The Larson Research Center

In the LRC seminar concluded two days ago, a question was asked in the virtual room about the chart of motions that we had been discussing in the meeting. I missed it, because I was saying goodbye to someone in the physical room at the time. However, the next day, listening to the recorded audio of the meeting, I finally heard it and realized that it was a very good question, and I think I will try to answer it here.

The question was, “Why do the base numbers in the rows of the chart of motion start with 1, while the column numbers in the chart start with 0? Shouldn’t the row numbers start with 0 too?” The answer is very simple, but also very enlightening. The reason the base numbers start with 1 is that the numbers of the chart aren’t arbitrary notations, but come from the mathematics of the tetraktys, the numerical expansion of dimension, which has the form:

* = 1
** = 2
*** = 3
**** = 4

Each successive row is a higher dimension, or power of numbers, starting with 0 power, which is always defined for any number as 1. However, zero is not a number and therefore can’t be raised to any power; that is, 0⁰, or 0¹, or 0², or 0³, or 0ⁿ is not a number.

On the other hand, 1 is a number. Regardless of the power we raise 1 to, it always remains 1. So 1⁰ = 1¹ = 1² = 1³ = 1ⁿ = 1. Yet, there is a spatial difference between a linear quantity of one, a square quantity of one, and a cube quantity of one, like the difference between a meter, a square meter, and a cubic meter, even though, numerically, they are all equal to 1.

Recognizing that this geometric difference, in the magnitudes of the powers of one, can have a spatial significance, leads us to conclude that it’s a magnitude difference, and since magnitudes of space don’t exist without time, we conclude that n-dimensional magnitudes are magnitudes of motion and nothing else.

It follows from this reasoning that we can generalize numbers and magnitudes, something that man has sought to do from ancient times, but has never been able to accomplish in a completely satisfactory manner. The invention of the imaginary and complex numbers, the modern approach to generalizing these two concepts, is a powerful method, but it ultimately runs out of gas in higher dimensions (or in successive rows of the tetraktys), in the form of hypercomplex numbers, because it’s an artificial, or contrived, procedure. We might call this the algebraic pathology of hypercomplex numbers.

It has long been recognized that plugging the number 2 into the tetraktys is a way to describe a sequence of algebras of real numbers, each with twice the dimension of the previous one. Producing algebras this way, known as the Cayley-Dickson construction, extends the complex numbers into hypercomplex numbers, the algebra of which is known as a Cayley-Dickson algebra. These algebras of complex and hypercomplex numbers conform to a concept of norm and conjugation, where the product of a number and the conjugate of the same number is equal to the norm of the number. The norm is simply a “size” of a vector, like the unit size of the radius in the unit circle of the complex plane, and adding the norm operation to a vector space produces a “normed vector space.” So the normed vector space used to define the U(1) Lie group, the unit circle on which we can define an infinite set of points and the complex numbers that correspond to these points, contains an infinite number of unique complex numbers that are all “size one” complex numbers, as we discussed in the seminar.

All this is just a complex way of saying that the radius of the unit circle is equal to the hypotenuse of a triangle in the circle, the sides of which are the numbers that, when plugged into the Pythagorean formula, produce a hypotenuse equal to 1. The set of the combination of numbers that will work is infinite, but it is also true that it is the operation of division that makes this possible, and, as it turns out, this operation is not possible, if we try to define more than three of these complicated vector spaces; that is, the complex numbers, forming one complex dimension (the first “normed vector space”) takes two geometric dimensions (2¹ = 2) and two numbers (z=a + bi); incrementing the complex dimension from one to two dimensions (forming the second “normed vector space”) requires two geometric planes of two dimensions (2² = 4), or four numbers (z=a+bi and z’=c+di); and one would think that incrementing the complex dimensions from two to three dimensions (forming the third “normed vector space”) requires three geometric planes of two dimensions, or six numbers (z=a+bi, z’=c+di and z”=e+fi).

However, here’s where the algebraic trouble arises in spades, because as we increase the complex dimensions to these hypercomplex numbers, the algebra of the higher normed vector space, which is so useful at one complex dimension, becomes pathological in the second and third instance. Indeed, it really becomes acute in the third such space, because for one thing, 2³ = 8, not 6. So, we really can’t use the third space as such, but have to concoct the third space out of two of the second spaces, but then this leads to other complications, and so it goes.

However, now, in light of the chart of motion, it’s easy to see the problem that is being encountered here and its solution. Instead of using the ad hoc invention of the imaginary and complex numbers to define normed vector spaces so that we can treat vectors as scalars, which have no algebraic pathology, we can, instead, recognize, following Grassmann and Clifford, that there are two interpretations of number, the quantitative and the operational interpretation.

Using the operational interpretation of number, we can define higher dimensional numbers without resorting to the concept of vectors, or vector spaces. In other words, employing the operational interpretation of number, we can obtain linear spaces of n-dimensions and the algebra of the elements in these spaces is not pathological! We do this by simply recognizing the symmetry of the tetraktys and that we can describe four dimensions of four numbers that are sixteen fundamental magnitudes. The first set of four magnitudes is:

M₁⁰ = 1⁰ = 1 potential point
M₁¹ = 1¹ = 1 potential line
M₁² = 1² = 1 potential area
M₁³ = 1³ = 1 potential volume

The second set of four magnitudes is:

M₂⁰ = 2⁰ = 1 vector point
M₂¹ = 2¹ = 1 vector line
M₂² = 2² = 1 vector area
M₂³ = 2³ = 1 vector volume

The third set of four magnitudes is:

M₃⁰ = 3⁰ = 1 interval point
M₃¹ = 3¹ = 1 interval line
M₃² = 3² = 1 interval area
M₃³ = 3³ = 1 interval volume

Finally, the fourth set of fundamental magnitudes is:

M₄⁰ = 4⁰ = 1 scalar point
M₄¹ = 4¹ = 1 scalar line
M₄² = 4² = 1 scalar area
M₄³ = 4³ = 1 scalar volume

Thus, now we can see that the “size one” of the normed vector spaces, which enables legacy physicists to do marvelous things with complex numbers, but eventually defeats them in the algebraic pathology introduced by the restriction to linear “vector” spaces that the concept of vector entails, when they are constructed with complex numbers, is easily accomplished without restricting ourselves to vectors.

In this new view, the n-dimensional “size one” magnitudes are composed of the four numbers in the tetraktys, 1, 2, 3, and 4. The “number” zero, in this number system is conspicuous by its absence, but it easily explained by our operational interpretation of number, because:

1/n, …,1/2, 1/1, 2/1, …n/1,

is equivalent to:

-n, …, -1, 0, 1, …, n.

This means that we don’t need to restrict ourselves to the infinitude between 1 and -1, which we have been availing ourselves of by means of the concept of rotation, which unites the vector concept with the interval concept of magnitude, in its complex form, but only to a limited extent.  Instead of proceeding along these lines, we can now examine n-dimensional magnitudes directly, in their natural forms.

However, how we might do this is another chapter in the continuing saga. The important point I want to make in this chapter is that the numbers 1, 2, 3, and 4, of the chart of motion magnitudes, are not arbitrarily chosen for purposes of notation, but are mathematically derived, fundamental, magnitudes of reality.

Selah.