The Larson Research Center

Since we can see the analogs of scalar and vector motion, in the tetraktys, one would expect that it also includes analogs of scalar and vector force and acceleration as well. Still, how can force and acceleration, which by definition are vectors, ever be scalar? The short answer is that, just as the motion generating the multi-dimensional spaces in the tetraktys can be reinterpreted as scalar magnitudes, even though it has always been defined as vector magnitudes in LST science, so too force and acceleration can be reinterpreted as scalar magnitudes, even though we aren’t accustomed to doing so in LST science.

Since, in terms of their dimensions, we can think of velocity and energy as inverses of one another, we should be able to use this fact to find the scalar analogs of the familiar vector magnitudes of force and acceleration. For example, whereas the equation of velocity is normally interpreted as a change of distance over time, v = ds/dt, a vector, the equation works equally well when we are thinking of the vector as a change in the position of a point itself, or in the change in the position of the midpoint of a line (the motion of an arrow), or a change in the position of the midpoint of a plane (the motion of a frisbee), or of a sphere (the motion of a cannon ball).

Likewise, when we reinterpret the equation of velocity to mean the scalar expansion of a point (change of size), instead of the vector translation of a point, the equation works equally well whether we are thinking of the scalar as a change in the size of the point itself (expanding balloon), or a change in only one or two dimensions of the point (telescoping pointer, ripples on a water surface). In other words, we can write the velocity equation to express the linear expansion of a line, the bi-linear expansion of a plane, or the tri-linear expansion of sphere. We could write the three linear equations as follows

1. v = ds/dt, the uni-linear form of the equation of velocity
   
2. v = ds²/dt, the bi-linear form of the equation, and
   
3. v = ds³/dt, the tri-linear form of the equation.
   

These three equations express one, two, and three dimensional scalar motion, as one, two, and three-dimensional expansions over time. Clearly, the midpoint of an n-dimensional object can only be moved one-dimensionally, so these equations are unusual, albeit consistent, formulations of the velocity equation. If we make the scalar motion a vibration over one unit, in lieu of a continuous expansion, the equations don’t change, but all the spaces of the tetraktys are formed, as shown in figure 1, below:

 

 

 

 

Figure 1. The Four Spaces of the Tetraktys Formed by Multi-dimensional Scalar Motion

Actually, all the spaces of the tetraktys are included in the expanding/contracting sphere. In a very real sense, then, we can say that the expanding/contracting sphere describes all the spaces of Euclidean geometry, the spaces of the octonion:

1. The point,
   
2. The point and the line,
   
3. The point, two lines and a plane,
   
4. The point, three lines, three planes and a cube.

But this is old news. The new point is that, if this is true, it follows that we can write the analogs of the scalar velocity equations, the inverse scalar velocity equations, in like manner:

1. v_i = dt/ds, the uni-linear form of the equation of inverse velocity
   
2. v_i = dt²/ds, the bi-linear form of the equation, and
   
3. v_i = dt³/ds, the tri-linear form of the equation.

Of course, to do this, we have to reinterpret the scalar inverse velocity equation as a vector inverse velocity equation, the opposite of what we had to do in the case of scalar velocity. This is the inverse conceptual problem. Whereas, with vector velocity, there is no multi-dimensional possibility for midpoint change of location, with scalar inverse velocity, there is only the multi-dimensional possibility for midpoint change of size. So, instead of having to go from vector to scalar analogs, as we do in the case of velocity, we have to go from scalar to vector analogs, in the case of inverse velocity.

To achieve this, one approach we can consider is to regard scalar inverse velocity equations as we do vectorial velocity equations; that is, the equations of vectorial velocity describe the time rate of change of position, regardless of the number of dimensions of the object changing locations, but, as we have seen, it can also be interpreted as the description of the time rate of change of the size of linear, area, or volume space units, and that all three of these velocities are contained in the changing volume, as one, vibrating, octonion, as illustrated in figure 1.

It certainly follows, then, that the analogous equation of scalar inverse velocity, the temporal vibrational velocity of volume, v_i = dt³/ds, describes the time rate of change of 1D and 2D time units, as well (inverse velocity equations 1, 2, 3 above). But, then, what are multi-dimensional time units? Since time has no direction in space, we don’t know what this means from a vector standpoint, but from a scalar standpoint, we can easily see that a scalar time expansion would be scalar in the sense of expanding in all directions of time, generating the temporal pseudoscalar of the octonion, assuming there are temporal directions. In other words, the directions of time are time-like, not space-like, but they are three-dimensional nonetheless.

In fact, we use multidimensional units of time regularly. For example, the change in the time rate of change of velocity is what we call acceleration, and the time/space dimensions of the acceleration equation are a = ds/dt². So, in terms of space/time dimensions, the inverse velocity equation of “area,” or bi-linear, time units is the inverse of the velocity equation of acceleration; that is, a = ds/dt² is the inverse of v_i = dt²/ds. If we interpret this equation as the inverse velocity of bi-linear time units, then it follows that v_i = dt/ds, with time/space dimensions of energy, is the equation for inverse velocity of uni-linear time units.

Though it sounds strange to do so, given the foregoing, it follows that we can consistenly refer to the space rate of change of time units, as inverse velocity, and the change of the space rate of change of time units, as inverse acceleration. Thus, the time/space dimensions of force, f = t/s², are actually the dimensions of inverse acceleration. Moreover, since we can conceive of scalar velocities as the 1, 2, and 3 dimensional spatial expansions of the octonion, we can also conceive of a change in these multi-dimensional velocities as accelerations of multi-dimensional spatial expansions. Therefore, we can write the acceleration equations as

1. a = ds/dt², the uni-linear form of the equation of acceleration
   
2. a = ds²/dt², the bi-linear form of the equation, and
   
3. a = ds³/dt², the tri-linear form of the equation.

And, if this is so, then we can also write the corresponding equations of inverse acceleration as

1. a_i = dt/ds², the uni-linear form of the equation of inverse acceleration
   
2. a_i = dt²/ds², the bi-linear form of the equation, and
   
3. a_i = dt³/ds², the tri-linear form of the equation.

However, where time in the material sector has no direction, all of the time terms in the inverse acceleration equations reduce to t. Yet, since there is an actual dimensional difference in these three inverse accelerations, there must be a dimensional difference in their three force analogs, as well, though we don’t recognize it in the force equations per se. We say, instead, that:

1. The force of equation 1 is one-dimensional, or electrical (F = q₁*q₂/r²).
2. The force of equation 2 is two-dimensional, or magnetic (F = Q₁*Q₂/r²).
3. The force of equation 3 is three-dimensional, or gravitational (F = M₁*M₂/r²).

Recall that the definition of energy, in the LST, though regarded as a scalar magnitude without direction, has the dimensions of inverse velocity, and velocity does have direction, in the LST. How can the dimensions of a scalar have the same number of dimensions as a vector? The answer to this contradiction is handled in the LST by defining energy in terms of work, and work in terms of force, applied over distance. Thus, if a force produces a displacement, either a positive or negative displacement, the product of the two vectors is the scalar work, and, thus, the scalar energy is manifest (via the inner product of two vectors), and it has the dimensions of a scalar, in one sense, but the dimensions of a vector in another sense.

This is known as the work-energy theorem. The ability to do work requires energy, and it is defined by virtue of the vectorial motion of mass called kinetic energy. Whether that work is to accelerate or decelerate a particle’s motion, is unimportant. Hence, the magnitude of the vector of the motion has direction, but the magnitude of the energy difference does not, making energy a scalar, associated with motion, a vector.

However, the dimensions of force, are the dimensions of energy per unit space, t/s x 1/s = t/s², regardless of the LST definition of kinetic energy, in terms of vectorial motion. Therefore, if we define force in this manner, then it follows that force is a quantity of acceleration, expressed in terms of energy per unit area, regardless of the dimensions of the physical entity generating the force. Whether it be electrical fields, magnetic fields, gravitational fields, or inertial fields, generating the force, it is all the same. Force is always a measure of energy per unit area, with dimensions dt/ds².

For example, the energy per unit area of electrical energy (disregarding constants of proportionality) is given by

f = q₁q₂ x 1/r² = t/s * 1/s = t/s²,

because the dimensions of area, even though area is length squared, are scalar, just as the inner product of two vectors is scalar. Actually, though, both forms of the dimensions of area are correct, depending on the sense intended, and neither the scalar nor the vector aspects of area space are sufficient to describe the space value alone. This is what GA’s geometric product is all about. In this case, we are interested in the magnitude of the area, not in the lengths of its sides, so the proper number of mathematical dimensions is zero, the magnitude-only dimension of scalars.

However, we are again running into difficulties with the inconsistent use of dimensions, because a scalar times a scalar is a scalar, and yet force is a vector. What’s going on? The problem can be even more clearly discerned when we consider the dimensions of acceleration,

a = ds/dt².

Now we know that time is scalar, so why do we express it in terms of square units? Well, the calculus saves us in this case, with its concept of derivatives, but the mixed up concept of dimension is still confusing, and we can’t use derivatives in the case of space, because energy per unit of area (force) is a different concept than the time rate of change of velocity (acceleration), expressed in terms of delta space per second, per second - a change of a change.

When you think about it, what is needed is the concept of three scalar dimensions, where a number raised to a higher power is a different kind of number, such that its dimensions are scalar dimensions (i.e. 2⁰ = 1), but its magnitude is multidimensional (2¹ = linear magnitude, 2² = area magnitude, 2³ = volumetric magnitude.) In other words, scalar lines, scalar areas, and scalar volumes, are descriptions of magnitude-only scalar values. In this way, the equation of force could express discrete units of energy per unit area, and still be just as much a scalar concept as is the concept of bushels of corn per acre.

On this basis, the equation for a single charge would be

f = q x 1/s² = t/s²,

as well, because the force (quantity of acceleration), is a measure of energy, whether or not it is measured in terms of work (where a direction is specified). This is an analog to potential versus kinetic energy in LST science, but charge is not understood this way in LST science. Instead, it’s understood in terms of charge density, Q, which density can be expressed in units of Coulombs per unit of linear, area, or volume space. Coulombs, in turn are defined in terms of amperes of current per second, but amperes are defined in terms of force!

One way to clarify this confusing maze of definitions is to think of force in the same way we think of acceleration, since the two concepts are reciprocal; that is, whereas we think of acceleration is a change of speed per second, force can be thought of as a change of energy per space, regarding the unit of space in the equation as a scalar value.  So we have

a = dv/t,

for accleration, and we have

f = dE/s,

for force.This makes sense for a couple of reasons.  First, we can readily understand from this formulation that force and acceleration are conceptual reciprocals, we might say, and second,  we see that force and acceleration are forms of energy exchange.  To Increase, or decrease, the speed of a mass, relative to some reference point, at a given time rate of change, requires an exchange of energy at a certain space rate of change.  Of course, this is well known, but we never put it in these terms, because we really don’t know what a “space rate of change” is.  We know what a force is, but what is a “space rate of change?”

I’ll tell you what it is.  It is the inverse square law. More on this later.  I have to give a presentation on scalar science tomorrow and I need to work on it.

In the previous post, I discussed how, in the LST system of vectorial science,  energy is defined as a scalar quantity, even though it has the space/time dimensions of inverse velocity, which is considered a vector (magnitude with direction), in vectorial science.  We found that the key to understanding this enigma is the recognition that vectorial energy is defined as a vectorial displacement from unity.  If the direction of a force vector neither veers to the left or to the right of a center line (the inner product is unity), it creates no displacement to the left or right of center, but, if it bears to the right or left of center, like the pointer on an amp meter, such displacement is regarded as positive or negative work, which is a manifestation of positive or negative energy.

The analogy to the time or space displacement of the unit scalar progression, leading to the SUDR or TUDR, is inescapable, and this gives us a clear picture of the relation of velocity (ds/dt) and energy (dt/ds), as inverse ratios of space and time. This also gives us a clearer understanding of the vectorial system of science vis-a-vis the scalar system of science. Vectorial concepts are analogs of scalar concepts; that is, there is a scalar and vector way to understand space and time and velocity and energy, and the understanding of the one helps in our understanding of the other.

However, lacking the knowledge of scalar motion, vectorial science has to use space, time and mass to calibrate the units of the vectorial system, such as the centimeter-gram-second system (CGS), or the meter-kilogram-second system (MKS), or its modern version, the International System of Units (SI). Nevertheless, given the knowledge of the scalar system, it is clear that a calibration of space and time units suffices for defining all other physical units.  Indeed, even without a knowlege of Larson’s scalar system of motion, the SI system of units can be redefined in terms of space and time, as shown by Engineer Saviour Borg of Blaze Labs.  He has recognized that understanding the dimensions of physical quantities is paramount to unifying physical theory, and that, through the use of dimensional analysis in the study of the key relationships of physical quantities, we can gain great insight into the concepts that underly the fundamental principles of nature.  He writes:

One of the most powerful mathematical tools in science is dimensional analysis. Dimensional analysis is often applied in different scientific fields to simplify a problem by reducing the number of variables to the smallest number of “essential” parameters. Systems which share these parameters are called similar and do not have to be studied separately. Most often then not, two apparently different systems are shown to obey the same laws and one of them can be considered to be analogous to the other.

The dimension of a physical quantity is the type of unit needed to express it. For instance, the dimension of a speed is distance/time and the dimension of a force is mass×distance/time². Conventionally, we know that in mechanics, every physical quantity can be expressed in terms of MLT dimensions, namely mass, length and time or alternatively in terms of MLF dimensions, namely mass, length and force. Depending on the problem, it may be advantageous to choose one or the other set of fundamental units. Every unit is a product of (possibly fractional) powers of the fundamental units, and the units form a group under multiplication.

However, the number of fundamental units of the SI system has been expanded, from the three of the MKS system, to seven fundamental units in the SI.  Generally, the more the fundamental units required to define a system, the less fundamental the system is. Borg comments on this:

We know that measurements are the backbone of science. A lot of work has been done to get the present self-coherent SI system of physical parameters, so why not choose SI as the foundation of a unifying theory? Because if the present science is not leading to unification, it means that something in its foundations is really wrong, and where else to start searching if not in its measuring units. The present SI system of units have been laid out over the past couple of centuries while the same knowledge that generated them in the first place have changed (sic), making the SI system more or less a database of historical units. The major fault in the SI system can be easily seen in the relation diagram shown here, officially issued by BIPM (Bureau International des Poids et Mesures). We just added the 3 green arrows for the Kelvin unit. One would expect to see the seven base units totally isolated, with arrows pointing radially outwards towards derived units, instead, what we get is a totally different picture. Here we see that the seven SI base units are not even independent, but totally interdependent like a web, and so do not even strictly qualify as fundamental dimensions.

Borg permits use of his published materials, given due attribution and notification, so I have included the diagram he refers to in figure 1, below.

 

 

Figure 1. Borg’s Diagram of the “Fundamental Units” of SI 

Borg explains how most of these units are redundant, some even included for historical reasons, but he has drawn the green arrows from mass, space and time to temperature, because temperature, in units of Joules, has the dimensions of energy. He writes:

Temperature can be seen as an alternative scale for measuring that characteristic energy [Boltzmann’s constant times temperature, which relates the energy portion of temperature to its units]. The Joule is equivalent to Kg/m²/sec², so for the Kelvin unit we had to add the three green arrows pointing from Kg, metres and seconds which are the SI units defining energy.

 Of course, we’ve been discussing the dimensions of energy as t³/s³ * s²/t² = t/s, in Einstein’s equation, E = mc², but Borg points out that, in the SI diagram, of all of the SI units, only two don’t have incoming arrows, showing their dependence upon other units.  Hence, the only two, independent, units are the unit of mass and the unit of time.  He writes:

How many dimensions can the SI system be reduced to? Looking again at the SI relations diagram, let us see which units DO NOT depend on others, that is which are those having only outgoing arrows and no incoming arrows. We see that in the SI system, only the units Seconds and Kg are independent. So, this means that the SI system can be reduced to no more than two dimensions, without losing any of its physical significance of all the involved units.

 Yikes, does this mean that only mass and time are fundamental units?  Borg explains:

But we know that there are a lot of other combinations that can lead to the same number of fundamental dimensions, and that Kg and Seconds might not be the most physically meaningful independent dimensions. Strictly speaking only Space and Time are fundamental dimensions …. so what are the rest? Just patches in physics covering our ignorance, our inability to accept that point particles, with the fictitious Kg dimension, do not exist.

Curiously, however, even though Borg recognizes that “only space and time are fundamental dimensions,” he doesn’t refer to the most important relation between space and time, indeed the only known relation of space and time, motion.  Nevertheless, he proceeds to redefine the SI units, in terms of space and time only, as shown in figure 2 below.

 

 

 

 

 

 

 

 

 

 

Figure 2. Borg’s Diagram of the Redefined “Fundamental Units” of SI

Notice that the arrow from time to space in figure 1 has been removed in figure 2, apparently indicating that the space and time units are regarded as independent units, not as the two reciprocal aspects of one component. However, recognizing, as Larson did, that space is just the reciprocal of time, in the equation of motion, the diagram of the SI units in figure 2, above, can be redrawn, as in figure 3, below.

 

 
Figure 3. Fundamental Units of SI Based on Scalar Motion 

Nevertheless, even though Borg doesn’t take the final step that Larson took, in concluding that scalar motion is the fundamental physical unit,  he is still able to show how the space/time dimensions of the fundamental unit of motion can be used to derive all other physical units, as shown in table 1 below.

 

Parameter	Units	SI units	ST Dimensions
Distance S	metres	m	S
Area A	metres square	m2	S2
Volume V	metres cubed	m3	S3
Time t	seconds	s	T
Speed/ Velocity u	metres/sec	m/s	ST-1
Acceleration a	metres/sec2	m/s2	ST-2
Force/ Drag F	Newtons	Kgm/s2	TS-2
Surface Tension g	Newton per meter	Kg/s2	TS-3
Energy/ Work E	Joules	Kgm2/s2	TS-1
Power P	Watts or J/sec	m2 Kg/s3	S-1
Density r	kg/m3	kg/m3	T3 S-6
Mass m	Kilogram	Kg	T3 S-3
Momentum p	Kg metres/sec	Kgm/s	T2 S-2
Impulse J	Newton Seconds	Kg m/s	T2 S-2
Moment m	Newton metres	m2 Kg/sec2	T S-1
Torque t	Foot Pounds or Nm	m2 Kg/sec2	T S-1
Angular Momentum L	Kg m2/s	Kg m2/s	T2 S-1
Inertia I	Kilogram m2	Kgm2	T3 S-1
Angular velocity/frequency w	Radians/sec	rad/sec	T-1
Pressure/Stress P	Pascal or N/m2	Kg/m/sec2	T S-4
Specific heat Capacity c	J/kG/K	m2/sec2/K	S3 T-3
Specific Entropy	J/kG/K	m2/sec2/K	S3 T-3
Resistance R	Ohms	m2Kg/sec3/Amp2	T2 S-3
Impedance Z	Ohms	m2Kg/sec3/Amp2	T2 S-3
Conductance S	Siemens or Amp/Volts	sec3 Amp2/Kg/m2	S3 T-2
Capacitance C	Farads	sec4Amp2/Kg/m2	S3 T-1
Inductance L	Henry	m2 Kg/sec2/Amp2	T3 S-3
Current I	Amps	Amp	S T-1
Electric charge/flux q	Coulomb	Amp sec	S
Magnetic charge/flux f	Weber or Volts Sec	m2 Kg/sec2/Amp	T2 S-2
Magnetic flux density B	Tesla /gauss/ Wb/m2	Kg/sec2/Amp	T2 S-4
Magnetic reluctance R	R	Amp2 sec2/Kg/m2	S3 T-3
Electric flux density	Jm2	Kg m4/sec2	ST
Electric field strength E	N/C or V/m	m Kg/sec3/Amp	T S-3
Magnetic field strength H	Oersted or Amp-turn/m	Amp/m	T-1
Poynting vector S	Joule/s/m2	Kg/sec3	S-3
Frequency f	Hertz	sec-1	T-1
Wavelength l	metres	m	S
Voltage EMF V	Volts	m2 Kg/sec3/Amp	T S-2
Magnetic/Vector potential MMF	MMF	Kg/sec/Amp	T2 S-3
Permittivity e	Farad per metre	sec4 Amp2 /Kg/m3	S2 T-1
Permeability m	Henry per metre	Kg m/sec2/Amp2	T3 S-4
Resistivity r	Ohm metres	m3Kg/sec3/Amp2	T2 S-2
Temperature T	° Kelvin	K	T S-1
Enthalpy H	Joules	Kgm2/s2	T S-1
Conductivity s	Siemens per metre	Sec3Amp2 /Kg/m3	S2 T-2
Thermal Conductivity	W/m/° K	Kg m /sec3/K	S-1T-1
Energy density	J/m3	Kg/m/sec2	T S-4
Ion mobility m	Metre2/ Volts seconds	Amp sec2/Kg	S4 T-2
Radioactive dose Sv	Sievert or J/Kg	m2/s2	S2 T -2
Dynamic Viscosity	Pa sec or Poise	Kg/m/s	T2 S-4
Fluidity	1/Pascal second	m sec/Kg	S4 T-2
Effective radiated power ERP	Watts/m2	Kg/m/sec3	S-3
Luminance	Nit	Candela/m2	S-3
Radiant Flux	Watts	Kg.m/sec3	S-1
Luminous Intensity	Candela	Candela	S-1
Gravitational Constant G	Nm2/Kg2	m3/Kg/s2	S6 T-5
Planck Constant h	Joules second	Kg m2/sec	T2 S-1
Coefficient of viscosity h	n	Kg/m/s	T2 S-4
Young’s Modulus of elasticity E	N/m2	Kg/m/s2	T S-4
Electron Volt eV	1eV	Kg m2/sec2	T S-1
Hubble constant Ho	H	Km/sec/Parsec	T-1
Stefan’s Constant s	W/m2/K4	Kg/s3/m/K4	S T-4
Strain e	-	-	S0 T0
Refractive index h	-	-	S0 T0
Angular position rad	Radians	m/m	S0 T0
Boltzmann constant k	Erg or Joule/Kelvin	Kg.m2/s2/K	S0 T0
Molar gas constant R	J/mol/Kelvin	Kg.m2/s2/K	S0 T0
Mole n	Mol	Kg/Kg	S0 T0
Fine Structure constant a	-	-	S0 T0
Entropy S	Joule/Kelvin	Kg.m2/s2/K	S0 T0
Reynolds Number Re	-	-	S0 T0
Newton Power Number Np	-	-	S0 T0

 

Table 1. Borg’s Derived Physical Units

There are some of these we would take issue with, but, in general, it’s a great, independent, demonstration of progress in the use of the new concepts of space and time.

We can gain a useful understanding of the conflict in the view of the dimensions of scalars, discussed previously in terms of the definitions of mass and energy, and see how the existence of non-zero scalar dimensions actually clarifies how a physical scalar value such as energy can have non-zero mathematical dimensions, by studying the dimensional properties of the Greek tetraktys and comparing/contrasting the meaning of these dimensions in terms of vectors, Clifford algebas, and proportions, using the operational interpretation of number.

In the vector view of the tetraktys, the 2⁰ points are scalar multipliers of 2¹ vectors, and a vector times a vector is another vector, a resultant vector. So we have a resultant vector as the diagonal between two orthogonal vectors, or two non-parallel vectors, times the magnitude of a scalar, or this product times another vector times a scalar, etc. All the possible combinations and the mathematics for these vectors, in the tetraktys, are described by the vector algebra, using the numbers in its hypercomplex number system, the set of reals, complexes, quaternions and octonions.

In the Clifford algebra view of the tetraktys, used to formulate Geometric Algebra, the 2⁰ points are again scalars, but vectors are directed, one-dimensional, lines, multiplied by the scalars, while the product of vectors is not another 1D vector, but a directed 2D bivector, or 3D trivector, again, multiplied by the scalars. All the possible combinations and the mathematics for these multivectors in the tetraktys are described in the Geometric Algebra, using the multi-dimensional number system, the set of zero, one, two, and three-grade blades.

By contrast, in the proportional view of the tetraktys, the “points” are also scalars, but, unlike in the previous views of 1D vectors and nD multivectors, the scalar in the proportional view is the source of the higher dimensional numbers, in the sense that all the higher-dimensional numbers in the tetraktys are expanded scalars, rather than rotated vectors, or multivectors. There are no vectors in this view, no vectors, no bivectors, and no trivectors, only “n-dimensional” scalars.

To illustrate how this works, we can use scalar values, such as colors, and walk them through the tetraktys. Each scalar value represents a relative proportion, which is either equal to, greater than, or less than, the reference proportion. We begin with the first element, at the top of the tetraktys (1/1=1 of line 0), the 2⁰ = 1 scalar, or the void. We assign the color black to it, a scalar value corresponding to a black “point,” if you will.

At the next higher dimension (11 of line 1), 2¹ = 2 scalar, we can expand the “point” scalar value in two “directions” to form a 1D value corresponding to a geometric “line,” with three scalar values, representing the expansion of the black scalar, expanded to a scalar value, or “point,” on either side of black, to a red value, or “point,” on the left, and to a blue value, or “point,” on the right. The blue “point” is a scalar value of greater proportion than the black “point” (2/1 > 1/1), while the red “point” is a scalar value of less proportion than the black “point” (1/2 < 1/1). We will give the set of these three values, corresponding to a 1D geometric “line,” defined between these three scalar values, or “points,” the color green, representing the one-dimensional equilibrium established by its three scalar numbers.

At the next higher dimension, 2² = 4 scalar (121 of line 2), again we have the zero-dimensional, black, “point,” but now we can expand it into two 1D scalar values, or “lines,” the new one of which we will color red. However, the difference in the color of the 1D values, represents a scalar difference in the symmetry of the two lines; one is symmetrical and one is not, the difference in symmetry defining two scalar “dimensions.”

There is a scalar difference of dimension between the red value and the green value, and this difference is manifest as the difference in the symmetry of the two 1D values; that is, the green 1D value is symmetrical, or balanced ((an/am)+(an/bn)+(bm/bn)), where a = b, but the red 1D value is unbalanced (a > b). Its symmetry is broken, we might say, in the red “direction,” representing the new, or second, dimension at this level. The product of these two 1D values, the green 1D “line” * red 1D “line”, is a yellow, “two-dimensional,” scalar value, corresponding to a geometrical “area.”

At the next higher dimension of the tetraktys, the 2³ scalar on the fourth line (1331 of line 3), we again have the 0D, black, “point,” but now we can expand it three ways, corresponding to the three vectors of the Clifford algebra tetraktys. One of these is the balanced scalar value, or the symmetrical 1D expansion (a = b), and the other two are the unbalanced scalar values, or two, 1D, non-symmetrical, expansions of the 0D black “point,” where a > b and b > a.

The first, balanced, 1D value is again colored green, while the second 1D value, unbalanced in the red “direction,” is again colored red. The third 1D value, unbalanced in the blue “direction,” is colored “blue.” Now, we can combine each of the three, 1D, scalar values, with each of the others, so we have three combinations of two, 1D, scalar values, forming a 2D scalar value, and these three combinations correspond to the three, 2D, bivectors of the Clifford algebra tetraktys:

1. green 1D “line” ^ red 1D “line” = yellow 2D “area”
2. green 1D “line” ^ blue 1D “line” = cyan 2D “area”
3. red 1D “line” ^ blue 1D “line” = magenta 2D “area”

Notice that, because these are scalar values, they are commutative; that is, the order of combining them makes no difference in the result. Now, at this, the bottom level of the tetraktys, there are also three more combinations, where we combine one of the three 1D values, with one of the three 2D values. However, there is only one result, regardless of the combinations, and it corresponds to the Clifford algebra, 3D, trivector, a “volume:”

1. blue 1D “line” ^ yellow 2D “area” = white 3D “volume”
2. red 1D “line” ^ cyan 2D “area” = white 3D “volume”
3. green 1D “line” ^ magenta 2D “area” = white 3D “volume”

Figure 1 below illustrates the three scalar dimensions of the scalar tetraktys.

 

 

 

 

 

Figure 1. Scalar Tetraktys

Again, since these values are scalar values, their algebra is associative; that is, it doesn’t matter how the three, 1D, scalar values and the three, 2D, scalar values are grouped to form the one, 3D, scalar value, the result is always a white, 3D, volume.

Of course, the point is that the scalar combinations of the scalar tetraktys correspond to the combinations of the scalar values of the red SUDR, and the blue TUDR in the development of the physical theory that we are working on at the LRC. The SUDR and TUDR, are initially joined together to form the green SUDR|TUDR combo. This combo (green 1D value) represents the one-dimensional, balanced, RN, the symmetry of which can be “broken” in two “directions,” by the addition of red SUDRs, and/or blue TUDRS, to the green symmetrical combo. Thus, we see that the units of scalar motion have three “dimensions,” and though these scalar “dimensions” are not the vectorial dimensions of Euclidean geometry, they are nevertheless consistent with three-dimensional mathematics. Not an insignificant result.

Once we understand this, we can see that the 1s running down the right side of the tetraktys in figure 1, have n “dimensions” (multicolors), while the 1s running down the left side of the tetraktys have 0 “dimensions” (black color), but they are all scalar values nonetheless.

Therefore, we see that the zero-dimensional units of mass, which we measure in kilograms, can also consistently be expressed as the three-dimensional units of scalar motion. Hence, all the physical dimensions reduce to consistent multi-dimensional units of space/time in two, reciprocal, scalar groups, when we provide the correct dimensions of the scalar values involved:

The energy group:

1. mass = t³/s³
2. momentum = t²/s²
3. energy = t¹/s¹

The velocity group:

1. inverse mass = s³/t³
2. inverse momentum = s²/t²
3. velocity = s¹/t¹,

where I’m explicitly indicating the one-dimensional values in the superscripts, for greater clarity. If 3D inverse mass is the mass of antimatter, then 2D inverse momentum is the momentum of antimatter, but it is also 1D velocity squared. So, multiplying 2D inverse momentum by 3D mass, yields 1D energy, as shown above, but, by the same token, multiplying 2D inverse mass (antimatter) by 2D momentum, yields 1D velocity,

v = s³/t³ * t²/s² = s¹/t¹

So, then, what is 2D momentum and 2D inverse momentum? We know 2D momentum is a product of 3D mass and 1D velocity, so 2D inverse momentum must be the product of 3D inverse mass and 1D inverse velocity, but 1D inverse velocity is energy, therefore, 2D inverse momentum is the product of 3D inverse mass and energy, or

(p) = (m)*E = s³/t³ * t/s = s²/t²,

where the parentheses indicate inverse. So, though we don’t know what it means at this point, at least we have a consistent and fundamental definition of velocity times velocity, or velocity squared.

More to come on that later, but in the meantime, since force (a quantity of acceleration) is required to produce a 1D velocity of mass (2D momentum), an inverse force (a quantity of inverse acceleration) should be required to produce a 1D inverse velocity of 3D inverse mass (2D inverse momentum).

So, the next question is, then, what are force and acceleration? In the LST vectorial system, force and acceleration must be defined without the knowlege of the space/time dimensions of mass, but in the RST this is not necessary.

In the new scalar system, force, is energy per unit space:

f = t/s * 1/s = t/s²,

while acceleration is velocity per unit time:

a = s/t * 1/t = s/t².

So, force extended over space is energy,

E = t/s² * s/1 = t¹/s¹,

while acceleration extended over time is velocity,

v = s/t² * t/1 = s¹/t¹.

Further, force extended over time is momentum.

p = t/s² * t/1 = t²/s²,

while acceleration extended over space is inverse momentum,

(p) = s/t² * s/1 = s²/t².

The only thing I want to emphasize at this point, is that these space/time dimensions are entirely consistent. Larson used this fact to great advantage, as you can see in Chapter 12 of “Nothing But Motion,” but he did so without knowledge of the scalar tetraktys. Now that we understand it better, we expect to be able to exploit these scalar equations of motion, both on the velocity side and on the inverse (energy side) of unity, to great advantage.

However, the answer to the question we considered last time, “why is energy considered a scalar, while its dimensions are not the zero dimensions of a scalar, but the one dimensions of a vector?” is now apparent. The LST community has defined scalar energy in terms of the scalar work it can do, but work is defined as force applied over distance, so when the distance is zero, work (energy) is also zero. In other words, energy is manifest as work, and the manifestation is a displacement, or potential displacement, caused by force. When we realize that, if, like the unit value of velocity, the unit value of energy is considered to be the datum of energy (the zero point), then displacement from unity is one dimensional magnitude, while non-displacement, if you will, is zero-dimensional magnitude.

Hence, while the dimensions of the RST definition of energy, dt/ds, are equivalent to a 0D scalar, when dt = ds, but are equal to a 3D scalar (pseudoscalar), when dt > ds, or ds > dt, the imbalance, if any, is caused by “directon” reversals in the time aspect creating units of time/space displacement. On the other hand, the dimensions of the LST definition of energy, seconds/meters, are equivalent to a 0D scalar, when force is applied to displace mass over a distance, but the vector of the displacement is orthogonal to the vector of the force, and no displacement takes place. Otherwise, the force displaces mass in one of two “directions,” which is a 1D scalar, or scalar with two “directions,” one positive and one negative. As explained in the Wikipedia article on work:

Mechanical work is a force applied through a distance, defined mathematically as the line integral of a scalar product of force and displacement vectors. Work is a scalar quantity which can be positive or negative.

As seen from the above definition, force can do positive, negative, or zero work. For instance, a centripetal force in uniform circular motion does zero work (because the scalar product of force and displacement vector is zero as they are orthogonal to each other). Another example is Lorentz magnetic force on moving electric charge which always does zero work because it is always orthogonal to the direction of motion of the charge.

Thus, while the dimensions of energy are the same, in both systems, their definitions differ, but the difference is found in the definition of motion, scalar motion in the RST, and vector motion in the LST!

Consequently, we conclude that, just as there is first scalar velocity, then vector velocity, so also there is first scalar energy (inverse velocity), then vector energy. You can’t have one without the other, in the RST; that is, in the universe of motion, without scalar velocity, matter wouldn’t exist, making vectorial motion of mass impossible, and without vectorial motion of mass, vectorial energy wouldn’t exist, making vectorial force impossible. In physics, therefore, it’s important to recognize the duality of scalar and vectorial velocity, and scalar and vectorial energy. Einstein’s equation,

E = mc²,

relates vectorial velocity to vectorial energy, through the one-dimensional difference between the vectorial dimensions of the vectorial motion of mass, along a 1D path (ds/dt), which is a 1D differential magnitude that must be squared in order to get the right result. However, it also relates scalar velocity to scalar energy, through the two-dimensional difference between the scalar dimensions of the one-dimensional scalar velocity and the one-dimensional scalar energy, which is a 2D differential; that is, to convert from the dimensions of s/t (velocity), to the dimensions t/s (energy), requires two multiplications:

1. The ratio ds/dt, where dt > ds, times the ratio dt/ds, where ds > dt, which makes ds = dt, and
2. The ratio of the result, where dt = ds, times the ratio dt/ds, where ds > dt, which makes ds > dt.

For example,

1. 1/2 * 2/1 = 2/2 = 1/1, and
   
2. 1/1 * 2/1 = 2/1

This is equivalent to two, 90 degree, rotations, from -1 (dt > ds) to i (dt = ds), then from i to 1 (dt < ds), or two positive additions, from (-1 + 1) = 0, to (0 + 1) = 1. The fact that mass has three inverse space/time dimensions, because it opposes velocity in any direction (inertia), and velocity and energy both have one space time dimension, which are inverses, makes it possible to use mass to convert between the two inverses, just as 1 = 3-2 and -1 = 2-3, makes it possible to convert from 1 to -1, and vice versa. The point is, while it’s about as simple a concept as there is, it’s not a simple task to explain it, but with patience, one can see why energy is related to mass through the square of unit speed, and why, even though energy is a scalar, it has two “directions” in one dimension, just as velocity does.

Hence, we conclude that velocity too, is a scalar, the inverse of energy.  The rising sun is dissipating the fog in our understanding of space, time, substance, mass, energy.

From another post on the BAUT forum series: 

 

Last night I watched a PBS special on E=mc², called “Einstein’s Big Idea: The Ancestors of E=mc².” I learned several things from the show.  For instance, I learned that the letter c, as a symbol for the speed of light, comes from the latin word for swiftness, “celeritas.”  I never knew that, I don’t think.

However, I learned something else as well, something much more significant.  I’ve written before that I can easily understand how energy and matter are related in the RST, since it’s so straightforward in the new system, but I really have trouble understanding how Einstein knew this.  Now, after seeing the show, the mystery has been cleared up for me.  Einstein accepted, as most others did by then, that energy could be defined by the mass of an object times its velocity squared, because Emilie du Châtelet, following Leibniz lead, had done it, and she demonstrated the veracity of the definition with the help of Willem ‘sGravesande’s clay penetration experiments.

So, when Einstein realized that the velocity of mass was limited to the value of c, he realized, at the same time, that its energy must also be limited by c.  In other words, what was being asserted was that the [i]maximum[/i] energy that a moving object could obtain, given the accepted definition of energy, was

E = mv²,

which, given the newly discovered speed limit of v at c, naturally leads to

E = mc².

However, it’s not clear to me that Einstein understood that this means that the mass of matter could be converted into energy by some process other than putting the energy into it, by accelerating it to c-speed (or as close as possible). In other words, it wasn’t necessarily understood right away that mass and energy are equated by the equation.  At least this seems to be the logical conclusion; especially, since the dimensions of mass, kg, are obviously not the same as the dimensions of energy, it would to be the most logical conslusion. Energy is measured in units of joules in the SI system of units (probably ergs in the cgs system of his day).  Therefore, the dimensions of the equation must have initially been interpreted as

ergs = mass x velocity x velocity,

but then what is velocity times velocity?  I don’t think Leibniz, or du Châtelet, or Einstein knew the answer to this question. I wish I could find out WHY Leibniz thought it was necessary to square the velocity in the energy equation, but all I can find so far is that he did think so, even though he couldn’t prove it.  Therefore, as far as I can tell, there is no theoretical explanation of the definition of kinetic energy as mv²/2.

On the other hand, using the RST dimensions of motion, energy clearly has space/time dimensions,

t¹/s¹,

and momentum has space/time dimensions,

t²/s²,

while velocity has space/time dimensions,

s¹/t¹.

However, the dimensions of the energy equation balance, only if mass has dimensions,

t³/s³,

as can be seen in Einstein’s equation of energy,

t¹/s¹  = (t³/s³) * (s²/t²).

But Einstein did not suspect that mass has these space/time dimensions. Indeed, LST scientists do not understand it this way, even today. Kilograms, the units of mass, like the units of temperature. or the units of apples, or bananas, are understood as scalar units, as units of an amount, or a quantity, of a substance. Energy is also a scalar quantity; that is, it has magnitude only, with no direction in space, but the mathematical dimensions of scalar units are zero, not three, as required by the dimensional analysis of Einstein’s energy equation show above.

Yet, as we have discussed previously, while the dimensions of a scalar quantity are zero, the dimensions of the pseudoscalar octonion, are three, so this admits the possibility that a substance can also have three dimensions and still be a “scalar.”  In this case, the units of the amount of substance would be units of volume.

However, as it turns out, even though energy is a scalar quantity, its definition in LST physics, where its divided into two categories, potential and kinetic, has the dimensions of work. In Wikipedia’s article on energy, we read:

The most common definition of energy in the context of mainstream science is “the ability to do work”. Thus in physics, energy is mathematically defined as a mechanical work done by a force (gravitational, electromagnetic, etc) and has many different forms that can be broken down into two main forms: kinetic energy and potential energy. According to this definition, energy has the same units as work; a force applied through a distance. The SI unit of energy, the joule, equals one newton applied through one meter, for example. (emphasis added)

Hence, in LST physics, the scalar value of energy is transformed, by virtue of its definition, into the scalar value of “work done by a force,” which is the scalar product of force and a so-called “displacement vector,” and, therefore, can have one of three values: positive, negative, or zero, depending upon the degree of orthogonality of the force vector.  In other words, energy is a scalar defined as work, but work is a scalar that is defined as something only manifest as a vector product, the inner vector product that can be either positive, negative, or zero. From the Wikipedia article on work, we read:

Mechanical work is a force applied through a distance, defined mathematically as the line integral of a scalar product of force and displacement vectors. Work is a scalar quantity which can be positive or negative.

As seen from the above definition, force can do positive, negative, or zero work. For instance, a centripetal force in uniform circular motion does zero work (because the scalar product of force and displacement vector is zero as they are orthogonal to each other). Another example is Lorentz magnetic force on moving electric charge which always does zero work because it is always orthogonal to the direction of motion of the charge.

Thus, energy, in LST physics, is defined as the projection of one vector upon another.  When the vectors are perpendicular, no shadow is cast by the vertical upon the horizontal, but, if the angle between them is greater than, or less than 90 degrees, the shadow (projection) takes on a definite positive or negative value - just like a scalar.  Maybe we could call this a pseudoscalar too, but not in the same sense.  It’s a physical pseudoscalar, we might say, not a mathematical one.

The interesting aspect of all this, in the present context, is the insight that it gives us into the relation of numbers and magnitudes.  LST physics deals only with vectorial motion magnitudes, as we have been explaining, so to define scalar motion magnitudes (in this case the motion of inverse scalar velocity, or energy), a way has been found to describe it in terms of vector magnitudes.  It would be considered ingenious were it the work of one individual, but really it’s a concept that has evolved unconciously, as it were, over centuries.  The driving evolutionary force, success in physics, like biological success in living and reproducing, is powerful, if not all that efficient.

Nevertheless, with this much understood, we are left to ponder the meaning of scalar.  If energy is a scalar quantity, and a scalar has no direction, then how can it have the mathematical dimensions of 1 in Einstein’s energy equation? As scalars in a three-dimensional physical system, one would think that mass and energy should have the mathematical dimensions of the octonion scalars (2⁰), or pseudoscalars (2³), but, while we can see that, in the RST at least, mass has the three dimensions of the fourth line in the tetraktys, the dimensions of the pseudoscalars, energy has the dimensions of the second line in the tetraktys, the 2¹, or one-dimensional, line, not the first line, the 2⁰ line.  What’s up with that?

I’ll try to answer that question in the next post.

In Larson’s development of the RST, what we call the RSt, his results are both qualitative and quantitative, but mostly qualitative, because, in part, while he had a new scalar system to work with, he didn’t have the scalar mathematics to go along with the scalar motion concepts that are necessary to develop the theory in a scientifically rigorous manner.

However, as the principles of scalar mathematics emerge, a true scalar science becomes possible in which scalar theory can be developed, exploiting the new principles of the system more systematically.

Larson’s qualitative theoretical results, including concepts of radiation, charge, magnetic moment, mass, energy, etc, enabled him to develop many quantitative results, including specifics of global interactions, such as the dynamics of inward, or gravitational motion of mass, combined with the outward, or expansion motion of galaxies, in a new cosmology.

This lead to extended concepts of the macrocosmic physical structure of the universe, such as the evolutionary sequences of star and galaxy formation, and the overall structure of the universe, as two interacting, inverse sectors, linking the isotropic emergence of high-energy cosmic rays, in the material sector, with the high-speed ejecta of material concentrated by time gravity, in the cosmic sector.

The incoming matter is then concentrated by space gravity, where it aggregates, concentrates, and eventually explodes and ejects matter at such high speeds that it re-enters the cosmic sector, as cosmic sector cosmic rays, and the same process begins anew, as gravity in that sector once again begins the process of aggregation and concentration of matter in that sector.

Consequently, we see a new cosmology, based on the reciprocity of space and time, forming the universe into an eternal cycle of radiation, matter, and energy, where the cosmic sector, based on 3D time and scalar space, “feeds” the material sector, based on 3D space and scalar time, and the material sector, in turn, “feeds” the cosmic sector, in one eternal round.

Not only is there no need for a “hot big bang” in this cosmology, but it comes with many features that match our observations, such as the flat geometry of the universe, the expansion of space and time, the properties of gravity, the ages of distant galaxies, the energy of quasars, etc.

Many people, when they read Larson’s works, are compelled by the development’s logical symmetry, and its consistency with observations, but they wonder about the quantitative side of the theory. Fortunately, as the vectorial motion of high-speed astronomical events is the primary motion under consideration, at this scale, the current LST methods of most calculations are not affected, except in the regime of greater than light speeds, which in the RST take the form of time motion.

It’s this concept of time motion, and the concept of the two interacting, inverse, sectors of the universe, which provide the basis for Larson’s new results, and they are primarily qualitative in nature, and he concentrates on their qualitative effects.

However, when it comes to the quantitative details of specific interactions, pertinent to the topic at hand, Larson uses the best data available at the time to deduce his conclusions. Of course, the scope of what one man can accomplish in such a vast field is very limited. In Volume III of his work, The Universe of Motion, Larson writes:

…the scope of the work, both in the number of subjects covered, and in the extent to which the examination of each subject has been carried, has been severely limited by the amount of time that could be allocated to the astronomical portion of a project equally concerned with many other fields of science. The omissions from the field of coverage, in addition to those having relevance only to individual objects, include (1) items that are not significantly affected by the new findings and are adequately covered in existing astronomical literature, and (2) subjects that the author simply has not thus far gotten around to considering. Attention is centered principally on the evolutionary patterns, and on those phenomena, such as the white dwarfs, quasars, and related objects, with which conventional theory is having serious difficulties.

Obviously, given a new system of physical theory, much of the early work is bound to be immature and preliminary. No one can expect anything else, yet the results Larson achieves, by applying the new system in many areas, are remarkable. Nevertheless, the value of these results lies primarially in serving as indications of the validity of the new system, and illuminating the general nature of the new processes involved.

To say that much remains to be done is an astronomically sized understatement. Certainly, however, enough has been accomplished to warrant further investigation of this scalar system, where Larson has been able to blaze the trail so to speak, and, by so doing, to deliver the outlines of exciting developments to come.

However, a significant part of that outline depends on the new definition of motion and the changes it introduces to the physical picture, but while Larson’s conclusions, in the field of physical cosmology, viz-a-viz the standard, or hot big bang cosmology, are fairly easy to compare and contrast with current theory, the same is not true at smaller scales.

Yet, much of what forms the basis of physical cosmology, the theories of the large-scale structure of the physical universe, depends upon the theories of the small-scale universe, and the comparison and contrasting of the two systems is not that easy at the microscale, because the terminology and language of the standard model of particle physics is much more daunting than that of cosmology.

Indeed, the world of particle physics has become so esoteric that it is almost impossible for anyone, at an undergraduate level, to master the key concepts, and the language used to describe them, well enough to get a clear picture of how it all works to even venture an opinion, regarding the issues arising from the processes involved.

This is because the experiments, the phenomenology, are described in terms of the theory, not in terms of observations. Therefore, instead of a universe of observed galaxies, clustered together, yet receeding away from one another and composed of various types of physical enties with a range of observable properties, describable in terms relatively easy to define, we have a universe of observed debris, created by manmade collisions, reacting to an artificial environment in which they move out at near light speed in the flash of nano-explosions, describable only in terms of esoteric equations.

These equations, describing the observed phenomena, are equations ultimately based on the principles of vectorial motion, and the language of mathematics that has been developed to express the laws of these principles. Clearly, what this does, in effect, is cloak the processes involved, because they can only be perceived through the equations that describe them.

While it’s true that the picture of reality that emerges from this study, called the standard model of particle physics, can be graphically depicted in a schematic diagram, it contains only a fraction of the information used to construct it. For outsiders, trying to understand the results of the experiments, it’s as if we had to study the stars and galaxies from pictures drawn in the sand.

The bottom line is that in order to develop the new scalar science of astrophysics, planetary physics, and cosmology, in many cases, we have to build a scalar science of particle physics, atomic physics, and nuclear physics first. To do that, we need to understand what the observers at the microscale are seeing so that we can compare our theoretical entities with the observed entities, but this task is complicated by the successes of the current theories, which tends to drive the descriptions of the observations in terms of the theories.

Thus, we have leptons and hadrons, which are all fermions, the basic building blocks of matter, but not bosons, which are the particles of radiation. Yet, these are all described in terms of fields, which for the physicist, are as real as the chair in which he sits. However, fields are not enough to totally describe the hadrons, because hadrons are not elementary as once thought, but are now considered to be composed of unobservable quarks.

There are three families of fermions, each of which is composed of different sets of two different leptons and two different quarks. The leptons and quarks of the first family, the members of which form the relatively stable matter of most of the elements in an earth environment, are the electron, and the electron neutrino, and the up and down quark, respectively.

It takes three quarks to form a hadron fermion in the first family. These hadron fermions each have three quarks, two ups and one down, or two downs and one up, the first forming the proton hadron fermion, and the second forming the neutron hadron fermion.

The atoms of successive elements of matter are composed of a successive number of hadron fermions and lepton fermions, in a fixed proportion: there is one lepton fermion for every hadron fermion that is a proton hadron fermion, and, generally, there is neutron hadron fermion for every proton hadron fermion in each atom, except the first, although this number may vary somewhat.

Now, what do we have in the RST? Larson’s RSt has subatoms of matter that consist of a system of motions: A linear vibration, which rotates in two of three dimensions, as one 2D rotation, and can also rotate in the third, orthogonal, dimension, as one 1D rotation, forms the basic system of rotations, in his system.

Atoms consist of two of these subatomic systems combined as one. Subatoms consist of one of these systems and are distinguished from one another by the number of discrete units of rotation in their system of rotations. The first system of rotations to emerge is identified with the electron, or the positron, depending upon the scalar “direction” of the rotation of its unit of motion. Successive subatomic entities emerge by adding units of rotation to the system of the previous entity. Hence, an electron becomes a proton, or a positron becomes an anti-proton, through the addition of the appropriate unit of rotation. These systems can take on different charges as well as different masses, depending upon the nature of the added unit of rotation.

So how does this RSt model compare with the standard model? Well, it predicts the periodic table of elements much better than does the standard model, and it explains the inter-atomic distances of elements better than does the standard model, but in other ways it can’t compare to the standard model.

The most important of these shortcomings is the inability to calculate the atomic spectra from it, which is the corner stone of the standard model. However, at the LRC, we think that we have identified a fundamental error in Larson’s development that will enable us to correct this deficiency, but at the cost of having to redo a lot of the theoretical development.

This would have never been possible to contemplate, if it weren’t for the discovery of the scalar mathematics that enables us to develop the theory mathematically. Though this discovery has only emerged in the last few years, it seems promising in many ways, but perhaps the most promising aspect of all is found in the shadows of the standard model that its light is casting.

We have discovered three scalar “dimensions” of motion that are analogous to the three vectorial dimensions of geometry, quite unexpectedly. These three dimensions come in several configurations similar to the configurations of quarks in the standard model (i.e. uud or ddu) that constitute units of motion that cannot be separated, or exist apart (one of the major mysteries of the standard model now considered solved by Gross and company’s concept of asymptotic freedom in QCD).

We have also discovered that the scale of these configurations forms three discrete families (another of the major mysteries of the standard model, still unsolved), and we have hints of how the electron and neutrino and photons of radiation fit into the picture as well, albeit perceived only dimly at this point.

With all this, many of Larson’s substantial contributions remain intact, especially the understanding of special relativity, the ubiquitous force of gravity, as the property of the intrinsic motion of mass, and the space/time dimensions of physical constants. In addition, we now have the new found understanding of the octonions and the Bott periodicity theorem, so we can see shadows of string theory’s Supersymmetry lurking about.

In short, there are lots of exciting qualitative results coming from the system, and we are working hard to extract the requisite quantitative results, which, if and when we do, we’ll post.