For mathematicians, the term "multi-dimensional scalars" is an oxymoron. Scalars, by their definition in the legacy system of mathematics (LSM), have no dimensions and the pseudoscalars are not normally thought of as numbers. Wikipedia defines pseudoscalars, when the term is used in physics, as "a quantity that behaves like a scalar, except that it changes sign under a parity inversion such as improper rotations while a true scalar does not," and, when used in mathematics, Wikipedia defines the term as "an element of the top exterior power of a vector space, or the top power of a Clifford algebra; see pseudoscalar (Clifford algebra). More generally, it is an element of the canonical bundle of a differentiable manifold."

But if such definitions satisfy the members of the LST community, they are not of much use to us, as members of the RST community. The reason, I suspect, is that the basis of LST science is vector motion, so everything, including the mathematics of the science, is cast in that form. Scalar motion science, on the other hand, cannot make much use of vector concepts, since scalar magnitudes have no direction, by definition.

But does this mean that scalar magnitudes have no dimension? No, "the top power of [the first] Clifford algebra" has no dimensions (2⁰), but the others do 2¹, 2², 2³, ...etc, as indicated by the power notation. As I have been pointing out for years now, the first four Clifford algebras form the mathematical domain of science, because they are the only known normed division algebras. The Bott periodicity theorem proves that there is no new phenomena beyond these four, which correspond to the three (four counting 0) known dimensions of the observed universe.

Quantum physics, relativity theory, string theory, all define their fundamentals within the bounds of these four dimensions. String theorists thought they had escaped this constraint, but then it was discovered that they were only working in copies of these dimensions, so they soon found themselves back in the domain of what I call the tetraktys, the Greek name for the first four numbers, 1, 2, 3 and 4.

Unfortunately, Larson did not have the advantage of understanding the implications of Clifford algebras, as the general knowledge of them did not begin to spread outside the esoteric realm of the doctors of mathematics, until the 1960 investigations of Hestenes began to come to light in the 1980s. By then, it was way too late for Larson to take advantage of the insight they provide. Had he understood the Clifford algebras, the Bott periodicity theorem, and the tetraktys, maybe his development would have taken a different course, but we will never know.

Yet, a careful reading of Larson's works reveals many concepts that are fundamental to these obscure mathematical subjects. For instance, he knew that there were two scalar "directions" for each scalar dimension that must be considered apart from the vectorial directions, defined by three dimensions. He also knew that there were two units, a positive unit and a negative unit in each scalar dimension. He also understood that, for 3D scalar motion, there are eight scalar "directions," which he illustrated by invoking the image of a 2x2x2 stack of 1-unit cubes, its eight "directions" extending diagonally from the center of the stack to its eight corners.

Realizing the importance of recognizing these ideas, as legitimate mathematical concepts, I began calling Larson's 2x2x2 stack of 1-unit cubes, "Larson's cube," years ago. But I didn't realize back then how central it would become to investigating the consequences of the RST. Since then, I've found that we can use Larson's cube to quantify the natural progression, and that the number 2 is the number of "directions" in each of the three dimensions of the cube, while the center intersection of the 2x2x2 stack represents the 0-dimensional reference point, from which the scalar "direction" "out" and "in" have meaning. It was also clear that the space/time dimensions of the expanding cube could be inverted, to time/space, and that the oscillation of the cube would constitute a space/time displacement of 1 unit (1/2), while the oscillation of the inverse cube would constitute a time-space displacement of 1 unit (1/2), but in the inverse "direction," giving us a positive and negative, 1-unit, speed displacement.

On this basis, I was able to do a lot of things that amazed me, in terms of combining these scalar units to form combinations of them, and it was always satisfying to know that it was a mathematical development, as well as a logical one that I was able to follow. I felt like the development along these lines was important, so I continued pursuing it, even though my course was not well received by all members of ISUS. One of the biggest objections to my line of thinking was lodged on the same grounds, as was that of objections to Larson's work: The fact that the physical expansion of space and time is continuous, with no discontinuities. Larson's cube, while mathematically sound, does not describe a physical expansion/contraction, which is demonstrably not cubical!

However, I soon discovered something called inversive geometry, which shows that the radius of a given circle is equal to one-half its inverse circle. This intrigued me, and after a while I realized that I could place two balls defined by Larson's cube, one on the inside, just contained by the cube, and one on the outside, just containing the cube, and put all these right lines and circles inside the inverse ball of the inner ball, satisfying the equation of inversive geometry, r'² = r * r''.

To my surprise and delight, I discovered that the dimensional magnitudes of this set of cubes and balls, i.e. their radii, diameters, areas and volumes, reflected the mathematical structure of the periodic table and the atomic spectra. I hope to be able to prepare a formal paper on this soon, but it was not until recently that I discovered that not only are these patterns of numbers found in this set of right lines and circles, but the numbers and geometry of the tetraktys is also hidden inside the relations of its elements.

That is to say, the natural progression of discrete numbers, existing in three dimensions, with two, reciprocal, aspects, produces continuous magnitudes (with a factor of π) the ratios of which are instances of 2⁰, 2¹, 2² and 2³ discrete magnitudes! This is a remarkable finding. Who would have guessed that the multi-dimensional, discrete, numbers of the tetraktys, so fundamental to all science and mathematics, would be found in the geometry of a 3D progression, in the geometry of Larson's cube!

More about this later, as time and circumstances permit.