The Larson Research Center

Just the thought of developing a 3D scalar algebra (SA) is enough to make me want to run home, climb in bed, and assume a fetal position. I am so intimidated by this prospect, yet I have to face the challenge no matter how daunting, if we are to have any hope of making significant progress toward our goal of calculating the atomic spectra, and the properties of matter, using RST-based theory.

The truth is that scalar motion is not vector motion, and the difference makes a 3D scalar algebra (and a scalar calculus) necessary. However, just as the scalar magnitudes are not unrelated to vector magnitudes, scalar algebra (and calculus) are not unrelated to vector algebra (and calculus.)

Our approach is based on using geometric algebra (GA) as a guide in the development of SA, because we have found the relation between the reciprocal numbers of the RSM, and the binomial expansion - based Clifford algebras that the 3D GA is built on, through the tetraktys. Nevertheless, the difference between the 1D vector motion in 3D space that GA describes, and the  multidimensional scalar motion in 3D space|time that the SA must describe, makes it imperative that the development of SA is based on logical deduction from first principles; that is, we must ensure that its principles follow as necessary consequences from the fundamental postulate that we have assumed: that all mathematics stems from order in two, reciprocal, progressions. This is an extension of Hamilton’s premise, as expanded by Larson.

As explained in previous posts below, the discovery that two groups are defined from one set of reciprocal numbers (RNs) by the operational interpretation of number, wherein the difference operation between the numerator and denominator of the RN forms a group under addition, while the division operation forms a group under multiplication, enables us to achieve a level of unprecedented mathematical integration, even though all the ramifications of that fact are not all understood yet.

What we are trying to do now is apply the RN to the tetraktys, so that we can use the analogy of vector geometry, that is the points, lines, areas, and volumes of Euclidean geometry, to shed light on scalar “geometry.” Clearly, the idea of scalar geometry seems absurd, until we realize that the “direction” of scalar poles is analogous to the direction of vector distance, and the independence of scalar combinations is analogous to the orthogonality of vector dimensions.

However, once this much is understood, we can proceed to analyze the way GA applies to vector geometry and vector physics, and use that knowledge to illuminate our path to understanding how SA would apply to scalar “geometry” and scalar physics. In the previous posts below, we saw that (n/n)⁰ represents a given number of steps of reciprocal progression, and that, with one step (1/1)⁰, we have no degrees of freedom in our inherent duality, the two opposing scalar “directions.” This, then, is analogous to the point in vector space, our initial number in the tetraktys, representing a single system of numbers (in the legacy system of mathematics (LSM), this is the class of real (1D) numbers.) 

But with two steps of progression (2/2)⁰, we have one degree of freedom, giving us an expanded system of numbers, with two classes of numbers analogous to points and lines in vector space (in the LSM, this is the system of complexes (i.e. 2D numbers.)) With four steps of progression (4/4)⁰, we have two degrees of freedom, expanding the number system further to include a third class of numbers, analogous to areas in vector space (in the LSM, this is the system of quaternions (i.e. 4D numbers.)) With eight steps of progression (8/8)⁰, we have three degrees of freedom, giving us the fully expanded system of numbers, with a total of four classes of numbers, analogous to points, lines, areas, and volumes in vector space (in the LSM, this is the system of octonions (i.e. 8D numbers.))

Of course, the LSM number system uses the invented concept of the “imaginary” number to accommodate the inherent duality of directions in vector geometry, which GA reinterprets in a specific and well-defined way, eliminating in the process much of the complexity that this ad hoc invention introduces into mathematics. It turns out, however, that the approach used in GA to do this is also based on a non-intuitive, ad hoc, invention called the geometric product, and thus this advance actually complicates our efforts to apply the concepts of GA to the development of SA, in a straightforward manner.

Nevertheless, we’ve made some progress, by taking advantage of the fact that Larson’s cube encodes the geometry of the tetraktys, which enables us to distinguish the difference between the geometry of duality in GA, and the geometry of the inherent, 3D, duality that SA must incorporate. We can clearly see that the difference stems from the contrast of vector and scalar motion. Vector motion is 1D motion in 3D space, while scalar motion is 3D motion (with 1D and 2D components.)

With the knowledge of this difference, we were able to discovery the degeneracy of the three dimensions in vector space, wherein one dimension must always be redundant, and thus hidden, in the three, orthogonal, axes of Larson’s cube. This is apparent, because, as we double the inherent duality of scalar space, in the tetraktys, the (8/8)⁰ point of the cube consists of the intersection of four (2/2)¹ “lines.” These are the four diagonals of Larson’s cube indicating the eight directions of 3D vector space, as well as the eight “directions” of scalar space.

The degeneracy is not important to GA, since the 3D volume (pseudoscalar) space of the tetraktys is the container of 1D vector motion. Of course, this is not the case for scalar motion, where the pseudoscalar is the inverse of the scalar, and constitutes the highest form of the motion itself. For this reason, we have to choose a different basis set of unit “directions” for SA, which immediately confronts us with the challenge of redefining multiplication in SA, bringing us to the consideration of the almost imponderable geometric product of GA.

Our new basis set is

e₀, the (8/8)⁰ scalar at the intersection of the 2x2x2 stack of unit cubes:

(2/2)⁰ + (2/2)⁰ + (2/2)⁰ + (2/2)⁰ = 8/8⁰, and

e₁, e₂, e₃, e₄,

the positive unit “directions” formed by the four 2/1 halves of the eight diagonals, each with their inverses, the four negative unit “directions” formed by the four, reciprocal, halves of the eight diagonals (1/2).

This is a big change, because not only is the number of elements in the 1D basis set one more in the SA set, than in the GA set, but the inverses (negatives) of each of these is an independent negative unit, on the other side of the intersection (unity) of the four diagonals. In contrast, the three unit vectors used as the basis set in GA, if I understand correctly, have their inverses construed as the reverse direction of the same unit; that is, the three positive directions diverge from the point, while the three negative directions converge to the point.

Thus, in effect, the geometry of GA is derived from one of the eight unit cubes in the 2x2x2 stack, called the dextral basis set, or right-handed set, of unit vectors. This is huge in our effort to understand how to “map” the GA algebra to the SA algebra. We know that we have four basis scalars, as we might call them, but they are 1D scalars, analogous to the four diagonals in the four unit cubes on one side of the 2x2x2 stack of unit cubes, diagonally opposite the four diagonals in the four unit cubes on the other side of the stack. Thus, we have to have, in effect, a different basis set for each of the eight 3D cubes in the stack. It’s hardly worth calling them basis sets, because there is no basis common to them all.  

When we think about it this way, we see that the dextral basis set in GA is used to describe points, vectors, and bivectors, in one cube, or in one 3D volume, of the eight 3D volumes that are available in the 2x2x2 stack, the geometric version of the tetraktys. In the RST-based theory, we have no need for a mathematical language, such as GA, to describe points, or the locations of points, or the orientations, directions and magnitudes of lengths between points, or the orientations and magnitudes of planes between lengths, in the dextral cube, of the stack. Instead, we need a mathematical language to describe the distribution of scalar magnitudes within the 2x2x2 stack, as a whole, within a fixed scale, although there still is an infinite set of possible scalar combinations in the stack, just as there is an infinite set of possible vector combinations in the dextral cube, with a variable scale.

That is to say, the basis values in GA, inside the dextral cube, represent the scale of lengths, which are then taken as the value of the scalar, or multiplier, α, used to multiply, or divide, individual vectors and bivectors in the algebra, while the basis value in SA, that which sets the scale of the system, must be the scalar itself! Thus, in SA, if we start with the minimum point (1/1)⁰, the scale of the system is set to the scale of a single point; If we double the scalar to (2/2)⁰, the scale of the system is expanded up to the scale of lines; If we double the scalar again to (4/4)⁰, the scale of the system is expanded up to the scale of planes; Finally, if we double the scalar once more, to (8/8)⁰, the scale of the system is fully expanded to include points, lines, planes, and volumes.

Of course, we can double again, and again, ad infinitum, in principle, but, as we do, we increase the density of points, lines, planes, and volumes, so to speak. It’s why Bott’s periodicity theorem limits us to period eight. We complete a 3D system with every factor of eight scalar expansions, as the scalar continues to double. However, we are getting ahead of ourselves. We need to fully understand the first tetraktys at this point. Later on, we double it in the second tetraktys.

In the first T, we have eight 3D cubes that make up one 2x2x2 stack. We know that the scalar, at the intersection of the stack, has the value (8/8)⁰, made up of the four diagonals, the four (2/2)⁰ values it takes to create four independent 1D numbers, each containing 1/2 and 2/1 “directions,” or one degree of freedom. Now we can see that taking four of these eight cubes on the same side of the scalar (intersection at the center of the cube) defines a plane of cubes, with an inverse plane on the other side of the intersection point (scale-setting scalar), and that there are exactly three ways to select a plane of four cubes in the stack, corresponding to the three, orthogonal, faces of the planes in the stack.

Interestingly enough, this gets us back to the 2³ = 8 total dimensions of the four linear spaces, giving us the binomial expansion of the fourth line of the tetraktys, associated with Larson’s cube; that is, the total numbers of these spaces is 1421 = 8, which is isomorphic to the standard 1331 = 8 (I guess we lose our connection with string theory’s 10 dimensions, though. Oh well!).

Clearly however, if this is the way to go from 1D to 2D geometrically, then we need to find an algebraic operation that will raise and lower the dimensions of these numbers. Yet, this seems problematic from the outset, because the 2D plane is made up of four 3D cubes! To do this we have to consider the four 1D lines, as 3D points, or 3D “directions,” not 1D “directions.”

Of course, that was exactly Larson’s point: The four diagonals are the four dual, or eight, 3D, “directions,” of scalar magnitudes.  This may give us a clue as to the needed operations of SA: Instead of multiplying, we divide, and instead of thinking of building up from the 0D scalar (actually 1D vector) to the 3D pseudoscalar (volume), we think of decomposing, from the 3D pseudoscalar to the 0D scalar (actually 1D line). On this basis, the plane is a two-part division of the 2x2x2 stack of unit cubes, the line is a two-part division of the plane of cubes, and the point is a two-part division of the line. Notice that the “direction” of each division operation is orthogonal, or independent, of the others: The line division “cut” is orthogonal to the plane division “cut,” while the point division “cut” is orthogonal to both.

This bodes well for the prospect of defining a scalar algebra for the scalar tetraktys.