The Larson Research Center

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.