The Larson Research Center

There is an obscure plaque on the Brougham Bridge in Dublin, which reads:

According to Wikipedia, “mathematicians from all over the world have been known to take part in the annual commemorative walk from Dunsink Observatory to the site [of the plaque]. Attendees have included Nobel Prize winners Murray Gell-Mann, Steven Weinberg and Frank Wilczek, and mathematicians Sir Andrew Wiles and Ingrid Daubechies.”

I’ve referred to the story of Hamilton’s discovery of quaternions many times over the years, because Hestenes makes it so interesting. At the point in time that I discovered Hestenes and his GA, I had no idea what quaternions even were. One evening at an ISUS conference in SLC, John Pratt tried to explain them to a group of us that stayed up until midnight to learn about them.

The crux of Hamilton’s “flash of genius” was the realization that, to move from 2D algebra (what has become known as the algebra of complex numbers) to 3D algebra (now known as quaternions, meaning four), requires four dimensions (here the word “dimensions” is to be taken in the sense of the mathematician, not in the normal geometric sense).

Hamilton had been trying to find a 3D algebra based on 3 mathematical dimensions, which he thought should require three independent numbers to define it (see Larson’s discussion of mathematical dimemsions here). It was at the bridge, walking with his wife, that he suddenly realized that a 3D algebra requires 4 dimensions, or 4 independent numbers to define it, not 3, as one would think.

Hamilton’s fascination with his quaternions from that point on kept him from doing anything else. He was obsessed with them until the day he died, but they eventually fell into obscurity and his complete devotion to them was generally regarded as a tragic waste of the genius’ career. 

That’s because everything that needed to be done in electrical engineering and theoretical physics could be done with the 2D algebra of complex numbers. It wasn’t until the last half of the 20th Century, that engineers began to see an advantage to the 3D algebra, mainly in robotics.

Some few theoretical physicists have sought to employ quaternions and even the eight dimensional algebra of octonions in their theoretical work, but these efforts have not enjoyed a lot of success. So, why are Nobel Prize winning physicists and mathematicians so interested in this commemoration of Hamilton’s insight?

It’s because of Clifford algebras and the binomial expansion, which captures the reality of three (geometrical) dimensions and the two “directions” inherent in each of those dimensions: Counting the “directions” of nature we get 20 = 1, 21 = 2, 22 = 4 and 23 = 8, the four numbers of the tetraktys, which constrains Euclidean geometry and should constrain theoretical physics, but doesn’t, because mankind can’t explain the so-called elementary particles of nature and all their interactions, within its three dimensions. They need extra dimensions, because the “point,” and the three dimensions of physical reality, are not compatible in their theories.

Well, we have struggled within the tetraktys as well, but have embraced its constraints due to the second fundamental postulate of the RST. But just as the LST community is stuck, in their theoretical development, so are we. And it turns out that we are stuck, not at the 0D point, but where Hamilton was stuck: We have been trying to use three independent mathematical dimensions to understand motion in three geometric dimensions, but it takes four to do it!

Recall that in our preon model of the standard model of particle physics, formed from S|T units, which are formed in turn from the SUDRs and TUDRs, or the initial 3D space oscillations and 3D time oscillations, deduced from the fundamental postulates of the RST, we describe bosons (S|T doublets) and fermions (S|T triplets). The triplets form the quarks and the leptons, which, of course, we use to form the atomic elements of the periodic table (see Wheel of Motion), but based on their RST scalar motion values, not the vectorial motion values of the LST.

In each of these preons, we use three colors to represent the space/time and time/space displacements of the S|T and T|S units. For instance, the up quark triplets contains one green and two blue S|Ts, while the down quark triplet contains two green S|Ts and one red S|T, as shown below in figure 1.                        

Figure 1. The Three S|T Units of the Up and Down Quark Triplets

The red and blue endpoints indicate that the S|T (red) and T|S (blue) units are inverses (red is a “lower” displacement, a value below 0 displacement, while blue is a “higher” displacement, a value above 0 displacement). The colors of the midpoints indicate the ratio of S|T to T|S units. A green midpoint color denotes a 1:1 ratio, while blue denotes a 1:2 ratio and red denotes a 2:1 ratio. 

It’s important to understand that these diagrams of the relative displacements do not contain all the information we would like to convey. For instance, since the actual physical oscillations are 3D, they would have to combine as balls, not lines (see here).

But two partially merged balls necessarily combine as a 1D entity (the line between their origins at their centers), while three partially merged balls necessarily combine as a 2D entity (the area defined by the plane formed from the three conected points of their origins). Moreover, there is no way to partially merge a fourth ball to change this geometry from the 2D area to a 3D volume, except by placing it on one side, or the other, of the three-ball plane.

This works out nicely in some ways, but when we completely merge the balls, this Euclidean geometry disappears. The question we’ve pondered has always been how far are the balls separated and how do we determine that distance? But with the new view of the 3D oscillation explained in the FQXI essay, the doubt arises as to why they even need to be partially merged. Maybe the S|T units are completely merged with the T|S units and, if this is true, how do we interpret the binding together of the red and blue entities?

It’s still true that the relative displacement from the perspective of each of the two displacements is akin to the balance scale: if the left is lower than the right from one perspective, then the opposite is true from the inverse perspective (if the left end of a teeter-totter is observed from the side view to be lower than the right end, then if the observer crosses over to the other side of the teeter-totter, turning around to view it from the opposite “direction,” the right end will be lower than the left end, even though it’s the observer’s right and left that have been swapped, not the teeter-totter’s).

However, in comparing two relative displacements (i.e. two teeter-totters), one displaced in the opposite direction of the other, i.e. 2:1 and 1:2, there are four points of comparison, which are the four endpoints of the two teeter-totters. 

We can illustrate this with the teeter-totter, or lever analogy. If two levers are on the same pivot point mid-way along their length, with equal weights on each of their respective endpoints, they will be balanced in the horizontal, but if there are two of the equal weights on one end, and one on the other, the lever will not lie horizontally, but diagonally, and if the relative imbalance of the two levers is opposed, then they will lie in inverse diagonals, forming an X pattern, when viewed from the side. 

Now, the question is, if the X pattern represents the relation (s:t):(t:s) and (s:t):(t:s), or two red-blue sets of the three in a triplet, then how do we include the third set in this same ratio? The answer is, I believe, we don’t. Just as Hamilton could not multiply triplets, but had to add a fourth mathematical dimension, we can’t compare three S|T units. In order to compare four S|T units, they must be combined as two sets of orthogonal Xs.

Fortunately, the two sets of Xs correspond to the eight corners of Larson’s cube, our 3D geometry generated by scalar motion, which just happens to perfectly integrate Euclidean geometry and our new algebra through the tetraktys. But now we need to ask, “Won’t this change in the fundamental combinations of S|T units ruin the three-fold symmetry of the preon model?”

Happily, it won’t. Not if its mathematical symmetry is interpreted geometrically, it won’t; That is, Larson’s cube can be extended (or unbalanced if you will) in each of its three dimensions, depending upon the relative number of its constituent S|T units in the three dimensions, which should, hopefully, enable us to preserve the existing preon model, while now building it by proceeding from 0D, to 1D, to 2D, to 3D, in the three iconic steps of the tetraktys.

This is an exciting advance in our development, if I’m not mistaken. Now, if I could only find a stone bridge to carve!