For vector motion, whether linear or rotational, the change in position or location over time that defines it must be uni-"directional." For linear motion, this means outward from a specified point, or inward toward a specified point. For rotational motion, this means a clockwise change of angle or a counter-clockwise change of angle. Unlike scalar motion, vector motion cannot proceed in the two "directions" of a given dimension, simultaneously. Think of changing sines and cosines describing rotation. They change in one "direction" and then reverse and change in the opposite "direction," as the rotation proceeds.
However, this is not so with scalar motion. A scalar expansion of a line expands in both "directions" of its dimension, simultaneously. When it reverses, it contracts in both "directions" simultaneously (2
1 "directions"). If it's two-dimensional scalar motion, it expands/contracts in both "directions" of its two dimensions (2
2 "directions"). If it's three-dimensional scalar motion, it expand/contracts in both "directions" of its three dimensions (2
3 "directions").
This much is very clear. Now, what is needed is to understand how we can find numbers with these same properties. It was problematic for the ancient Greeks that lengths could be found for every number, but numbers couldn't be found for every length. It took Western civilization a long time to meet that challenge, but it finally did, at least to the satisfaction of most people.
Yet, not long after that, it was discovered that nature sticks with whole numbers anyway, so our unit measures had to be adapted to nature somehow, which, as Xavier Borg shows, have the dimensions of motion; that is, the dimensions of space/time.
Consequently, given Larson's new system of scalar motion, and the LST system of vector motion, we need to differentiate them in terms of nature's space/time. Well, we know that vector motion requires an object to change location over time, while scalar motion does not, so it follows that scalar motion is prior to vector motion, and we've shown how that can be in our version of the standard model of particle physics.
Unfortunately, though, as we've seen demonstrated in this forum so recently, the vector motion number system has been crafted to describe nature's space/time in terms of vector spaces and vector algebra. The idea of scalar spaces and scalar algebra isn't even comprehensible to LST physicists and mathematicians.
Nevertheless, we have seen how genuine these concepts are, by demonstrating the one-to-one correspondence between the "directions" of the Tetraktys dimensions and those of Larson's cube. Moreover, we can show that the scalar expansion of the 2x2x2 cube contains multi-dimensional unit magnitudes.
These multi-dimensional units shed an entirely new light on the LST's multi-dimensional numbers, based on vector motion, used in the so-called "normed division algebras," upon which all the technology, engineering and science of Western civilization are based.
As incredible as it may seem, the fact is multi-dimensional scalar motion gives rise to multi-dimensional scalar numbers and the algebra to add, multiply, subtract and divide them. Years ago, UofU math professor, Bob Palais, attended one of our ISUS meetings in SLC, and he and I had several follow up discussions. I tried to explain the idea of scalar numbers to him, but, at that point, the subject was not so developed in my own mind. He listened politely, and then asked me what the square of some of the numbers were. When I tried to square them, I failed and that was that, as far as he was concerned. We never met again.
I don't remember why I failed, because it seems trivial now, but it may have been due to not understanding the concept of "unit." What we mean by multi-dimensional is multi-dimensional units, the three listed in the previous post.
It's not that they are multi-dimensional in terms of orthogonal directions. That's a vector motion based concept. And it's not that they are multi-dimensional in terms of the wedge products of multivectors in Geometric Algebra. It's that they are the concurrent unit magnitudes of the expanding/contracting 2x2x2 stack of unit cubes - the three radii.
Larson thought of scalar magnitudes in terms of space or time displacement: s/t = 1/1, the unit speed, or datum, of the system, has zero displacement, but "direction" reversals in the expansion of space or time (two observables) led him to quantify speeds in terms of time displacement s/t = 1/(n>1) and space displacement, s/t = (n>1)/1; that is, displacement from unity in the numerator or denominator.
Of course, this is just the set of rational numbers, but with a special twist, where the displacements are inverse units in two, opposite "directions," instead of the fractions of a whole:
s/t = 1/2, 1/1, 2/1, in terms of speed displacements, are equivalent to -1, 0, +1, not .5, 1, 2.
Therefore, in terms of our 1d unit, √(1
2+0
2+0
2) = √1, the possibilities are:
1(√1)/2(√1), 1(√1)/1(√1), 2(√1)/1(√1) = -1, 0, +1
but in terms of our 2d unit, √(1
2+1
2+0
2) = √2, the possibilities are:
1(√2)/2(√2), 1(√2)/1(√2), 2(√2)/1(√2) = -1(√2), 0, +1(√2),
and for 3d, √(1
2+1
2+1
2) = √3, they are:
1(√3)/2(√3), 1(√3)/1(√3), 2(√3)/1(√3) = -1(√3), 0, +1(√3),
However, while the use of polarity signs above indicates the opposite "directions" of the numbers, it hides the inverse character of the opposite terms. To avoid this problem and clarify it a bit, let's revert to the rational form of the numbers and place the appropriate inverses under the ratios and then the displacement values under those:
...............1d -------------------------------------2d---------------------------------------3d
√1/2√1, √1/√1, 2√1/√1 <------------>√2/√8, √2/√2, √8/√2 <------------>√3/√12, √3/√3, √12/√3
.5______1_______2 <------------------> .5____1_____2 <------------------> .5____1____2
-1______0______+1<------------------> -1____0____+1 <-----------------> -1____0____+1
Now, we can see the inverse nature of the opposite terms, and , when we square n to eliminate the radicals, and write the difference between numerator and denominator (i.e. the displacement), adding the polarity signs, we can see the dimensional progressions:
1d = -n
2, ...-3, -2, -1, 0, 1, 2, 3, ...n
2
2d = -n
2, ...-18, -8, -2, 0, 2, 8, 18, ...n
2
3d = -n
2, ...-27, -12, -3, 0, 3, 12, 27, ...n
2
One might argue that the squaring operation removes the negative signs, and that's true, but only because, as no less a light than Sir William Rowan Hamilton pointed out, the rules of algebra have been crafted as an art, not a science. He wanted to put algebra on as sound a footing as we find for geometry. He wrote in his essay on the
Science of Pure Time:
The thing aimed at, is to improve the Science, not the Art nor the Language of Algebra. The imperfections sought to be removed, are confusions of thought, and obscurities or errors of reasoning; not difficulties of application of an instrument, nor failures of symmetry in expression. And that confusions of thought, and errors of reasoning, still darken the beginnings of Algebra, is the earnest and just complaint of sober and thoughtful men, who in a spirit of love and honour have studied Algebraic Science, admiring, extending, and applying what has been already brought to light, and feeling all the beauty and consistence of many a remote deduction, from principles which yet remain obscure, and doubtful.
So it is here. The confusion of thought, which comes into play here is, as Hestenes has pointed out, that there are two interpretations of number, a quantitative interpretation and an operational one, where the first is a measure of how much or how many, and the second defines a relation between different quantities. It is the second, or operational interpretation, which Larson employs in his concept of "speed displacements" that we are using here. The difference between the denominator and the numerator can be either positive or negative, depending on which is larger, but that is an algebraic rule for convenience in tracking the "direction" of numbers, and it doesn't countervail the science of reciprocity.
In our case, we simply add the negative sign to the square of n, as a shorthand symbol of the scientific, reciprocal, "direction," rules of the art be damned. The important thing to understand here is that the scalar expansion of three dimensional numbers, as quantities (2x2x2), entails three "units," not just one.