The Larson Research Center

An LRC seminar is scheduled for next month in which I am to present the new concepts of the reciprocal system of mathematics (RSM) to a group of professional mathematicians, physicists, and engineers. Some are familiar with Larson’s reciprocal system of physical theory (RST), but others are not. Needless to say, the challenge seems formidable at this point.

I’m planning on placing the discussion in the context of empirical discoveries as much as possible, starting with the discovery of the Pythagorean incommensurables. I’ll talk about the historical effort from that point on to generalize the concept of number enough to identify its properties with the properties of physical magnitudes in the attempt to found a geometric algebra. I’ll point out the often overlooked, or misunderstood, significance of the mathematical implications of Newton’s third law of motion, which leads to the corollary that for every direction there is an opposite direction, or, which is tantamount to the same thing, that each dimension of space has reciprocal, or dual, directions.

Then I’ll want to show that Hamilton discovered that, for the science of algebra, as opposed to the science of geometry, considering the mathematical inequalities in the order in progression makes more sense than considering them in terms of increasing and diminishing magnitudes, but that this has gone unrecognized in the world opened up by Dedekind and Cantor.

I probably won’t be able to sufficiently, and succinctly, capture the compelling drama between the misdirected development of mathematical ideas, lamented by Hamilton, and the success of their physical applications, which obscures their imperfections, without confusing the audience. So, instead, I’ll just try to show that what Hamilton discovered, when combined with what Larson discovered, sheds light on what Hestenes revealed lies hidden in the Clifford algebra, based on the Grassmann algebra, when combined with Hamilton’s ideas: namely, that the generalization of number in terms of the definition of geometric algebra’s (GA) geometric product enables us to work with coordinate-free vectors in 3D space, under a new interpretation of imaginaries, interpreted as rotations, defined in terms of GA multivectors, arising out of the four vector spaces of the three-dimensional line of the tetraktys (1331), enabling the formation of an eight dimensional algebra, corresponding to the three-dimensional geometry of the tetraktys.

However, I will make the point that GA does not eliminate the idea of imaginaries in its formulation. It merely transforms this enigmatic, ad hoc, invention of the human mind, which was invented to deal with negative numbers, from the quantitative interpretation of the square root of -1, to the operational interpretation of a π/2 rotation, by introducing two products, one the dual, or the reciprocal, of the other.

Then, I hope to make clear that this idea of a π/2 rotation originated in the four dimensional quaternions, coming to Clifford from Hamilton, and thus on to Hestenes and the eight-dimensional GA of today. The great significance of this fact is that GA is founded on a mistaken notion that not only caused a hundred years of great confusion (before Hestenes’ work and the advent of GA), which is aptly described by Simon Altmann, but incredibly enough, goes on to connect the RSM to the tetraktys and its geometry found in Larson’s cube, the foundation of the RSM, and the basis for the RST-based theoretical development at the LRC.

To show what I mean, I will begin with a one-dimensional magnitude, a line segment, and treat its properties logically. To wit: If we divide it into two equal lengths, each length will be .5 times the original 1 unit, but the half-lengths have opposing direction with respect to the center cut of the original. Yet, combining them arithmetically, as usual, we generally ignore this difference in the direction to get

.5 + .5 = 1,

because, if we include the opposite direction information, we get

-.5 + .5 = 0,

which, in terms of physical length, makes no sense, because combining the two half-lengths should give us the original unit length again, not nothing, a point. Of course, if we put opposing arrow heads on the two lengths, representing two magnitudes with two opposing directions, the result of combining them is 0, as in the resultant zero motion of two opposing force vectors.

However, when Larson recognized the two “directions” of speed-displacement that the two reciprocal, space|time, progressions can take, we were able to see that the same two numbers can take a different form

-.5 + .5 = 1/2 + 2/1 = -1 + 1 = 1,

even though it makes no sense in the context of usual arithmetic, which we would normally understand as tantamount to writing

0 = 2.5 = 0 = 1.

Even so, this complete nonsense, in terms of grade school arithmetic, makes perfect sense in terms of recombining two half-lengths, if we understand that -.5 = 1/2 and .5 = 2/1, in terms of a unit displacement in two, opposite, directions, as when two girls are on one side of a teeter-totter opposite one boy, or vice versa. The displacement is one unit in one direction, or the the other: One is in a vertical position that is a positive unit, while the other is in a vertical position of equal magnitude that is a negative unit relative to the equilibrium condition. If we place two boy and girl teeter-totter triplets, one with two boys and the other with two girls, onto one teeter-totter, we will have three kids on each side of the teeter-totter, and, if they all weigh the same, it will be balanced. So, the arithmetic

1/2 + 2/1 = 3/3 = 1/1 = 1

is ordinary arithmetic after all, when we interpret the numbers as reciprocal numbers, with opposing directions.

The teeter-totter analogy is very useful in this respect. For instance, if we have, say, two boys standing in the middle of the device, they could walk in opposite directions, and, if they were careful enough, they could keep the beam balanced until they reached the ends and took their seats. As far as the condition of equilibrium is concerned, though, no change has taken place, although now there is a distance between them, which didn’t exist before. Clearly, this is analogous to the equation

1/1 = -1 + 1 = 1,

if we interpret the numbers in such a way that the positive and negative values offset one another, but do not annihilate each other. In this interpretation, the numerator and denominator of reciprocal numbers are interpreted as having opposing “directions,” where placing quotations around the word directions indicate that we mean the dual directions of polarity, not the dual directions of space.

Therefore, we can clearly establish that the reciprocal number has both quantity and the dual direction of dimension. After establishing this important concept in the presentation, I will try to show that these reciprocal numbers also have the third property of physical magnitudes, the property of three dimensions. Once that is established, I will want to go on to show how these numbers, with their three properties of physical magnitudes, quantity, dimension, and “direction,” define an eight-dimensional scalar algebra, that is remarkably similar to the eight-dimensional vector algebra, GA.

More on that next.