The Larson Research Center

We have started discussing the construction of a scalar algebra (SA), using Hestenes’ geometric algebra (GA), as a guide. However, whereas GA is formulated axiomatically, beginning with an abstraction of natural numbers from the traditional set theory of Dedekind and Cantor and continuing with a further abstraction regarding algebraic operations from Clifford and Hestenes, SA must be deduced from the purely inductive principles of order in progression.

This is a tall order, to say the least. First, Clifford and then Hestenes recognized Grassmann’s geometric product and Hamilton’s quaternions, as containing the seeds of an eight-dimensional, geometric, algebra. Nevertheless, few seem to share Hamilton’s concern with the axiomatic foundations of formalizations such as GA that disregard the fact that many times one cannot “rise to intuition from induction, or cannot look beyond the signs to the things signified” in these formalisms.

Hestenes tauts GA as a very useful language for physics, and has impressively demonstrated its power in that connection, but it seems that Hamilton would object on theoretical grounds, because, he would certainly protest that, while there is much of beauty and consistency “in what has already been brought to light,” there is still considerable confusion of thought in what the foundations of this algebra are. He writes in the preface to his essay on the Algebra of Pure Time:

As we have already noted extensively, Hamilton’s approach was based on turning from considering the “bounded notion of magnitude,” to what he called the intuition of order in progression, the principles of which he deduced from inductively reasoning upon “pure time.” However, this approach seemed complicated and unnecessary to formalists who were seeking to improve the “art of language,” and the fact that the accepted concepts of negatives and imaginaries work well in practice is apparently good enough for them. That a theorist like Hamilton, seeking to improve the science of algebra, might be bothered by “principles which yet remain obscure and doubtful,” seems hardly enough to turn the thoughts of mathematicians, let alone physicists, to the intuition of pure time.

Yet, we have seen that, with the ideas of Larson, beginning a century later, the concept of numbers and magnitudes, deduced from the properties of order, in assuming a space|time, reciprocal, progression, leads to a generalization of number that is remarkably consistent with the three properties of physical magnitudes, their quantity, dimension, and direction properties. Now, when we look at Clifford algebras, we can see that the idea of dimension is associated with the idea of direction; that is, as dimension increases, the two directions of the dimensions increase, as seen in the binomial expansion. Thus, in the tetraktys, the two, inherent, directions of dimension multiply from zero, to two, to four, and to eight, in its four levels, creating a corresponding algebra of directions, we might say, with each successive one including the previous ones.

Therefore, in the third dimension, fourth counting zero, we have a Clifford algebra containing four linear spaces of dimension 0 through 3, describing what mathematicians call an eight-dimensional algebra that should be useful for calculating geometrical magnitudes, except that, because of the obscure and doubtful principles of negatives and imaginaries, the 1D space is not ordered, the 2D space is not commutative, and the 3D space is not associative, and without these crucial algebraic properties, the operations of multiplication in the algebra break down.

What Hestenes revealed, in the last half of the 20th Century, is that these problems can be overcome with Clifford’s formulation of the geometric product, if it is used to define the inner and outer products of vectors, which can then be interpreted as the magnitude and direction components of vector products in 3D space. While this is a truly amazing development in the art of mathematics as a language for physics (it is finally beginning to be recognized and taken seriously by the scientific community, mostly because engineers have found it so useful), it would hardly have been satisfying to Hamilton. The principles involve yet remain obscure and doubtful. It seems like intellectual acrobatics, rather than clean theoretical deduction, useful, but unable to reveal the foundations of truth in the structure of the physical universe.

In some sense, our own feeble attempt has the same object as Hamilton’s initial efforts had; that is, “to improve the science of algebra,” even though there is an ulterior, less noble, motive behind our efforts, because we simply need a scalar algebra to develop RST-based physical theory, due to the fact that a vector algebra does not apply to scalar magnitudes. Our first step was to deduce a set of natural, reciprocal, numbers, in a set theoretical manner (see previous posts below). Next, we plugged these numbers into the tetraktys, which gave us the same four linear spaces, used in the eight-dimensional GA. However, we know that the geometric product of GA, and consequently, the derivation of the inner and outer products, cannot be directly applicable to a SA, because there are no vectors, bivectors, or trivectors in our four linear spaces.

In fact, there is something very disturbing about the four linear spaces of the scalar tetraktys. While (8/8)⁰ is composed of four (2/2) 1D units, our 1D linear space, like the linear space of the vector tetraktys, has room for only three of these! What are we supposed to do with the fourth set? In thinking why this has happened, it soon becomes clear that there is a degeneracy in the vector tetraktys, because of the way it is applied to Larson’s cube, in terms of vectors, bivectors, and trivectors. To see this, we need to think of 2³ = 8, as generated from two instances of 2² = 4, or the 3D space, as generated by a rotation of two dimensions out of the plane. There are two ways to do this: Rotate the second plane around the horizontal axis of the first, or else rotate it around the vertical axis of the first.

Either way, a 3D space is generated from a 2D space, by the rotation.  However, in both cases, the axis of rotation is common to both planes! They are tantamount to what Hestenes would designate as colinear vectors. Applying this to the scalar spaces, it’s as if the colinear axis were composed of the “distance” (2/4, 4/2), instead of the “distance” (1/2, 2/1), so one entire 2/2 unit is hidden, as a degeneracy.

The significance of this may take a while to fully appreciate, but in the meantime we can recognize that it will not do. We must deal with it, because rotation is not possible with scalars.  Either the degeneracy does not exist in our scalar spaces, in which case the dimension will change, or there is another mechanism that produces the same degeneracy in the scalar tetraktys. Biting the bullet, we are going to assume, at least for now, that we have to put the fourth unit back in, which would change the scalar tetraktys to:

1
1 1
1 2 1
1 4 4 1

or from an eight-dimensional algebra, to a ten-dimensional algebra (we will ignore the glaring connection with string theory for now!!!!) Of course, the $64,000 question is how do we do this, since 2³ = 8, not 10. I can provide a preliminary answer to this question, but whether or not it will be the prize winning answer remains to be seen. First, we need to understand that the 1331 = 8 numbers of the fourth line of the binomial expansion only have to equal 8. We don’t necessarily have to follow the equation of the binomial theorem, at least for now. So, another combination that equals 8 is 1421 = 8.

String theorists would look for a way to “hide” the missing two dimensions, in 3D space. We might be able to follow their lead and denote the fourth line as 14(4-2)1 = 1421 = 8, and say that two hidden dimensions are “compactified,” but then, we don’t have the same problem as they do, because we are not talking about dimensions of space, but rather dimensions of motion. So, we would say that two dimensions of motion are hidden, not two dimensions of space, which reminds us that Larson’s cube is an illustration of 3D motion, not 3D space.

As such, recall that Larson referred to the four diagonals in the 2x2x2 stack of one-unit cubes, which is not a 1331 map of the vector tetraktys. We are the ones who first mapped the 1331 tetraktys to the cube, by replacing the four orthogonal diagonals with the three orthogonal axes normally used in algebra and Euclidean geometry.   In fact, we took our clue form GA, and paid scant attention to the difference.

But now, we find ourselves forced back to the four diagonals of Larson’s original cube, by the new discovery of the degeneracy in the three axes of the coordinate system view. This seems almost poetic, because it is the 8 units of 3D scalar motion of the RST that we are interested in, not the 8 units of the 1D vector motion of the LST! The diagonals of the stack of 8 cubes is again illustrated, as shown in figure 1 below:

Figure 1. The Four Diagonals of Larson’s Cube 

Of course, the volume of the 2x2x2 stack of cubes is related to the volume of a sphere with a diameter equal to the length of the diagonals. Likewise, the area of a plane on the face of the stack is related to the area of a circle, with a diameter equal to the length of the diagonals. However, the scalar operation that generates a line from a point, and an area from a line, and a volume from an area, in scalar space, differs from the equivalent operation in vector space. In vector space, a line is generated from a point through translation, an area through translations of two objects, in independent directions, or through a rotation around an independent axis, and a volume through three translations, in independent directions, or a second rotation around another independent axis. These are the operations of addition and multiplication in vector algebra, the result of combining individual elements, considered as generated in one direction at a time, or by multiple objects in motion simultaneously.

In contrast, in our scalar space, a line is generated from a point through the expansion of the point, in two, opposite directions simultaneously, something that is impossible for a single object to do. An area is generated through the expansion of a line, in two, independent, opposing, directions, simultaneously, and a volume through the expansion of an area in two more, independent, opposing, directions, simultaneously. Thus, the four diagonals of the cube represent the 2, independent, opposing, directions, that the point can be expanded into a line, the 2, independent, directions that the line can be expanded into a plane, and the 2, independent, directions that the plane can be expanded into a volume, which is 2³ = 8, independent, simultaneous, opposing, directions of expansion, in all, because the 2x2x2 stack of cubes is generated by simultaneous expansion in 2x2x2, opposing, directions.

In other words, the 3D scalar space is a 0D point expanded into a 3D sphere, not a set of 8, one-unit, cubes, generated individually and assembled together. Even though the end result is numerically equivalent, in both cases, the operations generating the multidimensional spaces are radically different. Hence, this implies that the meaning of the algebraic addition and multiplication operations, in the two different spaces, differ as well.

It seems clear that each, independent, 1D, diagonal of the sphere can be expanded into a 2D plane, in two, independent, ways. In one way, the line is expanded along the independent diagonal sloped in one direction, and in the other, it is expanded along the diagonal with opposite slope. Thus, there are four, independent, areas in the volume. Each of these areas can be expanded into a sphere by expansion in two, independent, opposing, directions to form the volume of the pseudoscalar. 

Therefore, it appears, at least, that the scalar tetraktys is as faithful to the geometry of the sphere, as the vector tetraktys is to the geometry of the cube, and we can translate back and forth between them as required, in a clear and consistent fashion. If this turns out to be the case, the next step will be to identify the nature of the algebraic operations that can adequately express the expansion to higher dimensions, and the contractions to lower dimensions. I’m thinking that the dot symbol would work for the contraction operation (contracting toward a point), and the circle symbol would work for the expansion operation (expanding toward a sphere).  We’ll see.