We can gain a useful understanding of the conflict in the view of the dimensions of scalars, discussed above 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. 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), 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), 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,” while the red “point” is a scalar value of less proportion than the black “point.” We will give the set of these three values, corresponding to a 1D geometric “line,” defined between the 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), 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, but the red 1D value is unbalanced. 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), 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, and the other two are the unbalanced scalar values, or two, 1D, non-symmetrical, expansions of the 0D black “point.”

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 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. 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 apart from mass and momentum, 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.

Stay tuned.

Excal