinate” numbers are based on, and see if we can identify them with the magnitudes of observed physical phenomena.

A tall order, to say the least, but what could be more fun?

 

 

 

In beginning the quantitative development of our RST-based theory, we incorporate the ancient principle of numbers in what we designate the operationally interpreted (OI) rational number, developed in the new reciprocal system of mathematics (RSM). In this view of number, the value of the number is the value of the relationship of two quantities. In the reciprocal system of physical theory (RST), this takes the form of two, reciprocal, changing, quantities, where the OI number represents the relative change rates of space and time in the equation of motion. Symbolically, this is written as

ds|dt = n|m,

where n and m are numbers corresponding to the reciprocal change rates of space, ds, and of time, dt, and the operational symbol employed to represent this relationship is the pipe symbol, “|”, rather than the division symbol, “/”, in order to distinguish the customary quantitative interpretation of the rational number from its new, operational, interpretation. The difference in the two interpretations is significant in the fact that zero is not an explicit part of the new system of numbers, just as it’s not an explicit part of the system of physical magnitudes in the RST; In other words, there is no such thing as zero motion, only a zero difference between two magnitudes of change, just as there is no such thing as a zero quantity, only a zero difference between two quantities.

This fundamental change in the concept of number, employed in the theoretical development of physical magnitudes, has a great effect on the normally accepted physical concepts such as space, time, and locations in space and time, which are used to define physical magnitudes in the legacy system of the mainstream scientific establishment. This is most easily recognized when the concepts of the Chart of Motion are understood, as these concepts extend the idea of motion, from an exclusive concept of the motion of objects, changing locations, in a container of space and time, to higher forms of motion, corresponding to higher forms of OI numbers. In effect, the properties of numbers and magnitudes are unified in the Chart of Motion.

The implications of this change are enormous, including its philosophical implications, which will be explored for centuries to come, if these developments prove to be as valid as they now appear. The essence of these philosophical implications is found in the difference between inventive and inductive science, which is profound. In short, the discovery of the concepts embodied in the Chart of Motion, the concepts of Larson’s reciprocal system, expressed in terms of a new system of numbers, enable the systematic development of physical theory, through the logic of deduction, from a few, fundamental, assumptions, induced within the limits of observation, rather than conjured up from the limitless imagination of human thought.

The observations upon which the assumptions of the RST are founded are the observations of expanding time and space. That the OI numerical representation of this universally observed expansion corresponds to a reciprocal relation of numbers, which constitutes the only observed relationship between space and time, places the RST, as a new system of physical theory, on incredibly solid, and unprecedented, philosophical grounds.

However, the current state of theoretical development reflects this philosophical strength only insofar as it constitutes an inflexible development of correctly deduced consequences of the fundamental assumptions. Of course, the certainty with which we can ascertain the validity of these theoretical deductions depends on the degree to which they correspond to observed, or measured, magnitudes of physical phenomena, which is never 100% accurate. Nevertheless, the comparison of theoretical entities, combinations of these, and relations between them, to corresponding observed entities, combination of these, and relations between them, are the elements that must comprise the science of RST-based research at the LRC.

The task is not an easy one. As John Baez has pointed out, “it takes guts to do physics” (see here.) However, we are quite gratified with the progress that we’ve been able to make thus far, which has come from following the logical consequences of the fundamental assumptions of the RST, even though this course has meant that the chain of our conclusions follows a different path than did Larson’s initial chain of conclusions. Happily, however, the determination of which course of development is the correct logical path to follow is testable, by comparison to physical observations. Obviously then, to perform these tests, requires more than the qualitative development of logical consequence. It requires the quantitative development of numerical magnitudes, with physical dimensions, that can be compared to observed physical magnitudes, with physical dimensions.

Nevertheless, the current system of measurements, and the accompanying concepts of physical dimensions, have been developed over centuries of work within the Newtonian system of physical theory, and, consequently, a translation between the legacy and reciprocal systems is required. One of Larson’s greatest contributions to science is his work in meeting this requirement. In his monumental Structure of the Physical Universe (SPU), in three volumes, he shows how the conventional cgs system of units, based on mass, length and seconds, can be expressed in the terms of the RST’s natural units of space and time. It should also be mentioned that, more recently, a theoretically unrelated, but similar, effort, with respect to the space and time dimensions of the international, SI, system of units, has been published independently by Engineer Xavier Borg (see here.)

In Larson’s theoretical development, he derives the natural units of motion from two, observed, constants of nature, the Rydberg constant and the speed of light constant. On this basis, the natural unit of space is calculated as

4.558816 x 10^-6 centimeters, and the natural unit of time is calculated as

1.520655 x 10^-16 seconds.

With these natural units, the natural units of acceleration, force, energy, etc, can be expressed in turn (see Chapter 13 of SPU, Volume II, Nothing But Motion). However, as the Chart of Motion shows, there is a distinction to be made between n-dimensional space|time units of M₁, M₂, M₃, and M₄ motion, which Larson did not explicitly recognize. Moreover, explicitly recognizing these different types of motion enables us to generalize the concept of number sufficient to conform to the concept of magnitude, unifying the two concepts in a most remarkable manner.

Armed with this additional knowledge of numbers and magnitudes, in the Chart of Motion, we should be able to develop the increasing degrees of freedom of fundamental numerical magnitudes to produce the fundamental entities of the RST-based theoretical development, in quantitative terms, that are related to, but different than, the corresponding entities in the LST-based theory, in certain cases. The most straight forward examples of this are seen in the cases of the concepts of velocity, energy, and momentum. They are very different concepts than those of LST physics.

For example, in RST-based theory, velocity is one-dimensional motion, or speed, with dimensions ds|dt, while energy is the inverse of this, or one-dimensional inverse speed, with dimensions dt|ds. Momentum, is two-dimensional energy, or inverse speed, with dimensions (dt)²|(ds)², and is, therefore, a quantity of motion that is the difference between one-dimensional inverse speed and three-dimensional inverse speed. In the RST-based theory, matter is composed of units of three-dimensional speed, with dimensions (ds)³|(dt)³, while mass consist of inverse units of this motion, or three-dimensional inverse speed, with dimensions (dt)³|(ds)³. Meanwhile, two-dimensional speed, or inverse momentum, with dimensions (ds)²|(dt)², is simply a quantity of motion that is the difference between one-dimensional speed and three-dimensional speed.

Clearly, this is a different system of units than that employed in LST physics. However, conformance to the LST system of units is not a requirement of nature. The only requirement is logical consistency. We know that, in the main, the LST system of units is logically consistent. Indeed, our modern technology is dramatic testimony of that fact. Nonetheless, it doesn’t follow from this that it is the only, or even the best, system of units conceivable.

The Chart of Motion reveals that both numbers and physical magnitudes correspond to geometric points, lines, planes, and volumes, or to 0, 1, 2, and 3-dimensional entities. The different types of motion in the chart, M₁, M₂, M₃, and M₄ motion, happen to correspond to the numbers 1, 2, 3, and 4, and the different characteristics of the four motion types stems from the different characteristics of the corresponding numbers, and the way motion can be defined using them.

What happens is that, as OI numbers, what we can term the “quantum degrees of freedom” of each of the three properties of the numbers, i.e. quantity, duality, and dimension, increase in ascending order. Thus, the number 1|1 has no quantum degree of freedom, but 2|2 has one quantity and one dual quantum degree of freedom (one in each dual “direction”), while 3|3 has two quantity and two dual, and 4|4 has three quantity and three dual quantum degrees of freedom. These four quantum degrees of freedom are shown below:

1. 1|1 = 0 degrees of freedom
   
2. 1|2, 2|2, 2|1 = 1 degree of freedom
   
3. 1|3, 2|3, 3|3, 3|2, 3|1 = 2 degrees of freedom
   
4. 1|4, 2|4, 3|4, 4|4, 4|3, 4|2, 4|1 = 3 degrees of freedom
   

In addition to the quantity and duality quantum degrees of freedom, shown above, there is also the dimensional quantum degree of freedom, and there are four of these, as well:

1. (1|1)⁰, (1|1)¹, (1|1)², (1|1)³
2. (1|2)⁰, (2|2)⁰, (2|1)⁰; (1|2)¹, (2|2)¹, (2|1)¹; (1|2)², (2|2)², (2|1)²; (1|2)³, (2|2)³, (2|1)³
3. (1|3)⁰, (2|3)⁰, (3|3)⁰, (3|2)⁰, (3|1)⁰; …(1|3)³, (2|3)³, (3|3)³, (3|2)³, (3|1)³
4. (1|4)⁰, (2|4)⁰, (3|4)⁰, (4|4)⁰, (4|3)⁰, (4|2)⁰, (4|1)⁰; …(1|4)³, (2|4)³, (3|4)³, (4|4)³, (4|3)³, (4|2)³, (4|1)³

It’s not that there can’t be more than three quantities, or more than three dimensions, or more than three dual magnitudes.  Obviously, there can be.  However, Raul Bott proved that there is no new phenomena above three dimensions (see This Weeks Finds 105), and the perfect symmetry of 0-3 makes me suspect that the same thing holds for all three properties of numbers and magnitudes; that is, these are fundamental numbers and magnitudes, from which all other numbers and magnitudes can be constructed, but someone else will have to prove it.

In the meantime, I’m going to refer to this as the “fundamental magnitude conjecture” (FMC) in our RST-based theoretical development.  The FMC is shown graphically in the four figures below:

 

Figure 1.  FMC of Number 1

 

Figure 2. FMC of Number 2

 

Figure 3. FMC of Number 3

 

Figure 4. FMC of Number 4 

In figure 2 above, the 3D magnitudes of number 2 are drawn in perspective.  However, to avoid unnecessary complexity, the 3D magnitudes of numbers 3 and 4 are not drawn in perspective.  Nevertheless, the 3D magnitudes of number 3 should be understood as a 4x4x4 cube, and the 3D magnitudes of number 4 should be understood as a 6x6x6 cube.  Of course, the 6 unit line, the 6x6 unit square, and the 6x6x6 unit cube also contain the respective, n-dimensional, magnitudes of the 1, 2, and 3 numbers, as well.

The significance of the FMC is that all other magnitudes can be constructed from these fundamental magnitudes, but how they are constructed depends upon the type of motion employed.  Beginning with M₁, or unit motion,  we see that this is not a zero magnitude of motion, but only a magnitude of motion without any degree of freedom.  Without it, no other motion could exist.  Then, by adding degrees of freedom to M₁ motion, we make the other magnitudes of motion possible.

For example, adding one degree of quantum quantity to the ds|dt = 1|1 magnitude of motion, increases it to the ds|dt = 2|2 magnitude, which is equivalent to the 1|1 magnitude of motion, but the difference is that adding a degree of quantum duality won’t change the result of the 1|1 magnitude, but it will change the result of the 2|2 magnitude of motion.  Similarly, adding a degree of quantum dimension to the 1|1 magnitude will not change the result of 1|1, but it will change the result of the 2|2 magnitude.  Hence, we conclude that the 2|2 magnitude of motion is not the same as the 1|1 magnitude of motion, when the degrees of freedom of its properties are increased.

Nevertheless, the transition, from the traditional numerical magnitudes, based on the quantitative interpretation of number, to this fundamental concept of numerical magnitude, based on the operational interpretation of number, requires the recognition of the central change of concept: We are transitioning from the foundational concept that plays the central role in the Newtonian system of physics, to the foundational concept that plays the central role in the Larsonian system of physics.

That is to say, we are going from a static frame of reference, where the datum of physical and numerical magnitude is zero, to a dynamic frame of reference, where the datum of physical and numerical magnitude is not zero, but one.  We still refer to zero magnitudes, but only as a difference, a delta, measure, not an absolute measure.  The absolute magnitudes of RST-based theory do not include zero, as the four figures above show.

So, how do we translate between the two systems?  Well, on the scalar level it’s trivial, but with increasing degrees of freedom, it’s a little more complicated, but not really a whole lot.  What we have to recognize is that there is a difference between the direction of angle and the “direction” of duality.  Hence, Increasing the dimensional degrees of freedom, increases the number of dual magnitudes, in non-unit magnitudes, as a power function.

For example, when the first quantity DoF is added to the 1|1 magnitude, to form the 2|2, it remains a point, indistinguishable from the 1|1 magnitude. Subsequently, adding the first duality DoF to the 2|2 magnitude creates the -1 and 1 magnitudes of the traditional, zero-based, integer system, except that the -1 magnitude of the new system is not an ad hoc invention, but actually the inverse of the positive 1 magnitude. Yet, when this is not properly recognized, only the positive magnitudes are treated as functions of dimensions.  In other words, because the duality property of numbers is not recognized in the legacy system of numbers, the dimensional DoF is only applied to the positive magnitudes. 

For example, in the binomial expansion, which is the dimensional expansion of the number 2 (equivalent to the number 3|1 in the new system), 2⁰ = 1, but 2¹ = 2, not 4, and 2² = 4, not 16, and 2³ = 8 not 64, as in the new system (see figure 3 above.)  This is due to the fact that the duality property of number is not recognized in the legacy system.

What this means, then, is that the four numbers of the tetraktys, and the first four numbers of the binomial expansion, and the associated Clifford algebras of these numbers, and the legacy concepts of real, complex, quaternion, and octonion number systems, and the topological and physics concepts based upon them, are all missing a power of 2 worth of numerical magnitudes, which they would otherwise have, if the duality property (reciprocity) of numerical magnitudes were properly recognized.

For us, though, this amounts to recognizing that the scalar and pseudoscalar of the octonion, where (2/1)⁰ = 1 is the scalar point, and where (2/1)³ = 8 is the pseudoscalar volume, which contains the scalar point, represent only a part of the fundamental numerical magnitudes, in the universe of motion.  There are actually the following magnitudes in the eight quadrants in the (3|3)³ graphic of figure 3:

1. (3|1)x(3|1)x(3|1) = 2³ = 8
   
2. (1|3)x(3|1)x(3|1) = 2³ = 8
3. (1|3)x(1|3)x(3|1) = 2³ = 8
4. (1|3)x(3|1)x(1|3) = 2³ = 8
5. (3|1)x(3|1)x(1|3) = 2³ = 8
6. (3|1)x(1|3)x(1|3) = 2³ = 8
7. (3|1)x(1|3)x(3|1) = 2³ = 8
8. (1|3)x(1|3)x(1|3) = 2³ = 8

Consequently, we actually have dual Charts of Motion, in our new system, each with four types of motion that can be identified as different modes of magnitude, which we will now designate as M_a, M_b, M_c, and M_d, to ensure that we distinguish between these, as definitions, and the numbers of the chart themselves.

What we mean by mode of motion is the way a change of space and time is affected in the definition of motion.  Of course, motion, by definition, is a change of space and time.  In the first mode, M_a motion, the space|time change is the constant, continuous, all-directional change of the unit space|time expansion.  In the second mode, M_b motion, time (space) expands continuously and uniformly, but the change in the space (time) aspect of this motion is the measured change of distance between two, or more, separate, objects.  With one object at one location, and one or more objects at separate locations, this motion can be defined as a change of distance between the objects. 

Hence, with no objects demarking spatial locations, or even just one object, demarking one location, this motion cannot be defined. With a minimum of two, separated, objects, however, the M_b motion between the respective locations is defined by either an increase in the distance, or a decrease in the distance between the objects, in a given time interval.

In the third mode, M_c motion, time (space) expands continuously, but the space (time) aspect of this motion is the measured change between three, or more, separate, objects. With one object at one location, and two or more objects at separate locations, this motion can be defined as a change of interval between a minimum of three objects, occupying three separate locations.

In the fourth mode, M_d motion, both the space and time aspects of the motion expand, but only one aspect does so uniformly, while its reciprocal aspect continuously expands and contracts.  This motion does not depend on the locations of objects for defining the required change.

With these four modes of motion, we can describe the magnitudes of the Chart of Motion in different ways.  More on this later. 

 

 

In the LST theories of the nuclear atom and quantum mechanics (QM), they speak of “filling the shells” of atomic orbits, with electrons of different energy levels, corresponding to the varying electrons in neutrally charged atoms of the periodic table of elements. Physicists characterize these “orbitals” in terms of the speed, angular momentum, magnetic moment, and spin of a given electron. Of course, these are properties of M₂ motion (except the enigmatic motion of spin), the only type of motion recognized by LST physicists. However, in RST-based science, we are able to avail ourselves of the M₄ type motion (see: here and subsequent posts in the New Math blog.)

Nevertheless, In Larson’s RST-based development, the atomic spectra were never calculated from the properties of M₄ motion. Larson spent a lot of time on it (of course, he referred to scalar motion, not to M₄ motion per se, which is an LRC designation), but, ultimately, he felt that he had to move on. He was afraid of becoming bogged down in the mountains of spectral data, while he had many other areas of physical theory that he wanted to cover.

Even in the LST, though, the atomic spectra calculations are only approximate, and then only for the simplest cases, the so-called “hydrogenic” atoms that are actually ions. The rest of the calculations are just too difficult to actually carry out, so it’s referred to as one of those achievements based on principle; that is, if it weren’t so complicated, the calculations could be carried out, but, as it is, the more complicated equations can only be solved “in principle.”

Whether or not we will ever be able to do better here at the LRC, remains to be seen, but the first step is “filling the shells” with more and more electrons, to build the periodic table of elements. Larson did this very successfully by adding more and more units of scalar rotation, where the difference between one atom and the next higher one in the order is a (double) unit of 1D (electric) rotation, as shown in figure 1 below.

Figure 1. The Wheel of Motion Showing Larson’s 4n² Relationship of Scalar Motion

As the Wheel of Motion in figure 1 shows, the periodic table of elements is actually a 4n² relationship of units of scalar motion, where n varies from 1 to 4, resulting in 4 scalar magnitudes in the first group (three of which are subatoms), 16 in the second group, 36 in the third, and 64 in the fourth group. This differs from the quantum mechanics (QM) concept where the relationship is understood in terms of 2n² and n is the unlimited (in principle) number of atomic energy levels Thus, in the QM concept, n is the first, or the principle, of the four quantum numbers determining the electronic “shells” and “orbitals” of the atom. The first shell only has room for two electrons, when n = 1, with up and down spin, but the second shell has room for eight, given the possible values of the other quantum numbers, when n is 2.

In the LST concept, certain selection rules determine the order that these shells and orbitals are filled with electrons. This results in a bizarre order, when viewed in the context of the Wheel of Motion. In Larson’s development, each element in the Wheel results from the addition of one more unit of (double) scalar rotation. However, in the LST periodic table, using the selection rules of QM, only half of each concentric circle of elements in the Wheel is filled first, in ascending order, then the remaining half is filled, in descending order! That is to say, 2 of 4 in the first circle, 8 of 16 in the second circle, 18 of 36 in the third circle, and 32 of 64 in the fourth circle, are filled first, then the last half of the fourth circle, the last half of the third circle, the last half of the second, and the last half of the first, are filled next.

The reason for this is the 2n² energy relationship of QM numbers. Without the 4n² of the RST-based model, it’s impossible to fill the orbitals any other way, because there are not enough magnitudes to do so. On the other hand, the 4n² relationship, which comes naturally to the RST-based model due to the n-dimensional nature of its scalar magnitudes of rotation, has just the correct number of magnitudes and no others. This is especially noteworthy, when one recognizes that the incorrect mathematical relationship introduces errors into the shell model of the LST, predicting more and more non-existent elements, as n increases beyond 4, to accommodate the last half of elements in each circle, or row, in the periodic table. Figure 2 below shows the anomalies.

Figure 2. Anomalies in LST Nuclear Model of the Atom

As the table in figure 2 above shows (see: the Chemogenesis Webbook), there are 164 unobserved elements predicted by the QM model of the nuclear atom. No such problem exists in Larson’s scalar rotation model of the atom, as shown in figure 1 above. However, using Larson’s model, we can only predict the atomic spectra of hydrogen. We cannot go beyond that point, which is a glaring difficulty, or “lacuna” in the RST-based theoretical development.

In the research program at the LRC, we’ve taken a new approach to RST-based theory development in that we’ve eliminated the concept of n-dimensional scalar rotation that Larson employed, in favor of a 3-dimensional scalar vibration concept, which leads to the four, n-dimensional, magnitudes of motion explained in the Chart of Motion.

Happily, in our new RST-based model of the atom, we have been able to identify theoretical entities with photons, neutrinos, electrons, positrons, and the quarks of protons, and neutrons. We even have the anti-entities of all these, along with their chiral versions, which we can formulate in terms of preons called S|T units (see previous posts below.)

In treating these S|T units as preons, we find that the difference between fermions and bosons, which are all comprised of three S|T units each, is in their respective triplet configurations. The constituent S|T units of a boson triplet occupy the same space|time location, while the constituent S|T units of fermions occupy three distinct, or adjacent, space|time locations that are joined together so-to-speak, in a triangle configuration, rather than the parallel configuration of the boson triplet.

This is an exciting breakthrough in the qualitative aspect of our efforts. Nevertheless, to be convincing, we need to be able to find the same level of success in the quantitative treatment of these theoretical entities, as Larson found in his quantitative development of the periodic table of elements, using units of n-dimensional scalar rotation. As previously mentioned, this is a daunting challenge, but we are determined to press forward, confident that the success that we’ve enjoyed to date in the qualitative endeavor was just as unexpected and as daunting a prospect, in the beginning, as this quantitative effort seems to be now.

It’s clear that in moving from the preon level of study to the atomic level, we need to condense our graphic representations of the theoretical entities involved, because attempting to show all the constituent preons of protons and neutrons linked together with those of electrons, in a given atomic unit, quickly becomes unwieldy.  Interestingly enough, though, thinking about how to do this has led to the discovery of a triplet we didn’t understand: the nucleon triplet we will call it.  It is the result of combining the three quarks of the neutron with the three quarks of the proton, as shown in figure 3 below.

 

 

 

Figure 3. Nucleon Red and Blue Triplet 

Since combining red or blue with green doesn’t affect the color (like adding equal weights to both sides of an already unbalanced pan balance), combining the quarks of the proton (upper row), or the quarks of the neutron (middle row), results in a red and blue triplet, as shown in figure 3 above.  Likewise, combining the proton and neutron, as two red and blue triplets (bottom row), doesn’t affect the color, only the intensity, of the combined red and blue triplet.  Hence, we can qualitatively represent all the quarks of a nucleus as a single red and blue triplet.

The three nodes of the nucleon imply that the physical representation of the fermion triplet, as discussed in the previous post below, represents three spheres partially merged together. Since the only possible geometry for this representation is two-dimensional, it follows that the nucleon has two, opposed, faces, one to the left and one to the right, or one to the front and one to the back, or one up and one down. Thus, it follows that multiple instances of these triplets can be pancake stacked, one on top of the other, and, if conditions are right, merged into one set of three locations (again recalling that all fermions occupy three adjacent locations, represented by the three partially merged spheres.)

In order to merge (correlate) the three space|time locations of the constituent quarks of the protons and neutrons together, we must merge the three adjacent locations of each together. This is a departure from the way we have previously combined them. However, this two-dimensional concept of combination, rather than the earlier three-dimensional concept, will not only enable us to represent them more compactly, but represents a logical alternative that is much simpler, and therefore, preferable. Although, the fact that there are now only three consolidated space|time locations may mean that we are headed toward undesirable complications downstream that will cause us to rethink this decision.  For now, however, we will explore it’s possibilities.

Next we will investigate how the nucleon triplets can be combined with the electron triplets to form atomic entities. However, following the 4n² relationship of the Wheel of Motion means that hydrogen should be the last entity (first complete element) of those in the first circle, where n = 1, while helium should be the first entity of the second circle, where n = 2. This is not so in the LST QM model, where the principle quantum number of hydrogen and helium is n = 1 in both cases, and the difference is found in the value of the last quantum number, the direction of quantum spin.

This is an important difference between the two models.  The 4n² pattern of Larson’s RST-based model avoids the 2n² pattern anomalies of the LST-based model, but, on the other hand, the QM model predicts the atomic spectra, at least in principle. Hence, this is going to be interesting.

 

In the legacy system of physical theory (LST), the difference between bosons and fermions is encapsulated in the Pauli exclusion principle: Bosons can occupy the same quantum state, but fermions cannot. In the reciprocal system of physical theory (RST), on the other hand, which doesn’t employ the wave equation and the associated QM concept of state space, the difference is that bosons occupy the same space|time location, but fermions do not. In the LST, the different states in the state space are determined by quantum numbers. Thus, bosons may have the same quantum numbers and, when these numbers constitute the ground state, a set of atomic bosons in this state are then referred to as a Bose-Einstein condensate. Interestingly, atomic bosons can be configured as fermions, and fermions as bosons, in experimental apparatus (see here and here).

The same behavior in the subatomic entities of the standard model (SM) can be understood in terms of the S|T triplet version (see previous posts below) of fermions and bosons. In the boson triplets, the three constituent S|T units occupy the same S|T location, indicated by their parallel configuration, as shown in figure 1 below:

Figure 1. The Boson S|T Triplets of the Standard Model

On the other hand, the S|T units of the fermions do not occupy the same space|time location, as indicated by their non-parallel configuration, as shown in figure 2 below.

Figure 2. The Fermion S|T Triplets of the Standard Model

In the fermion triplet, the spatial coordinate position of the time-displaced (red) component, of one S|T unit, is co-located with the spatial scalar position of the space-displaced (blue) component, of an adjacent S|T unit. Conversely, we can say that the time coordinate position of the space-displaced (blue) component, of one S|T unit, is co-located with the time scalar position, of an adjacent S|T unit.

Of course, there is a major difference between these subatomic bosons and fermions, and their atomic counterparts: Unlike their atomic relatives, subatomic bosons propagate at the speed of light and subatomic fermions gravitate (this is the reason why neutrinos must have some mass, though very little, propagating almost at the speed of light).

It follows, then, that since the only difference in the boson and fermion S|T triplets is that the three constituent S|T units of the boson occupy one in the same S|T location, while the three constituent S|T units of the fermion occupy three adjacent locations, this difference must account for why one propagates and the other doesn’t (or does so at a slower speed). The next question is, therefore, what space|time changes must take place to convert a boson into a fermion, or vice versa?  In the LST community’s 11D Supergravity theory, such a conversion constitutes motion, according to Lee Smolin (see his Trouble with Physics), and this is why it and string theory require Supersymmetry, which would double the current number of observed particles, by proposing a boson equivalent of every fermion and vice-versa.

However, in our RST-based theory, the difference between what makes a boson a boson, and a fermion a fermion, clearly prevents this.  For instance, if a down quark were converted into a boson, its three adjacent locations merging into one location, it would become a single red S|T unit (green + green + red = red).  It’s not possible to have two green and a red S|T unit in a parallel configuration, any more than it’s possible to have a scale that is both balanced and unbalanced at the same time.  This is why Supersymmetry has never been observed and never will be observed.  Indeed, we can say that our theory predicts that CERN’s large hadron collider (LHC) will not discover Supersymmetry.  It is impossible.

Nevertheless, the question of quantum gravity remains the holy grail here, and, just as it is for the LST community, our biggest clue is this difference between bosons and fermions, because fermions gravitate, while bosons don’t.  To understand the difference more clearly, we should recognize that the figures above, graphically showing the space|time relations of the theoretical entities, should be understood as schematic representations, not physical representations, of S|T units.  The physical representation of the S|T unit is an expanding/contracting ball, or point, as shown in figure 3 below.

 

Figure 3. Physical Representation of S|T Unit  

In figure 3, the two chiral versions of the S|T unit, one having the inverse space|time dimensions of the other, are shown  The size of the point at the center is exaggerated.  Because time has no direction in space, and space has no direction in time, the point actually has no physical extent. The red graphic of the spherical entity on the right represents the space unit expansion, which contracts to the point and then expands to the unit sphere, oscillating, at the speed of light.  As was pointed out in a previous post, the expanding/contracting motion of the unit sphere entails the 720 degree angle change of spin, which is so enigmatic in terms of rotational motion, clearing up a major mystery of quantum mechanics (QM).

On this basis, the red sphere represents the physical SUDR, while the blue sphere represents the physical TUDR.  The blue point, at the center of the red sphere, is the blue (time oscillation) TUDR component of the S|T unit, from the space perspective, while the red point, at the center of the blue sphere, is its red (space oscillation) SUDR component, from the time perspective.  If we could view the S|T unit from both the space and time perspectives simultaneously, we would see an alternating red-blue expansion/contraction, but we can only observe one component of the displacement magnitude at a time.  Larson called these inverse sectors the material (coordinate space) sector and the cosmic (coordinate time) sector.  Thus, the S|T unit is represented by the graphic on the left in figure 1, in the material sector, and the graphic on the right, in the cosmic sector.  In either case, the colored spheres contract and expand, or oscillate, at the speed of light, simultaneously.

Moreover, the oscillating S|T unit also necessarily propagates relative to SUDRs and TUDRs, at the unit speed of light, as shown in the world line chart, discussed in previous posts below.  However, while the magnitude of the speed of oscillation and propagation are fixed at unit speed, the total speed-displacement, which is determined by the number of SUDRs and TUDRs that are joined together in a given S|T unit, determines the energy of the unit, and thus its frequency in terms of E = hv.

In our S|T triplet model of bosons, there are an even number of SUDRs and TUDRs in the three, constituent, ST units of the photon boson, which we encode as green S|T units, while in the negative W boson, the three constituent S|T units consist of more SUDRs than TUDRs, so we encode them red.  Of course, the S|T units of the positive W boson consists of more TUDRs than SUDRs, so we encode them blue. The enigmatic Z boson is neutral, because it supposedly consists of a combination of the negative and positive W bosons, but we have yet to think extensively about that.

We could color the spheres to indicate the relative number of SUDRs and TUDRs in a given boson.  In this way, a photon would be a green sphere, a negative W boson would be a red sphere, and the positive W boson would be a blue sphere.  However, this would not buy us much in terms of clarity over the schematic representations in figure 1 above, which we are in the process of simplifying, as explained in a previous post below.  Plus, it would be doubly difficult to represent the fermions as three, slightly merged, balls, or spheres, which is the form that their physical representation takes, as shown in figure 4 below.

 

Figure 4. Physical Representation of Three S|T Units Merging

I borrowed the graphic, in figure 4 above, from a z-merge tutorial, because I don’t have that capability in my graphics software. As already pointed out, the three colors combined together like this are impossible as a boson triplet, which must either be balanced, or unbalanced in one “direction,” or another, but it might be possible in the fermion triplet, though, if so, this configuration would be tantamount to a prediction of a new particle in our RST-based theory.  Nevertheless, until I have more time to think about it, we’ll just have to shelve it for the time being.

The important thing right now is to determine, if we can, what difference this combination of adjacent positions in the fermion S|T triplet makes viz-a-viz the mass/gravity question.  In order to address this question, we will have to incorporate a more advanced concept of space|time magnitudes than that discussed so far, and we will have to do so both in terms of the reciprocal system of mathematics (RSM) equations, and the graphics.  Hence, it will be helpful to understand both the physical and the schematic representation of the S|T units.

We’ll start to take a look at this in the next post. 

 

 

   

It is really Heisenberg’s concept of isospin, and Gell-Mann’s concept of hypercharge, that form the basis for the quantum mechanical observables that are the foundations for the quarks and gluons in the standard model (SM), which Gell-Mann was first able to formulate, via the properties of the SU(3) Lie group. These two concepts led to Gell-Mann’s “eightfold way,” which provided the initial breakthrough for classifying the myriad particles observed in high-energy particle interactions. However, the concept of isospin is simply an abstract mathematical space, an “internal symmetry,” with no connection to real physical space, and the fact that it is conserved, and that its use has been so crucial to the success of the SM, greatly astounds many physicists, let alone philosophers, to this day.

Actually, there are two concepts of isospin used in the SM. One is based on SU(2) symmetry, the concept of weak isospin, and one is based on SU(3) symmetry, the concept of strong isospin, if you will, which has evolved into the internal symmetry space of “color charge,” where the “color” of quarks is preserved through charge interactions. In the previous post below, I introduced the S|T triplet versions of the first generation red, green, and blue quarks, which combine to form the first generation baryons of the SM, the proton and neutron.

In Heisenberg’s concept of isospin, the neutron and proton were seen as nucleons, with the value of nuclear isospin, determining the degree of “protoness,” or “neutroness,” that the nucleon, as a whole, possessed. However, this original concept of isospin eventually had to be extended, until the current concept of three color charges, working via QCD, and its eight differently colored bosons, called gluons, managed to explain how quarks are confined to nucleons, through the principle of asymptotic freedom, and, in the process, demonstrated the power and beauty of gauge theory.

At some point in time, in The New Math blog, I hope to be able to explore more deeply the connection between the mathematical principles of the Reciprocal System of Mathematics (RSM), and those of Lie groups, used in gauge theory, that are involved in the SM.  The connection is extremely interesting and informing, because, in the RSM, we don’t have recourse to the ad hoc inventions of the real and complex numbers, which are the foundation of quantum mechanics and gauge theory, but, of course, we still have to be consistent with observation and mathematics. 

Nevertheless, right now I would like to continue with the discussion of the S|T triplets, which we have been discussing in this blog, in order to show how the conceptual logic of our RST theory might be extended to the combinations of protons and neutrons, as nucleons, in terms of the S|T triplets that represent the quarks and leptons of the SM. 

Again, however, I would like to stress the tentative nature of these developments.  Naturally, everything is subject to change and revision, as we learn more about the RST, but the fact that we can still proceed in a logical manner, to develop the consequences of our fundamental assumptions from the beginning, and that we are able to “discover” theoretical entities, with the observed properties of the SM entities in doing so, is compelling to say the least.

When we look at the double tetrahedron form of the proton and neutron combinations of S|T triplets, discussed in the previous posts of this blog, we see that we can identify three, external, “faces.”  Accordingly, we continue to refer to the topological features of these combos, but all the while insisting that such features do not exist in extension space, as it were, but that they only represent the relative space|time magnitudes of the constituent S|T units, just as the “distance” between different numbers is a delta magnitude, not an actual n-dimensional line, area, or volume of space.

In figure 1 below, we denote each “face” of the a and b tetrahedra, the two halves that comprise the double tetrahedron, as faces 1, 2, and 3, as shown, which we can then treat separately, for purposes of clarification.

 

Figure 1. Three “Faces” of S|T Tetrahedra

Given the identification of the three faces in figure 1 above, and the separation of these three faces of both the a and b parts of the S|T double tetrahedron, we can then separate and denote the six S|T triplets that emerge from the combination, as shown in figure 2 below.

 

Figure 2. The Six Faces of the S|T Double Tetrahedron as Six S|T Triplets 

The three circles, representing the middle, or inner, term of each triplet’s three constituent S|T units, representing the three sides of the triplets shown in figure 2 above, are omitted for the sake of clarity. However, it’s clear that an additional S|T triplet can be added to the center of each of these six S|T triplets, forming three more “faces” in each case, and that this addition of triplets can continue, ad infinitum.  Indeed, we see from this that the “geometry” of these combos is clearly fractal, where, again, the “distance” is purely numerical, not spatial. Therefore, the magnitude of the triple “surface,” in each iteration, is not reduced, as must be the case with the area of the geometric representation of these combos.

The important point to notice at the moment, however, is that each S|T triplet, constituting one of the three quarks in the proton or neutron, creates the necessary “space” to add three more quarks to the S|T combination, each of which, in turn, makes “space” for three more, and so on.  Of course, what this implies is that these additional “spaces” make it possible to combine protons and neutrons.  However, the problem with this is that the geometric truth that makes this possible (the fractal geometry) requires that we treat the S|T magnitude “spaces” as actual spaces, when, in fact, these faces are only spatial representations of combinations of space|time ratios, or magnitudes of speed.

This has to give us pause at this point, because, if we proceed, we will find ourselves in the same kind of predicament as the LST physicists who are not able to provide a physical interpretation for the mathematical concept of isospin, or color charge “space.”  In our case, we start with the unit space|time progression, and then we logically proceed to the SUDR and TUDR space|time displacements of this progression, which we find corresponds to the positive and negative, operationally interpreted, rational numbers of the RSM.  Moreover, once we have these discrete, reciprocal, units of M₄ motion, we see that the nature of the space|time progression produces the possibility that they will combine, forming the S|T combo that then progresses in both space and time at unit speed, but as a union of two, reciprocal, physical entities, which looks an awful lot like a photon, when we plot the combo on the space|time world line chart (The world line chart was discussed in previous posts below, but far enough below now that it’s obvious that I’m going to have to start archiving these posts so that I can provide direct links to them, which is something that I promise to do soon.)

Obviously, the S|T units, since they progress in both space and time, come into contact with other SUDRs and TUDRs, but what about coming into contact with other S|T units?  How can this happen?  See the problem is that, if a discrete location of space and a discrete location of time are plotted on the world line chart, as a combined space|time location, the location’s position on the unit progression line is unique; that is, different locations on the line cannot “move” relative to one another: They are eternally separated, as shown on the world line chart in figure 3 below.  

 

Figure 3. Two S|T Unit Locations on World Line

The S|T units plotted on the chart in figure 3 above, where p is the xyz space location, and t is the xyz time location, are, by definition, propagating as separate locations in both space and time. Combining them is tantamount to eliminating them as separate units.  Therefore, the question is, how can we combine three S|T units, as three constituents of a triangular triplet, since the three vertices of the triplet must be separated somehow in space|time?  Surprisingly, the answer not only assures us that it is possible to do this, without violating the world line chart, but, at the same time, it explains why two, or three S|T units can combine, but not more than three, and, consequently, why there are two types of hadrons, the baryons and mesons of the SM, and also why bosons can easily combine, but fermions cannot.

To understand how this works, it is necessary to recognize that only one of the three space coordinates, or only one of the three time coordinates, in a given S|T unit, need differ, in order to separate them.  Therefore, to plot these separate locations, it’s only necessary to convert the two-dimensional world line chart into a three-dimensional chart.  This permits us to plot the space|time progressions of three S|T units in parallel, as shown in figure 4 below.

 

Figure 4. World Line Plot of Parallel S|T Units 

In the 3D plot (here, we’ve omitted the green unit progression line for clarity), the z component of the p coordinate of a given S|T unit may differ from the z component of an adjacent unit.  Similarly, the z components of the t coordinates of two units may differ. The difference constitutes a relative offset of two locations, in one direction only, just as the locations of the center point of two adjacent spheres can only be in one direction.  Of course, for every possible offset position, there is a corresponding offset position diametrically opposite it.  Consequently, there only exists two potential z “slots” on either side of any given location, where the x and y coordinates are identical.  In other words, if we fix any two of the three components of a coordinate location, there can be only two, reciprocal, directions that the non-fixed component can take, in an adjacent location. This is the geometric version of the “odd man out” principle.

However, just as many spheres may surround a given sphere, there are many possible locations where one of the three components of the p and t coordinates of the associated locations may vary, but, in each case, there is only one set of these locations, and there is always a corresponding set that contains locations that are diametrically opposite to any location in the reciprocal set.  Moreover, in the case of S|T magnitudes, instead of lengths of distance measure, these two, reciprocal, sets correspond precisely to the reciprocal red and blue S|T magnitudes.

Therefore, in the universe of motion, it turns out that there is a physical reason why there can only be a maximum of three colors, and it is a mathematical one: any S|T magnitude, except the minimum, which happens to be a 1|2 or 2|1 (-1 or +1), is a sum of smaller values, in two, reciprocal, “directions.”  Again, it’s the ancient idea of balance.  There are only three discrete possibilities:

1. Balanced postitive and negative magnitudes
2. Negatively unbalanced magnitudes
3. Positively unbalanced magnitudes

Here, then, is the explanation for why there are boson and fermion S|T units: bosons consist of combinations of S|T units with all three constituent S|T unit magnitudes on one, or the other side, of unity, or they consist of equal magnitudes (balanced magnitudes) of both sides, but, in addition to this, the sum of the SUDRs and TUDRs that constitute the S|T units all occupy the same space, or time, location; that is, just as the pans on either side of a balance may be balanced or unbalanced in two “directions” with any conceivable number of units on either side, that may be arbitrarily subdivided into pairs of balanced and unbalanced units, many bosons can be combined into one boson for the same reason. They are the analog of a combination of n one-dimensional lines that are easily bundled together,

However, fermions consist of combinations of S|T units where one of the three components of the constituent S|T units is allowed to vary in one of two “directions.”  Thus, the locations of the constituent S|T units are not coincident, but offset in a given direction, as explained above, hence the fermion S|T triplets are not easily combined, unless they are oriented in a special way, just as planes can only be easily stacked, if they are oriented so that they are parallel.  In the case of spatial lengths, we don’t make this distinction, because one-dimensional lines also have to be parallel in order to be bundled, but, in the case of S|T magnitudes, one-dimensional combinations of magnitudes have no freedom of orientation; that is, S|T units are spherical, and thus invariant under rotation. 

The two-dimensional combinations of fermions (two-dimensional as in the two dimensions of a triangle), however, are orientable relative to one another! that is, the S|T unit positions 1, 2, and 3, in the triplet, may coincide, or not coincide. If they coincide, the orientation is equivalent to parallel planes, but if they don’t coincide, the orientation is equivalent to non-parallel planes. In Bilson-Thompson’s braid model, there is no provision for the non-coincident case, but, we recognize that it exists in the S|T model, though we won’t pursue the consequences at this time.  Only suffice it to say now that since the constituent quarks, in a proton, or neutron, or combination of them, are set, certain constraints should follow from this fact.

Certainly, we must conclude, in view of all of these developments, that there is yet much more to say. Yet, clearly, it is easy to see, from what has emerged so far, how the physical principles of symmetry, and thus the mathematical principles of Lie groups, can be used to describe the laws of conservation of “charge,” or what we would term “displacement,” that are inherent in all of this, which then obviously makes for a consistent system of physics, albeit one much more complex than what we are dealing with here.

The direct and inescapable implication is that there exists a much simpler system of constructing physical theory, a new system of physical theory, if you will.