The Larson Research Center

The sixth post in the BAUT forum follows.  These posts are a continuation of the RST thread in the “Against the Mainstream” forum of BAUT.

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Since reciprocal numbers (RNs) are composed of OI numbers, and OI numbers are isomorphic to integers, they do not have the property of direction. However, they do have the property of “direction,” because just as an increasing negative integer increases in the opposite “direction” with respect to increasing positive integers, an increasing OI number on one side of unity increases in the opposite direction with respect to increasing OI numbers on the other side of unity. Thus, 1/2 + 1/2 = 2/4 = -2 is “farther away” from unity than 1/2 = -1, but in the opposite “direction” with respect to unity than a similar increase on the positive side of unity would be.

However, given two RNs such as, for example,

(1/2 + 1/1 + 2/1) = 4/4 and (2/4 + 2/2 + 4/2) = 8/8,

there is no orthogonal relationship of directions that can be defined when combining them, just as there is no orthogonal relationship of directions that is definable when physical scalars such as points, or pseudoscalars such as volumes, are combined. Consequently, when multiplying RNs, though we multiply each term in the multiplicand by each term in the multiplier, as we would if they were vector magnitudes, the product “n times m” literally means that n is repeated m times, not that the scalar unit is changed to an “n by m” area unit by the operation. Also because the numbers are OI numbers, not fractions, we multiply numerators by numerators and denominators by denominators in each term, so the product of the two RNs is

(1/2 + 1/1 + 2/1) * (2/4 + 2/2 + 4/2) =
(4/4) * (8/8) = 32/32 nm,

because multiplying three terms by three terms,

(1/2 + 1/1 + 2/1)
(2/4 + 2/2 + 4/2) =

2/8 + 2/4 + 4/4 + 2/4 + 2/2 + 4/2 + 4/4 + 4/2 + 8/2,

gives us nine terms. Then, combining like terms together, we get

(2/8 + 2/4 + 2/4) + (4/4 + 2/2 + 4/4) + (4/2 + 4/2 + 8/2) =
[(6/16) + (10/10) + (16/6)] = 32/32 nm,

but the result is a scalar number, not a vector; that is, we are multiplying points here (actually pseudopoints, because RNs have volume). This means, in the language of GA, that the multiplication operation doesn’t affect the grade of the pseudoscalar, or 3-blade, anymore than it affects the grade of the scalar, or 0-blade.

So, while a vector times a vector is a bivector, a higher grade blade, and a vector times a bivector is a trivector, a higher grade blade, a scalar times a scalar is a blade of the same grade. Stating the same thing symbollically,

A¹* B¹ = C² (in the form of the “sum” of the inner and outer product, the geometric product), and

A¹ * B² = C³,

for the vectors, and

A⁰ * B⁰ = C⁰ ,

for the scalars, but what’s suprising is that

C³ * C³ = C³, not C⁹

for the pseudoscalars! This is because the three dimensions of the pseudoscalar are the internal dimensions of the volume, represented by the three terms of the RN, and the nine terms of the product are always just scalar expansions of their three terms, as I have shown above.

As far as I know, this has never been noted before, because GA isn’t designed to use the 3-blades, except as what is called the unit trivector, symbolized by the upper case letter, I. This view of the trivector is a vector view, but it is also called a pseudoscalar, because its outer product commutes with everything, as a scalar does. Yet, unlike a scalar, it also changes sign under inversion! What does that remind you of?

What we have discovered here is that the dilation of the scalar, or origin of a coordinate system, in the form of the pseudoscalar, can be expressed in the usual terms of the three dimensions of vectors, as a trivector, or alternatively, in terms of the three “dimensions” of the RNs, as a pseudoscalar. Moreover, this leads us to view the binomial expansion, which is the dimensional expansion of duality, as the expansion of points, which don’t expand; the expansion of vectors which expand vectorially, and the expansion of pseudoscalars, which expand scalarly. That is why the numbers down the left and right edge of Pascal’s triangle are always 1, but the numbers between these two are always more than 1, but also always symmetric.

However, there is a difference between the 1s down the left side, which are scalars (2⁰ = 1), because they have no direction, but magnitude only, and the 1s down the right side, which are pseudoscalars (2ⁿ), because they have all directions possible in n dimensions. Hence, the number of directions in the pseudoscalar is the same as the number in the scalar at n = 0, namely zero, but at n = 1, the number of pseudoscalar directions is two, represented by the scalar expansion in all the directions of a line from the perspective of the scalar point in the middle (linear dilation). At n = 2, the number of pseudoscalar directions is four, represented by the scalar expansion in all the directions of a plane from the perspective of the scalar point in the middle (quadratic dilation), and at n= 3, the number of pseudoscalar directions is eight, represented by the scalar expansion in all the directions of a cube from the perspective of the scalar point in the middle (bi-quadratic dilation), as clearly depicted in Larson’s Cube.

There is much more to say about all this, but suffice it to say for now, that these are concrete mathematical results that are the subject of the mathematical research at the LRC. The application of these scalar math concepts, in developing a suitable language for the LRC’s scalar physics research, is a major goal of the LRC. The prospect of using the new language effectively is promising, given that string theory, the standard model’s guage theories, and the fundamentals of group theory and symmetry principles, all converge to the same place: the space of the octonions.

It has long been suspected that the octonions have a deep connection with physics for many reasons, but so far the connection has escaped physicists. Now that there is a new kid on the block, though, things may change relatively quickly. Our first objective is to understand the RSM, and we’ve made a lot of progress since first discovering it. Yet, none of this means anything, unless we can connect it to observations. We have hints on how to do that with photons of radiation, but our primary goal is to use the RNs to find the properties of neutrinos, electrons, positrons, protons, and neutrons, and learn how they relate to each other, and to photons, in the atomic spectra.

If we can do that, we will be on our way, because the origin of the material properties of mass, charge, and spin, are unexplainable in vector physics, but they should be relatively easy to explain in scalar physics, where the three “dimensions” of RNs, correspond to outward motion in space, outward motion in time, and inward motion in space, or in time, the magnitudes of which are functions of events and their associated probabilities.

Thus, since forces, such as gravitational, electrical, and magnetic forces, are properties of motion, we expect to discover how they are related to the three dimensions of motion in the RNs. An exciting prospect, indeed.

I guess this concludes my long-winded initial answer to antoniseb’s question. If there are followup questions, I would be happy to try to answer them.