The Larson Research Center

In the previous entry below, we found that the four linear spaces of our new scalar algebra (SA) do not have the same number of dimensions as the four vector spaces of geometric algebra (GA), the 1+3+3+1 = 8 dimensions of dual directions, using the word dimension in the mathematical sense of independent variables. Instead, they contain 1+4+4+1 = 10 dimensions. The same number of space dimensions in string theory (m-theory).

Yet, that these 10 mathematical dimensions are contained in the 3D geometric space of the tetraktys is now clear, when we see them as the 2x2x2 stack of cubes, which Larson used to describe the 3D units of scalar motion. String theorists, unaware of the concept of scalar motion, have tried for many decades to employ these dimensions in terms of the usual concepts of vectorial motion, without success, leading to confusion of thought and perplexing complications in their topological approaches (see the Calabi-Yau manifold), with serious theoretical, philosophical, and even sociological ramifications (see our Trouble With Physics blog).

However, if we compare the four linear spaces of GA, with the analogous spaces of SA, in terms of the geometric properties of Larson’s cube, we get a view of the scalar spaces in terms of linear, circular, and spherical expansions and contractions, as discussed last time.  Nevertheless, it’s not clear yet what we gain by this transformation, and we are still investigating it. In the meantime, though, it’s not difficult to see that the four “lines” of the 1D scalar space can be used algebraically to generate four groups of 2D scalar space, and that these groups have exactly 12 2D entities, which correspond to the 12 2D panels in the three intersecting, orthogonal, planes of the stack.

That these 12 2D scalar values correspond to the 12 2D areas of the vector space is also very intuitive, but, the comparison of them has revealed something odd about GA, which is not very intuitive. To explain this, we need to analyze the geometry of the 2x2x2 stack of unit cubes. Using the four diagonal lines of the stack to generate the 12 2D “areas,” we can describe them as the 2D products of the four sets shown in the graphic below.

Figure 1. Four Sets of 2D Scalar Products 

As can be seen above, we first denote one end of each diagonal in the stack, as a, b, c, and d, starting with the upper left corner of one face of the 2x2x2 stack, as indicated, in the left-most face in the graphic. The inverse of each 1D diagonal (reflection symmetry in 1D space), is the opposite end of the diagonal, in the diagonally opposing, 3D, unit cube (recall there are eight of these).  Thus, the unit cube containing -a is diagonally opposite the unit cube containing a, while -b is diagonally opposite b, and so on.

Next, we designate each 2D area, the product of two 1D diagonals, with numbers, beginning with 1 and continuing clockwise. On this basis, the first face contains four 2D products, but the point where the diagonals cross is in the center of the cube, so the face is indented at the center. Hence, the face of the stack of cubes is actually the projection of the diagonal lines onto the stack face, while the lines themselves only contact the face at the corners a, b, c, and d, forming a four-sided pyramid in the interior of the stack, with the apex at the center of the stack, and each numbered facet constituting a congruent isosceles triangle.

Rotating the stack clockwise, as viewed from the top, to view the next face, shown on the right of the first in the graphic, illustrates the second set of pyramid forming triangles, with the four points at the base of the interior pryamid (projecting the next face of the stack) now rotated into the plane of the page, and its apex coinciding with the apex of the first pyramid, at the point of intersection, at the center of the stack.

Notice that there are only three unique, numbered, facets in this set, since one facet is common to the first set (facet number 2, of the first face, now rotated into the page). A second rotation brings up the third set, which, again, has a common (thus not counted) facet, with the preceding set.  Finally, the fourth set has only two unique triangles, in the pyramid, since it has common facets with both the beginning set and the preceding set of facets. 

It’s interesting to note that the degeneracy of the three axes, in the vector spaces (see discussion below), which we have removed by switching to the four diagonals of the 2x2x2 stack, seems to reappear in the redundant form of the four common facets. However, there seems to be something else, even more significant, revealed by the switch to these scalar spaces (recall that the directions inherent in the stack of 8 unit cubes are actually scalar “directions” in this view.) This revelation is the fact that we would expect that the algebraic properties of these scalar spaces are ordered, commutative, and associative, since they are scalar spaces with “direction,” not vector spaces with direction.

In GA, a 1331 basis is selected, where e₁, e₂, and e₃ are regarded as the three 1D unit vector directions. In this way, the e₁^e₂, e₁^e₃, and e₂^e₃ bivectors define three 2D unit outer products (we’ll ignore the inner and geometric products for now), and any of these wedged with the odd man out, (e₁^e₂)^e₃, (e₁^e₃)^e₂, (e₂^e₃)^e₁, define one of three, 3D, trivectors. In other words, the pseudoscalar can be defined in three, equivalent, outer products.

Notice, however, that the scalar is not derived, and neither are the three basis vectors, but, as we have seen, they represent the assumed 0D point space and the three, orthogonal, dimensions of geometry, from which all else is derived. The scalar is designated α, and it is used to multiply the unit vectors in the algebra. All these spaces taken together, form a multivector, which has proven very useful for doing physical calculations.

However, in the scalar spaces of the SA, everything is defined, beginning with the 0D scalar and then proceeding to the 3D pseudoscalar. In this algebra, the scalar is defined and the basis is actually derived, because it grows, as the scalar grows. Thus, at zero dimensions, (1/1)⁰ and (8/8)⁰ are both a point, and there is no distinction, but at three dimensions, (8/8)⁰ and (2/2)³ form an assembly of a point expanded in three “directions,” defining eight, one-unit, cubes, in the 2x2x2 stack. So, we can say that (8/8)⁰ really is the potential of (2/2)³; that is, one is transformable into the other, both numerically and physically.    

So, what can we take as a basis for these multi-dimensional scalar units in SA? The logical answer is the four, two-unit, scalar, “distances,” (a to -a), (b to -b), (c to -c), and (d to -d) above. This would give us the bases for the 10 dimensions of the 1441 scalar tetraktys:

e₀; e₁, e₂, e₃, e₄; e₁^e₂, e₂^e₃, e₃^e₄, e₄^e₁; (e₁^e₂)^e₃(^e₄); (e₂^e₃)^e₁(^e₄); (e₃^e₄)^e₁(^e₂); (e₄^e₁)^e₂(^e₃);

where the parentheses around the fourth dimension indicate that the basis of the 3D scalar can be formed in one of two, equivalent, ways, as a product of each 2D scalar, with either of the two remaining 1D diagonals, forming the 3D scalar (pseudoscalar). Now, the interesting thing about this is, following GA, each n-dimensional scalar should have its inverse. For instance, the 1D scalars|inverse scalars would be:

e₁|-e₁; e₂|-e₂; e₃|-e₃; e₄|-e₄; 

but where the negative sign does not indicates the opposite direction of a vector, but the opposite “direction” of a scalar; that is, positive and negative in a real sense. In other words, using reciprocal numbers, we get four pairs of (1/2)|(2/1), that are equivalent to the four diagonals.

e₁|-e₁ = (1/2):(2/1)|(2/1):(1/2),
e₂|-e₂ = (1/2):(2/1)|(2/1):(1/2),
e₃|-e₃ = (1/2):(2/1)|(2/1):(1/2),
e₄|-e₄ = (1/2):(2/1)|(2/1):(1/2),

where the colon symbol is used to indicate the “difference” between the positive and negative reciprocal numbers (RNs), and the pipe symbol is used to separate the “directions” of the positive and negative bases. Therefore, we see that the eight scalar “directions” are perfectly analogous to the eight vector directions, relative to the center intersection of the four diagonals, in the 2x2x2 stack.

However, this is where the odd aspect of GA begins to show up. No one would suggest that this set of four diagonal directions would be taken as a basis set in vector algebra, because they are not independent, or orthogonal.  Yet, as we see in figure 1 above, the projection of them onto the respective faces is orthogonal. In fact, the change in the location of the intersection, from the center of the face to the center of the stack, at the apexes of the four pyramids, constitutes a change in an independent direction and therefore does not affect the magnitude in the two, orthogonal, dimensions. Thus, there is no inherent reason why we couldn’t use the set of four diagonals as a basis set, but only a practical reason not to: It simply complicates matters from a vector point of view, and with no motivation for doing it, it has never been done, at least as far as I know.

Indeed, in GA, if I understand correctly, the basis set, the dextral, or right-handed, set, taken as the basis set, is not the set of vectors defined at the interior intersection of the stack, as I have always assumed, but it is the set of vectors defined from one of the exterior corners of the stack! Thus, the negative basis represents a reversal of the outward direction, a reversal of the outward direction from one corner, to the inward direction toward the corner. In other words, in GA, the geometric interpretation of the positive basis set diverges from the dextral corner, while the negative basis set converges to the dextral corner.

Therefore, the difference between the basis set of three directions in GA, and the basis set of the four diagonals we are contemplating using for SA, is a divergence|convergence of the three directions at the corner, as opposed to the divergence|convergence of the four directions at the intersection.

This difference has many implications that we need to explore.