My apologies for not replying in so long. I haven't been able to keep up with the board, due to work scheduling. Hopefully next week, I'll have some more time to catch up and try to wrap my brain around this method.

__________________
"Reality is that which, when you stop believing in it, doesn't go away."
Philip K. Dick, Do Androids Dream of Electric Sheep?
"A lie can travel half way around the world while the truth is putting on its shoes."
Mark Twain

Avatar courtesy of Bunny.

Well, I'm back, after almost a year of being a away. I had to drop my contributions to this thread due to time constraints. A lot has happened in the past year, but Mike525 recently initiated a thread on the RST perspective of planet formation, which raised the subject of the RST in general, and offered me a chance to enter the dialog again.

However, to avoid hijacking his discussion, I thought it best to return to this thead, where we can discuss the RST in general. In his latest post to Mike525's thread, antoniseb replied to my post over there with the following request:

The above response was prompted in part, because, in answering an earlier request to "show how simple physics is expressed using Larson's ideas," I explained that it is important to recognize that Larson's theories are developed under a new system of physical theory, not Newton's system, where everything depends upon the function x(t) and its derivatives.

The difference is that, while the fields of modern physical science, from particle physics to cosmology, are based on the familiar concepts of the vectorial motion of objects, the new system is based on scalar concepts of motion defined without objects, and these ideas are new and unfamiliar.

However, while there is an important connection between these two concepts of motion, generally speaking the laws of physics, involving the motion of objects measured in terms of fixed spatial reference systems, and the familiar transformations of these, are not affected by the advent of the new system.

The domain of the new system is within the entities of radiation, matter, and energy, dealing with their inherent properties of mass, charge, spin, etc., which give rise to the effects of these properties, when scalar and vectorial motion between aggregates of matter exists, which sets the stage for the emerging concepts of force, acceleration, and momentum.

In the new scalar system, therefore, force is understood as a property of motion, not something that can exist autonomously, but this is not so in the vectorial system. The grand goal of Newton's program of research is to explain nature through the classification of a few elementary particles, by focusing on a few "fundamental" forces of interaction between them.

The mathematical and geometrical concepts used to clothe the concepts of vectorial motion, in a suitable formalism useful for conducting vectorial science, are naturally vectorial concepts, and since vectorial motion is the only motion recognized by the physicists and mathematicians that developed these mathematical and geometrical concepts, we tend to think that these concepts are all that exists, that the existing notions of mathematics and geometry, which pertain to vectorial motion, are concepts regarded as "the best of all possible worlds."

Well, Larson's works are changing all that, because the change of context that antoniseb inquires about takes us from a context in which the frame of space and time are paramount, into a new context where the frame of motion is paramount, motion defined in terms of space and time, but where they have no significance apart from their reciprocal relation in the equation of motion.

This means that the distance between the objects in a set of objects, which defines the concept of "space" in one, two, and three dimensions, is really only an emergent concept, the meaning of which is found solely in the history of the motion of the objects that separates them.

It's hard to overemphasize the impact that the redefinition of the traditional concept of space has on physics, and I'm sorely tempted to begin pointing out some of the more salient implications at this point, but I can't, if I want to limit the size of this post to any reasonable length.

If you read the previous posts in this thread, you will see that we were discussing Clifford algebras and Hestenes' Geometric Algebra (GA), and how Hestenes found that there is a surprising geometric interpretation of numbers that can be exploited to form an amazing algebra, not based on the use of imaginary numbers to form the one-dimensional complex numbers, or the two-dimensional quaternions, but rather based on the use of a new definition of vector operations, called the "geometric product," to form the three-dimensional octonions of the Cl3 Clifford algebra!

According to John Baez, the octonion numbers are the "crazy uncles kept in the attic" by traditional mathematicians (with some intriguing exceptions), because, though they are one of only four known normed division algebras, they are neither ordered, commutative, or associative, making them of little use to physicists, who have stuck with using complex numbers, the essential language of quantum physics.

However, Hestenes has shown that, with a geometric interpretation, these numbers form a set of four n-dimensional numbers, called blades, that can be used to great effect in the equations of theoretical physics, expressed as multivectors that contain one or more n-dimensional blades. The blades correspond to the four dimensions of the octonions (recall the binomial triangle at the fourth (2^3) line, explained in the previous posts of this thread), and constitute four grades of numbers:

• 0D number blades (2^0 = 1 grade, or scalar numbers)

• 1D number blades (2^1 = 2 grade, or vector numbers)

• 2D number blades (2^2 = 4 grade, or bivector numbers)

• 3D number blades (2^3 = 8 grade, or trivector numbers)

Therefore, an equation consisting of a multivector can contain all four of these blades in one! The utility of using this algebra is shown by Pezzaglia Jr, in his paper, Clifford Algebra Derivation of the Characteristic Hypersurfaces of Maxwell's Equations, where he combines all four of Maxwell's equations into one multivector, with each of the four blades yielding a fundamental law, or concept, of vectorial physics:

• The scalar blade yields Gauss's Law

• The vector blade yields Ampere's Law

• The bivector blade yields Faraday's Law

• The bivector blade yields the magnetic monopole

However, many mathematicians and physicists have trouble accepting the definition of the geometric product, because it combines the scalar of the inner product of vector multiplication with a special form of the outer product of vectors. How do you combine vectors with scalars?

Hestenes' GA shows how it can be done and in the process reveals the underlying connection between the numbers of algebra and the magnitudes of geometry, but it also reveals a fascinating drama in the evolution of mathematics as the language of physics.

In order to answer antoniseb's question, "Is there some way to apply Larson's concepts in a way that can be expressed mathematically?" I have to first set the context by delving into this mathematical drama, but I can promise you, if you will bear with me, it will be well worth the wait.

Excal

Thanks ExCal. I look forward to seeing the next installment.

__________________
Forming opinions as we speak

What Hestenes did was to take what is known as the operational interpretation of number, first recognized by Clifford, as opposed to the usual quantitative interpretation of number, and he used it to reinterpret the physical meaning of the imaginary number 'i'.

As I think we discussed last year, Michael Atiyah has characterized the imaginary number as the "biggest single invention of the human mind." He (and almost everyone else) is amazed by the fact that a pure invention, originally conceived to explain the negative roots of quadratic equations, ends up being essential to describe nature in quantum mechanics. It's just part of a great mystery as to how this concept could be essential in the appropriate language for physics, if mathematics is no more than a formalism, which can have little to do with a common underlying reality shared with physics.

Wigner referred to the "unreasonable effectiveness of mathematics in physics," as a "gift we neither understand nor deserve," but Hestenes sees the advent of the imaginary number as part of a deliberate attempt, lasting centuries, to generalize the concept of number to the point that algebra could be used to express the n-dimensional concepts of geometry.

In the final analysis, it is the operational interpretation of an imaginary number that makes Hestenes achievement possible. As he writes in his New Foundations for Classical Mechanics,

The way the two approaches are brought together in the geometric product is through the use of orthogonality. In the geometric product, orthogonality is expressed in the duality of the inner and outer vector product, which is equivalent to a certain rotation through an angle, as well as the orthogonal directions of a vector in a plane. In this interpretation, since i^2 is -1, which represents a 180 degree rotation through the complex plane from 1 to -1, then a 90 degree rotation is represented by i^1, a 270 degree rotation is represented by i^3, and i^4 brings us back to the beginning, or to i^0, on the unit circle.

However, GA does more than provide us with a neat package for unifying the concepts of the vectors and spinors in one language. As impressive an achievement as this is, it is crucial to recognize that, in the process of giving geometric meaning to the ad hoc invention of 'i', GA also unifies the Clifford algebra, Cl3, with the concepts of vectorial motion and Euclidean geometry, in that it permits numbers to represent n-dimensional magnitudes with direction.

This long-sought generalization of number unifies the binomial expansion of n-dimensional numbers, the associated Clifford algebras, and the elements of Euclidean geometry, when it is realized that the expansion of the numbers in the associated algebra are isomorphic to the expansion of a zero-dimensional point into a one-dimensional line, a one-dimensional line into a two-dimensional plane, and a two-dimensional plane into a three-dimensional cube.

As the expansion grows, from 0 to 3 dimensions, the number of elements in the algebra grows as the number of elements in a cube; that is, at 2^0 = 1, we have only the concept of points in the geometry and only scalar numbers in the algebra, but at 2^1 = 2, we have both the concept of points and lines in the geometry, and scalar and vector numbers in the algebra. At 2^2 = 4, we have the point concept, two line concepts (orthogonal lines), and the concept of the plane in geometry, and scalar, vector, and bivector numbers in the algebra.

Finally, at the 2^3 = 8 dimensions, we find the culmination of all these in the maximum dimensions of the geometric cube: There are the concepts of points, lines, planes, and cubes in 3D geometry, and the corresponding concepts of scalar, vector, bivector, and trivector numbers in the 3D algebra. Moreover, thanks to the operational interpretation of number and Hestenes work, the scalar, vector, bivector, and trivector numbers of the algebra can also be thought of as scalar magnitudes, linear magnitudes, quadratic magnitudes, and octonic magnitudes, generated by rotation, i.e. vectorial motion.

Thus, Hestenes has unified the concepts of vectorial motion, mathematics, and geometry, in his GA. However, the significance of this achievement is not just academic, but has huge ramifications in physics that I don't believe the mainstream physics community fully appreciates, at this point in time, preoccupied as they are with the string theory controversy, and the fundamental crisis in theoretical physics that gave rise to string theory, but which it seems only to be able to exacerbate.

Nevertheless, the mathematical and geometrical concepts clarified by GA, illuminate the central problem at the heart of the current crisis in theoretical physics: the issue of reconciling the concept of the continuum, upon which the fields of the general relativity theory of gravity are based, and the concept of the quantum, upon which the fields of the standard model theory of particle physics are based. The essential clue that GA provides is that while there is a fundamental disconnect (no pun intended) between the two concepts, they both suffer from the same ailment: the dreaded singularities, or infinities caused by the numerical/physical concept of zero.

To see how GA relates to the problem of infinities requires us to now turn our attention from the subject of mathematics to the subject of physics. In the previous posts of this thread last year, we discussed the ideas in Lee Smolin's paper on background independence, where he identifies five major, unsolved, problems in theoretical physics, and how these are related to the famous debate between Newton and Leibnitz over the nature of space, which is reincarnated in the argument for the background dependence of quantum physics.

Since then, Smolin has published a book entitled, The Trouble With Physics: The Rise of String Theory, the Fall of a Science, and What Comes Next. In the first part of the book, he again reviews the five unsolved problems of theoretical physics and in this connection writes:

referring to the revolution of the last century that started with the discovery of the quantum. This revolution is unfinished, according to Smolin, in contrast to the previous revolution that started when the continuum secret of the Pythagoreans' square root of 2 got out in the open, and finally culminated with Newton's overthrow of the Aristotelian theories of space, time, motion and cosmology that competed with the ideas of the Copernican revolution.

The challenge of reconciling these two concepts, one discrete and the other continuous, did not seem unsurmountable in the early days of the unifinished revolution, but today it is more intractable than ever. The problem is not that we don't have theories that work, but that the theories we have work too well, yet are incompatible. Smolin writes:

However, though this is the reason most cited for why we need a unified theory, it is not the only reason. There is more to it than that, and it has to do with the infinities that plague both the continuum based theory of gravity and the quantum based theory of particle physics. Regarding the former, Smolin writes:

Of course, this has lead to the effort to find a theory of quantum gravity, but the effort has been unsuccessful to date, and though quantum physics dealt with its own problems of infinities long ago, through the suspicious procedure of "renormalization," that has effectively swept them under the rug, it is hoped that a unified theory can be found that will include gravity and, at the same time, tame the infinities of quantum theory without recourse to renormalization. Smolin continues:

Clearly, the problem of infinities is related to the continuum. Given Zeno's paradox, there is no limit to how finely a continuum can be subdivided, whether it's the continuum constituting 1D linear space, or it's the continuum of a 3D volume of space. The reason string theory has been developed and highly touted is precisely because the nature of a string excludes the concept of zero, which corresponds to the concept of a point.

If particle physics can be described without recourse to the concept of point particles, or the infinite variables of a field, and the warping of a spacetime continuum, required to describe gravity, can be accomplished by the same means, then that way out of the impasse seems very promising.

However, the problem with the string theory approach brings us back again to the nature of space and time, but, in this case, the plague of the infinities of the continuum attack again, albeit from a different angle: string theory requires 10 dimensions of space, but there are only four dimensions of spacetime in current theory, leaving six dimensions that have to come from somewhere else. The thought is that these extra dimensions could be "compactified," but there are essentially an infinite number of ways to compactify them!

Thus, we have a fundamental crisis in theoretical physics on our hands that is being caused by the fact that the dual of the point is infinity, and though nature knows a way to deal with this fact, evidently, we don't.

In the meantime, Hestenes has found a way to unify the concepts of vectorial motion, algebra, and geometry. Could it be that this is a prerequisite to unifying the laws of physics? We'll discuss it next time.

Just as a reminder, this is the third post, which constitutes the latest part of my answer to antoniseb's question:

In the first post, I explained that the scalar system is different than the vector system and that this difference is reflected in the mathematical and geometrical concepts developed to apply the system in a program of scientific research. In the vectorial system, motion is measured in terms of an object's change in locations, and locations are separated by distances in terms of magnitude and direction, the "space" between objects.

However, we see that the RST redefines space, as simply the reciprocal quantity of time in the equation of motion. Consequently, with this redefinition of space, the "space" between objects has no independent meaning; that is, it is emergent. Nevertheless, it is this "space," defined by a set of points, that satisfies the postulates of geometry, and forms the basis for vectorial motion. Newton referred to this, when he wrote that geometry has nothing to say about how "right lines and circles" are drawn. These matters are outside the domain of geometry, and must come from mechanics, or the vectorial motion of objects.

So, we turned to the subject of how mathematics has developed historically, as a language of physics, in terms of the effort to generalize the concept of number from a limited concept of counting, to a more general concept of magnitude, the magnitudes of geometry, which magnitudes are points, lines, areas, and volumes.

In the second post, we considered how numbers were provided with the geometric property of direction, by means of the ad hoc invention of imaginary numbers, and how this approach was considerably improved by Hestenes, who was able to show that, by exploiting the operational interpretation of number, it's possible to define a direction property of numbers, without recourse to the ad hoc invention of imaginary numbers.

We discussed how the consequences of Hestenes' achievement are deeply significant, that his achievement constitutes more than an efficient method for writing equations, that it actually unifies the concepts of vectorial motion, algebra, and geometry in three dimensions for the first time. Then, we discussed the conclusion that this unification of the concepts of motion, numbers and geometry goes right to the heart of the modern crisis in theoretical physics, the challenge of reconciling the dual nature of the physical structure of the universe, the nature of the continuum and the nature of the quantum.

The basis of this conclusion is what we will discuss in this post. It has to do with the infinities that plague current theory, and the observation that, while these infinities are obviously the property of the continuum, the infinite values of magnitudes in the continuum are necessarily associated with the concepts of vectorial motion. Of course, the implication is that, with the advent of scalar motion, the seemingly insurmountable problem with infinities in the equations of motion, automatically disappears.

When I first discovered Hestenes' GA, in the context of my study of scalar motion, I thought it might prove to be the basis for a new mathematical language useful for expressing Larson's scalar motion concepts mathematically. Unfortunately, this was not the case. Ironically, however, it was GA's clarification of the concepts of vectorial motion that actually enabled me to discover that there exists a scalar concept of n-dimensional numbers that enables us to express the concepts of scalar motion mathematically.

In discussing this discovery, it's important to recognize that the insights involved came to me in fits and starts, and many times they were not clear until viewed in retrospect, after pressing ahead with nothing but the assumptions of the RST to guide me. Thus, this is the newly minted coin of a new territory we will be discussing here. It is consistent with Larson's new system, but was completely unknown to Larson, since it was only recently developed, more than a decade after his decease.

The key that opened the door to n-dimensional scalar mathematics, which I call the Reciprocal System of Mathematics (RSM), for obvious reasons, was the same key that unlocked the door for Hestenes, enabling him to formulate the n-dimensional numbers, or k-blades, of his algebra, the combinations of which constitute the multivectors of GA. That key is the operational interpretation (OI) of number.

Hestenes used it to define the geometric product in terms of the inner and outer products of vectors. The geometric product is an amazing innovation, because it defines a numerical value in terms of the relation between the directions of vectors, and thus it makes it possible to construct an algebra of these numbers, which doesn't suffer from the problems of numbers constructed with the traditional combinations of real and imaginary numbers, the complex and quaternion numbers.

The significance of this new algebra, then, is that it is literally a three-dimensional algebra, whereas the algebra of complex numbers is one-dimensional, and the algebra of quaternions is two-dimensional, but the significance of this fact cannot be fully appreciated, until we recall that vectorial motion is always one-dimensional.

This fact sets the stage for the drama we are experiencing in mathematics and physics today, because, in order for us to conceive of one-dimensional motion, three-dimensional algebra, and three-dimensional geometry, coexisting in some notion of a unified structure, the n-dimensional numbers of the algebra and the n-dimensional "space" of the geometry, have to contain the one-dimensional motion.

When one-dimensional motion is contained by n-dimensional "spaces," it is possible to define the vectorial motion of objects directly with three-dimensional algebra. In contrast, the one-dimensional algebra of complex numbers cannot define vectorial motion directly, it can only define it indirectly, by defining the points that are necessary to describe the historical path of vectorial motion.

Thus, using the algebra of complex numbers, we can define the function x(t), as a set of points in the complex plane, and we can define the differentials and integrals associated with the curve of the function, plotted with the numbers of the algebra. In other words, the one-dimensional complex algebra exploits the infinite points of the continuum to enable the calculus, which is the foundation of the modern world's science and technology.

We can do the same thing with GA, but without having to explicitly define the points of the curve, because GA's multivector can represent rotation directly, in the form of a bivector. Nevertheless, doing this is sort of like using a wheel to measure distance; it's a great way to measure length in some ways, but it hardly characterizes the potential contribution of the wheel to technology.

Our ability to exploit the extended power of GA will be limited until we gain greater insight into its significance, but the fact remains, it is literally two orders of magnitude more advanced than the algebra of complex numbers. In its one-blades, we have 1D numbers equivalent to magnitudes of linear motion, with two directions (positive and negative). In its two-blades, we have 2D numbers equivalent to magnitudes of rotational motion, with two directions (clockwise and counterclockwise), and in its three-blades, we have 3D numbers equivalent to magnitudes of scalar motion, with two directions (in and out).

However, currently, as far as I know, the magnitudes of the bi-directional scalar motion of the three-blades are not recognized for what they are. The 3-blade is labeled a trivector, and its space magnitude is viewed as a volume, but the concept of this number's equivalence to an outward/inward magnitude of motion is not generally discussed. Clearly, however, GA's linear 1-blade numbers are capable of expressing 1D magnitudes of simple harmonic motion (SHM), in one-dimensional equations of motion. Likewise, its linear 2-blade numbers are capable of expressing 2D magnitudes of torsional SHM, or rotational vibration, in one-dimensional equations. Consequently, it's just one more small step for us to conclude that its 3-blade numbers are capable of expressing 3D magnitudes of SHM, in one-dimensional equations, though I don't know if there is a name for this motion per se.

Nevertheless, if we describe these three magnitudes in terms of directions, the 1-blade number is equivalent to magnitude in one specific direction within the three dimensions of a volume, the 2-blade number is equivalent to magnitude in all the directions of one specific plane, within the three dimensions of a volume, and the 3-blade number is equivalent to magnitude in all the directions, within the three dimensions of a volume. Hence, when the GA magnitudes are applied to the description of vectorial motion, the 1-blade motion is the familiar translation of an object, the 2-blade motion is the familiar rotation of an object, and the 3-blade motion is the familiar expansion of an object.

Of course, SHM is alternating motion in the two "directions" of each blade, where, by placing the word "directions" in quotes, we are referring to the duality of each blade; that is, the negative and positive "directions" of the 1-blade, the clockwise and counterclockwise "directions" of the 2-blade, and the outward and inward "directions" of the 3-blade, constitute the duality property of each blade.

Clearly, as the dimensions of the k-blades in GA is three, there are three independent 1-blade numbers (three 1D numbers with orthogonal directions, or three vectors), three independent 2-blade numbers (three 2D numbers with orthogonal directions, or three bivectors), but only one 0-blade number (one 0D number with no directions, or scalar), and one 3-blade number (one 3D number with all directions, or pseudoscalar).

Of course, the members in each set of numbers, or blades, follow from the inherent duality of the numbers characterized by the grades: Since the duality of the scalar, or 0-blade, has no direction, it is always characterized by 2^0 = 1 duality, while the positive/negative duality of 1-blades is two, so it is always characterized by 2^1 = 2 duality, and the duality of the 2-blade is four, because it is composed of two 1-blades, so it is always characterized by 2^2 = 4 duality, and the duality of the 3-blade is eight, because it is composed of three 1-blades, so it is always characterized by 2^3 = 8 duality.

Hence, it's this dimensional expansion of duality that gives GA its power, in effect, defining 1 + 3 + 3 + 1 = 8, mathematical dimensions, within the 0, 1, 2, and 3 dimensions of geometry. If the geometric dimensions are counted, there are four of them, and then their sum is 1 + 2 + 3 + 4 = 10, which is the sum of the 3D binomial expansion, and the spatial dimensions of string theory. Some think that this is just coincidental, but perhaps not. If it is not, then could the compaction of these eight mathematical dimensions into three geometric dimensions, be another, more successful approach to string theory? I believe it is, but, regardless, that fact is just a distraction at this point, because our goal is to get to a new scalar science, through a new definition of space, as the reciprocal of time. String theory is a concept of vectorial science, and therefore we don't want to go there, but it sure is tempting (maybe in another life!)

Nevertheless, this neat packaging of algebraic numbers, with geometric magnitudes, exhibiting an almost breathtaking symmetry of duality and revealing a deep and mystifying connection of mathematics, geometry and vectorial motion, suggests a common underlying reality that has always been suspected, but only teasingly revealed in modern science.

Could it be that the scalar motion concepts of Larson are also part of this new-found unity? Is it possible that discrete values of GA's n-dimensional numbers could be used to represent discrete values of n-dimensional magnitudes of scalar motion? Although Larson never heard of GA, and probably knew nothing of Clifford algebras and the symmetry of the expansion of duality, the binomial expansion, his development of the world's first general theory of the universe, using his scalar system of physical theory, came very close to answering this question, breathtakingly close.

(to be continued)

Larson proceeded to develop his scalar motion theory from the basic postulates of the RST, without the benefit of Hestenes' work. Therefore, the concepts of duality and dimension, and how they are related to each other, as revealed in the binomial expansion, and the concepts of Clifford algebra that encapsulate these ideas in terms of n-dimensional numbers, though they were certainly available to informed specialists in the field of mathematics, during Larson's time, they were scarcely visible to many uninformed specialists, let alone non-specialists. They didn't become generally known until Hestenes' efforts to use them as the basis for a new, unified, language of physics, began to gather steam in the eighties and nineties.

Today, we know that the purpose of GA, according to Hestenes, is to facilitate physics educational and research efforts. In delivering his 2002 Oersted Medal Lecture, "Reforming the Mathematical Language of Physics," Hestenes told the audience:

Then he listed five features of the new language:

1. GA seamlessly integrates the properties of vectors and complex numbers to enable a completely coordinate-free treatment of 2D physics.
   
2. GA articulates seamlessly with standard vector algebra to enable easy contact with standard literature and mathematical methods.
   
3. GA Reduces "grad, div, curl and all that" to a single vector derivative that, among other things, combines the standard set of four Maxwell equations into a single equation and provides new methods to solve it.
   
4. The GA formulation of spinors facilitates the treatment of rotations and rotational dynamics in both classical and quantum mechanics without coordinates or matrices.
   
5. GA provides fresh insights into the geometric structure of quantum mechanics with implications for its physical interpretation. All of this generalizes smoothly to a completely coordinate-free language for spacetime physics and general relativity to be introduced in subsequent papers.

However, what few people knew, or could have even cared about at the time, was how the language that simplifies and clarifies the structure of vectorial physics, would also illuminate the work of Larson eventually. Indeed, its contribution to the science of scalar physics, may eventually prove to be more valuable to science in general than its contribution to the science of vector physics, since physicists and educators, well trained in the traditional language of vector and complex algebra, are relunctant to replace the known and familiar language of physics, with an unknown and unfamiliar one.

It may seem ironic that a new system of vector algebra, designed to improve the existing mathematical language of the mature science of vector physics, ends up providing a new mathematical language for the immature science of scalar physics. Yet, it is easy to understand it, when one looks at GA, not as a geometric algebra, but as a multi-dimensional number system that contains, as an integrated whole, advanced concepts of number that correspond to both scalar and vector magnitudes.

In starting our discussion of this expanded view of GA, it's important to recognize that there is a pleasing, but mysterious, symmetry in the grouping of its n-dimensional basis numbers: 1 + 3 + 3 + 1 = 8, which seem to pair vectors and bivectors (3+3), and scalars and trivectors (1+1). When we interpret this mathematical symmetry geometrically, as the 2x2x2 cube, with the scalar basis point at the intersection of the three orthogonal basis vectors and the three orthogonal basis bivectors, a point which coincides with the center of the eight corners of the pseudoscalar basis trivector, the physical beauty in the symmetry of its form, the conceptual beauty in the symmetry of its algebra, and the intellectual beauty in its complete and flawless union of the meanings of direction, duality, and dimension, is simply startling in its power to make one marvel.



Figure 1. Larson's Cube

I've dubbed it "Larson's Cube," because he initially used it to illustrate the eight units of three-dimensional scalar motion in the RST, but, now, when I look at it, it's as if I see the physical, mathematical, and intellectual structure of the entire universe contained in its simple, three-dimensional, figure. It seems to me to contain the great secret of nature's unity, as it were, the secret to how she unifies the continuum and the quantum, the secret that is the Holy Grail of theoretical physics.

Of course, I can't prove these things yet, but I'm so convinced of their veracity that I have organized and incorporated the "Dewey B. Larson Memorial Research Center," or "Larson Research Center," for short, to employ the mathematicians, physicists, and other professionals who can perform the scientific work necessary to prove it, to the extent that it may be possible to do so.

The reason that I see so much beauty in this cube is that it seems to contain every element that is necessary for a theory of everything, but nothing more. It integrates the concepts of motion, mathematics, and geometry into one graphic, which can be viewed as a single expression of that motion, which in the RST, we assume is the one component of which the universe consists. To see it expressed in one graphic, is tantamount to looking at the very foundation of the universe, in all its awesome power.

Clearly, however, I'm getting ahead of myself. To understand the power of Larson's Cube, we have to understand how it integrates the science of vectorial motion, with the science of scalar motion, and it all begins with the power of one, but not the quantitative interpretation of the number one, but the operational interpretation (OI) of the rational numbers as one.

As the ancient Greeks understood it, the number one (monad) represents eternity, and they probably learned it from the Egyptians, who in turn probably learned it from the Hebrews. In our OI number one, 1/1, we can understand this as signifying infinity/infinity is equal to 1/1; in other words, the number one contains all the numbers, 1/1, 2/2, 3/3, ...n/n, where n goes to infinity.

Normally, this fact is regarded as trivial, until, that is, we realize that this unity, apeiron, to the ancient Greeks, is a principle of reality that subsumes (i.e. contains, encompasses) peiron, or that which is limited, bounded, finite. These are the notions of continuum and quantum, the very notions that are plaguing theoretical physics. Larson's RST necessarily begins with this unity, which he calls the "unit progression" of space and time, as the initial condition of the universe of motion, because he defines space as the reciprocal of time, as motion, and motion as the sole component of the universe.

However, in describing the ancient Greek idea of this, we read in Wikipedia:

Larson never spoke of ancient Greek philosophy in this connection, as far as I know, but came to his conclusions based on his work to determine the inter-atomic distances in elements, in which he found that, only if he assumed a reciprocal relation between space and time, would his calculations work out:

However, Larson's initial reaction to this thought was negative, probably because he was thinking of space in terms of the distance that separates objects:

Intrigued, he pursued the idea:

In just this manner, his work began to bear fruit. One of the first things Larson had to do was to understand how, given this idea of the unit value in a universal motion in which space progressed as the reciprocal of time, deviations from this unit value could occur. In other words, he needed to know how n units of space, or n units of time, could come to be associated with one unit of its reciprocal aspect, so that a deviation from the unit, 1/1, condition occurred. Without such a deviation, where ds/dt = 1/n or n/1, the universe of motion is a great void, the perfection of nothing.

According to the Wikipedia article, this same challence faced the ancients:

Again, however, while Larson’s approach is much less philosophical, it amounts to the same principle in the end:

Hence, just as the ancients insisted, the play of apeiron and peiron commences according to principles of harmonia.

(to be continued)

Larson gave us very few diagrams to illustrate the principles of scalar motion, but in the previous posts, I introduced an algorithm, called the progression algorithm (PA), that produces a visual output that is very helpful in studying the scalar space/time progression ratio. The unit progression PA is



Figure 1. Unit Space/Time Progression PA

The unit progression is the datum, the point of reference for the scalar motion of the universe of motion, the apeiron, the perfect symmetry of the Reciprocal System's initial state, the space/time ratio equal to a 1/1 space/time progression. We call it the natural reference system of motion, the physical "zero" of the system, from which all physical activity is measured in the RST, and we use the PA in figure 1 above to generate this state of scalar motion as the unit progression ratio of space/time.

In the previous posts of this thread, I've explained how the symmetry of this initial state is spontaneously broken by "direction" reversals in the uniform progression of the space or time aspects of the motion, producing a second state of the system that deviates from the natural reference system by one unit in one of two possible "directions," one of which is designated "negative" and the other "positive." The PAs in figures 2 and 3 below generate these two progression ratios.



Figure 2. The Unit Time Displacement PA



Figure 3. The Unit Space Displacement PA

These two units of magnitude exist on either side of the unit ratio, and, taken together, the three of them correspond exactly to the initial three numbers of the integer number system, generated by the operational interpretation of the magnitude of rational numbers. In fact, we can easily summarize all this information with just a few numbers. The three numbers:

1/2, 1/1, 2/1,

which, operationally interpreted, are equivalent to three integers

-1, 0, +1,

because the change in the rates of progression of space and time, caused by the "direction" reversals in one aspect or the other, confines the progression of the oscillating aspect of the progression to be confined to one unit. In scalar physics, then we have a process, "direction" reversals, that, given a unit space/time progression, ds/dt = 1/1, producees two new ratios, ds/dt = 1/2 and ds/dt = 2/1, which constitute discrete, unit, magnitudes of oscillations, or motion, the peiron, emerging from the apeiron.

Now, one of the first things we want to know is, "What can we do with these numbers?" We want to know if we can add, subtract, multiply, divide, raise their powers, and extract their roots, because, if we can, then we have an algebra of scalar motion, and it will go a long way towards helping us develop a scalar science around the scalar system.

Larson was asked, many times, if there were a new mathematical formalism to go with his new system, and he insisted that there wasn't, because there wasn't any need for one, that the primary contribution of the RST is a clarification of the concepts of motion, not an addition to the already vast field of mathematics.

However, we now see the mathematics of a new system of numbers emerging from the mathematical equivalent of the RST assumptions. Whereas, in the RST, we assume that one component, motion, with two reciprocal aspects, space and time, is the initial condition of the physical universe, in the RSM, we assume that one number, an operationally interpreted ratio, with two reciprocal aspects, numerator and denominator, is the initial number of the mathematical universe.

Given this operational interpretation of the ratio of integers, as a signed integer itself, several things happen. First, we remove the necessity of qualifying zero as a number, which of course, it isn't. Zero is an important concept in its own right, to be sure, but it isn't a number. At least now it isn't. Second, we have no need to use signs with OI numbers, where the number itself is unique, and unambiguous. Third, we have no concept of orthogonality. The dimensions of OI numbers cannot include the notion of multiplication in the sense of magnitudes that are the product of two, orthogonal magnitudes of direction, because OI numbers are scalar numbers, and scalar numbers do not have direction, or othogonality.

However, given their equivalence to integers, we already know a lot about how they constitute a mathematical field and all, because they are isomorphic to the integers; that is,

1/n, ..., 1/2, 1/1, 2/1, ..., n/1,

is isomorphic to

-x, ..., -1, 0, 1, ..., x,

so anything we know about the integers, holds for the OI numbers, as well.

Therefore, if we add 1/2 = -1, to 2/4 = -2, we get 3/6 = -3, and, in general, m/n + m/n = 2m/2n, and so on.

Nevertheless, there is a big difference when we add equal OI numbers on opposite sides of unity, because not only are they equal in magnitude, but they are inverses as well. Therefore, like equal weights on the opposite sides of a pan balance, they balance each other, they do not eliminate one another. For example, a one pound weight on one side of a balance exactly offsets an equal weight on the opposite side, but together they weigh two pounds. In contrast, with integer numbers, a negative number cancels an equal positive number, requiring the concept of zero, the bane of modern theoretical physics.

Hence, we when we add 1/2 + 2/1, we get 3/3 = 1/1, not zero, but the operation is isomorphic to -1 + 1 = 0. It's just a different animal. Now, since n/n = 1/1, then

2/2 + 2/2 = 4/4 = 1/1,

a balanced OI number. Nevertheless, in the case of the ds/dt = 1/2 case of the time displaced PA, for every two units of time progression there are still two units of space progression, the difference being that one of these two space units is an increase (arrow points to the left), and one is a decrease (arrow points to the right). Thus, if we add the respective numbers of the two unit displaced PAs, we get

(1/2 + 2/1) = 3/3,

because we have not counted the decreasing space unit (arrow to the right) on the time displaced PA, nor the decreasing time unit (arrow to the left) on the space displaced PA. Adding a term to account for these units, gives us

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

where "nm" is the unit of "natural motion." This equation is a fundamental equation of the RSM, and is called the first, reciprocal, number (RN) of the RSM. Again, given that we are assuming that the OI number system is isomorphic to the integer number system, the RN, because it is a sum of OI numbers, should also be an integer, and it is, though 4/4 = 1/1.

However, if we square (1/1), we get (1/1), but if we square (4/4), we get (16/16), and if we cube (4/4), we get (64/64), and even though 4/4 = 16/16 = 64/64 = 1/1, 4/4 units of motion squared is

(1/2 + 1/1 + 2/4)² = (4/8 + 4/4 + 8/4) = 16/16 nm,

and

(1/2 + 1/1 + 2/4)³ = (16/32 + 16/16 + 32/16) = 64/64 nm

a completely different result.

There's much more to say about OI numbers, but I'll have to wait until next time to continue the discussion, including how we get the important results above.

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 1s are more than 1, but 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 dipicted 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, and 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 how they relate to each other and photons, in 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 find 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.

Excal

Hmmm. I'm not sure how to interpret the silence. I know this is primarily an astrophysics/cosmology forum, but the science of astrophysics and cosmology rests on the foundation of particle physics. The whole idea of big bang, cosmic inflation, nucleosynthesis, etc, is developed on the basis of the standard model and general relativity theories, so these fundamental sorts of things are hopefully seen as vital here. Again, the trouble with physics right now is that these two theories must be treated separately, as if the quantum and the continuum were unrelated, though we know that they are not, and this conflict emerges most clearly in today's concepts of cosmology and astrophysics.

However, to make my point a little more explicit and to explain things a little further, I'll refer to John Baez's characterization of Geoffrey Dixion's work in Week 59 of his online mathematics tutorials:

Baez admits that the answer is, yes, these very "special structures" do play important roles in physics, all but the octonians. These are the "crazy uncles" of the math world that are "kept in the attic." Yet, he writes:

Then, in response to questions arising with respect to applications of octonions, Baez wrote:

Which is just to make the point, as I wrote previously, that the world of theoretical physics converges on the structure of these algebras, or hypercomplex numbers, as they were once called. The physical part of this focus has to do with group theory and especially the representations of the groups U(1), SU(2), and SU(3), which form the basis of the standard model of particle physics. However, the mathematical focus is on the abstract algebraic structure itself.

The main mathematical challenge of this structure is best described by the loss of its elements' properties, as the dimension of the hypercomplex numbers increases; that is, in the words of one commentator on Baez's site:

Where the "hexadecanions" are the next dimension (2⁴ = 16) up from the octonions, in the binomial expansion. Yet, as Baez points out, there is a mysterious connection between these higher dimensional numbers and the octionions, found in the SU(5) GUT, and in the 10 dimensions of string theory, expressed by the Bott periodicity theorem.

In Week 211, Baez writes:

Then he gets into the details of describing Bott periodicity in terms of rotation groups in the form of Clifford algebras, which is very interesting and informative, but I can't possibly do it justice in this post. Instead, I will go back to his Week 105 and quote what he says there:

(continued in following post)

(continued from previous post)

What I have done, in answering antoniseb's question, seeking to understand the context in which Larson's RST is applicable to physics, is to show that a new, operationally interpreted, view of the numbers in this hypercomplex structure, reveals that there is an astonishing simplicity existing in it, when it is expressed in terms of "non-rotatable" (i.e. non-vector), but nevertheless n-dimensional, scalar numbers, called "Reciprocal Numbers" (RNs).

The power of the RNs should be clearly manifest in this, because, on the one hand, Baez asserts that, referring to the perplexing structure of hypercomplex numbers where n < 3, "You can classify all these things," but when n is equal to, or greater than, 3, then the classification breaks down, because "3-dimensional manifolds are a lot more complicated: nobody knows how to classify them; 4-dimensional manifolds are a lot more complicated: you can prove that it's impossible to classify them - that's called Markov's Theorem."

Yet on the other hand, we can show that they are easily classified, in the new system of reciprocal numbers. The first four dimensions (counting the zero dimension) are classified by the first four RNs:

1. 2⁰ (1/1) = 1/1
2. 2¹ (1/2+1/1+2/1) = 4/4
3. 2² (4/8+4/4+8/4) = 16/16
4. 2³ (16/32+16/16+32/16) = 64/64

where

1. = RN⁰
2. = RN¹
3. = RN²
4. = RN³

The next four dimensions are then:

1. 2⁴ (1/1) = 256/256
2. 2⁵ (256/512 + 256/256 + 512/256) = 1024/1024
3. 2⁶ (1024/2048 + 1024/1024 + 2048/1024) = 4096/4096
4. 2⁷ (4096/8192 + 4096/4096 + 8192/4096) = 16384/16384

where

1. = RN⁴
2. = RN⁵
3. = RN⁶
4. = RN⁷

and so on, up the ladder of dimensions, ad infinitum. We can see the structure more clearly when the ascending powers of the RNs are expanded in terms of the powers of the first RN:

1. = (1/2+1/1+2/1)⁰ = 1/1
2. = (1/2+1/1+2/1)¹ = 4/4
3. = (1/2+1/1+2/1)² = 16/16
4. = (1/2+1/1+2/1)³ = 64/64
5. = (1/2+1/1+2/1)⁴ = 256/256
6. = (1/2+1/1+2/1)⁵ = 1024/1024
7. = (1/2+1/1+2/1)⁶ = 4048/4048
8. = (1/2+1/1+2/1)⁷ = 16384/16384

The reason for the mysterious connection of Bott periodicity and the rotation groups that mathematicians work with in topology, is that, as Raul Bott proved, there are no new phenomena beyond three geometric dimensions in nature. However, as we can now see, that applies to the vectorial aspect of the structure only, it does not apply to the scalar aspect.

From the scalar point of view of n-dimensional numbers, the Bott periodicity theorem's assertion that there is a limit at three dimensions, after which things repeat, is clearly seen when we factor out the value of 2⁴ = 256 from the RNs in each group. This shows us that each 3D group is based on powers of 256:

256⁰

1. 2⁰ (1/1) = 1/1
2. 2¹ (1/2+1/1+2/1) = 4/4
3. 2² (4/8+4/4+8/4) = 16/16
4. 2³ (16/32+16/16+32/16) = 64/64

256¹

1. 2⁰ (1/1) = 1/1
2. 2¹ (1/2+1/1+2/1) = 4/4
3. 2² (4/8+4/4+8/4) = 16/16
4. 2³ (16/32+16/16+32/16) = 64/64

256²

1. 2⁰ (1/1) = 1/1
2. 2¹ (1/2+1/1+2/1) = 4/4
3. 2² (4/8+4/4+8/4) = 16/16
4. 2³ (16/32+16/16+32/16) = 64/64

.
.
.
256ⁿ

1. 2⁰ (1/1) = 1/1
2. 2¹ (1/2+1/1+2/1) = 4/4
3. 2² (4/8+4/4+8/4) = 16/16
4. 2³ (16/32+16/16+32/16) = 64/64

which is why Bott's period 8 periodicity theorem holds.

It is the lack of recognition, on the part of the physicists, of the nature of space, as the reciprocal aspect of time, in scalar motion, together with the lack of recognition, on the part of the mathematicians, of the operational interpretation of rational numbers, as integers, that is the root cause of the present confusion, both physical and mathematical.

What we have discovered is that the vectorial science is related to the scalar science; that is, vectorial science emerges from scalar science, just as vectors emerge from scalars. The key is to understand and recognize that the duality of numbers is built in, it doesn't have to be an add on through an ad hoc invention such as imaginary numbers.

The contribution that Hestenes has made, in bringing to light the advantage of using Clifford algebras in dealing with vectors, extends beyond the utilitarian aspect, regarding the use of different approaches to vector science, and ultimately lies in the unification of vectorial and scalar mathematical concepts.

However, this is not clear until the reciprocal ideas of Larson are applied to numbers, as well as space and time, and this is accomplished through following Hestenes' example in exploiting the idea of the operational interpretation of number. For this reason, the value of Hestenes' work is not fully appreciated yet, because the focus has been on GA defined as a better language for vectorial physics, not as a generalization of the number concept to accommodate the magnitude concept.

This is why Baez observes:

What we have found is that advocating GA as THE approach to teaching vectorial physics probably has great merit, because it vastly clarifies the relation of duality and dimension, when these aspects of numbers are properly understood. However, the only way to see what is really going on in three dimensions, is to see that the octonions contain all the properties of 3D numbers, including the "directions" of the scalars and pseudoscalars, not just the 1D and 2D directions of vectors.

In other words, the fact that the 3D dimensions of the octonions makes for a great way to do 2D physics, is greatly eclipsed by the fact that they make it possible to do 3D, or scalar, physics!

What we want to show next is how, even though restricted to integers, nature provides us with a continuum, and how that union actually exists in an integer number system like the RSM, reflecting Kronecker's astute observation"

Stay tuned.

Excal

What we are doing at the LRC is investigating how these multi-dimensional RNs correspond to material particles such as electrons, neutrinos, positrons, protons, neutrons, anti-protons, etc, and photons. Larson did the same thing, only he didn't have the mathematical development that we have with the RNs.

He reasoned that the oscillations, 1/2 and 2/1, were periodic displacements; that is, that they were not "stuck" at 1/2 and 2/1, but could take on other space/time progression ratios, beginning with 1/3, or 3/1, and going to 1/4, 1/5, 1/6, ... 1/n, or going from 3/1 to 4/1, 5/1, 6/1, ..., n/1. The increasing values of the displacements in these time and space displacements represent lower and lower frequencies of photons (1/n), relative to the speed of light, and higher and higher frequencies of photons (n/1), relative to the speed of light.

These n speed-displacements, as he called them, on either side of unity, represented a continuous spectrum of radiation from infinitely below infrared to infinitely above ultraviolet. They are actually a discrete series of frequencies, but because they can mix and match on their way out of matter aggregates, which is their source, they appear as a continuous spectrum, unless filtered appropriately through a prism to separate them out.

Now Larson also reasoned that, since these oscillations were confined to one unit of space (ds/dt = 1/n), or one unit of time (where ds/dt = 1/1 in each, undisplaced, dimension), which he assumed was a one-dimensional unit of space or time, these units could be rotated, like an oscillating vector can be rotated. This concept of the speed-displacements, as 1D vectorial oscillations, also enabled him to account for the propagation of the photons, because, if the oscillation were only effective in one of three dimensions, that left two dimensions in which no displacement was present (ds/dt = 1/1). Thus, such a photon was "free" in these two remaining dimensions; that is, it proceeded outward at unit speed, relative to a fixed refererence system, in one of these two free dimensions, which accounts for the observed outward propagation of different frequency photons, in every direction from a source, at the constant speed of light.

This was Larson's scalar theory of radiation, if you will. If you consider that it was developed in the same time frame roughly that the quantum theory of light was being developed, during the 1930s and 1940s, when so little was known about radiation, especially among amatuer investigators, it is really an impressive model of radiation.

However, unlike traditional physicists, who were entirely focused on accounting for the energy levels of atomic spectra, using the Bohr model of the atom and traditional vector motion concepts, Larson deduced his model from first principles, by simply assuming that space is the reciprocal aspect of time in the equation of motion and nothing more.

It's hard to imagine a more radical break from traditional physics than that which this concept represents, because it goes right to the heart of the system of physics used by centuries of physicists to account for nature, and, while they were able to calculate the energy of the atomic spectra, thanks to Heisenberg's non-commutative product in the Taylor series, they had no idea how to explain the constant speed of light, or how to explain its spectrum of frequencies.

Indeed, they still can't do this. In this respect, light is still a complete mystery, because while the difference in energy of the separate states of electrons in the atomic structure can be shown to correspond to the energy of the atomic spectra, using quantum mechanics, there is no understanding at all as to how this energy happens to be manifest as an oscillation, or how this oscillation, once it exists, propagates outward in every direction away from the atom, at the constant speed of light.

In contrast, Larson's concept was consistent in this regard, but he couldn't calculate the atomic spectra, try as he might. He finally gave up trying to solve the problem in the late 1950s, deciding to move on with his theory and to return to the problem at a later date. Of course, he never did return to it, and this has become a rather embarassing gap, or lacuna, in his theoretical development.

At the LRC, we are convinced that the reason Larson couldn't calculate the atomic spectra values from his theory is symptomatic of a fundamental error in his development, and it is our objective to correct this error. However, while it's easy enough to explain the error that we have found, it is a little more difficult to clarify its implications in the consequent development of the theory.

Therefore, I think that the best way to approach it is to employ Hestenes' GA and the concepts of Clifford algebra, even though Larson knew nothing about them. Recall that GA is based on the fourth Clifford algebra, Cl₃, the fourth dimension, 2³ (starting with the zeroth dimension of the scalars), of the binomial expansion, or octonions, which is known as the 3D Euclidean algebra, but which mathematicians regard as eight dimensional, because 2³ = 8 (whew!)

If you start from the left side of the 2³ expansion, 1 3 3 1, , you have four, independent, linear "vector" spaces (the space of traditional mathematics): the 2⁰ = 1 scalar space, the 2¹ = 2 vector space, the 2² = 4 bivector space, and the 2³ = 8 trivector space. Recall that, in GA, these four Clifford algebra spaces correspond to Euclidean geometry's concepts of points, lines, areas, and volumes, so that the 2⁰ space is the space of real numbers (in GA), the 2¹ space is the space of vectors in three dimensions (therefore three, orthogonal, vectors), the 2² space is the space of bivectors in three dimensions (therefore three, orthogonal, areas), and the 2³ space is the space of trivectors in three dimensions, which is a pseudoscalar, or an expanded point (volume). Mathematicians formulate these four, independent, linear spaces, in terms of basis vectors, written as

1. e₀;
2. e₁, e₂,e₃;
3. e₁₂,e₁₃, e₂₃;
4. e₁₂₃

Now, we can view Larson's development as beginning at the left with the first linear space, the space of scalars, and proceeding to the right toward the space of trivectors; that is, we start with the unit progression, or universal expansion of space and time, where ds/dt = 1/1, a scalar, and we generate a vector through the "direction" reversals of one aspect of the scalar (ds/dt = 1/n or n/1.)

However, regarding the oscillation thus produced, as a one-dimensional oscillation, as Larson did, affects only one of the three, orthogonal vectors, in the adjoining vector space, the remaining two vectors in this space are not confined by oscillation to one unit, which accounts for the propagation of the single oscillation, relative to a fixed reference system, as explained above.

The subtle contradiction that the two, undisplaced, vectors in the vector space are therefore, by definition, scalars, not vectors, is not readily apparent when GA is not available to illuminate what is happening, but we will return to this point later. In the meantime, Larson reasoned that once a vector exists in the vector space, by virtue of the 1D oscillation in the scalar, that this vector could then be rotated, transforming it from the vector space to the adjoining bivector space to the right of the vector space. Since rotation of the vector is about the mid-point of the vector, and two such rotations are possible, the motion in the bivector space consists of two bivectors, say a^b and a^c, constituting a 2D rotation.

Rotating this compound bivector in the b^c plane, completes the compound rotations, which now consist of [(a^b + a^c) + b^c], which is equivalent to a^b^c, a trivector, in Larson's development. Notice, however, that the rotation of the one vector, in two dimensions about the mid-point, produces the second and third vector (b and c); that is, Larson reasoned that the rotation was a displacement of a different type, a rotational displacement we would say, so that, while we start with only one vector, rotating it in two dimensions about its mid point, this is tantamount to generating two more vectors, and the wedge products of these vectors can be used to represent two rotations, one a 2D rotation, and one a 1D rotation of the 2D rotation. The concept is illustrated using GA generated graphics in figure 1 below:



Figure 1. One 2D + 1D Combination of Rotations as 3 Vectors, 3 Bivectors, and 1 Trivector

However, the problem referred to earlier, wherein, the vectors in the vector space are really, not vectors, but scalars, is now exacerbated by the fact that they are transformed into vectors through displacement by rotation in the bivector space, if you will, so that they don't become vectors directly, but indirectly, by virtue of rotation. I say that this is subtle, because it is consistent, if one considers that the initial, 1D, vibration is an object that is free to rotate as an object would, or at least, it seems to be consistent.

For instance, an extended object, such as the single blade of an aircraft propeller, rotates in a plane orthogonal to the plane of the thrust it produces, but this plane of rotation rotates as the aircraft rotates around its latitudinal axis, thus, in a sense, the combined rotation of the propeller in a given plane of rotation, and the rotation of the plane of rotation itself, is a two-dimensional rotation. Meanwhile, the compound rotation of the rotating propeller (let's detach it from the aircraft) can also be rotated around the third axis as well.

If units of rotation are equated with units of displacement from unity, in separate scalar magnitudes in this manner, the compound rotations of figure 1 can be equated to independent 2D and 1D scalar motions, which is exactly what Larson does. In this way, he can then add and subtract various units of 1D and 2D "scalar rotation" to form composites that constitute theoretical entites, corresponding to observed physical entities such as electrons, positrons, neutrinos, protons, antiprotons, neutrons, etc, and because the non-rotated oscillation propagates at unit speed, as radiation, he has all he needs to construct the world's first general theory of the universe, a theory of everything, so-to-speak, where the entire physical structure of the universe consists of nothing but motion.

The concepts of force, such as electrical, magnetic, and gravitational force, as properties of the constituent motions of a given theoretical entity, emerge from the development in a natural and very compelling manner. At the same time, the concepts of space and time forming this system lead to an entirely different cosmology, where the big bang concept of infinite matter/energy, expanding outward in cosmic inflation, producing the elements of matter in the process through nucleosynthesis, and aggregates of matter through gravity, which just happens to be perfectly balanced at this particular time in the evolution of the universe, so that the density of matter/energy is neither curving the spacetime fabric outward, nor inward, is replaced by an entirely new cosmology.

In the RST cosmology, the universe is cyclic, but not in the sense of a singular evolutionary process of matter/energy, dominated by laws of enthropy, but in a parallel evolutionary process dictated by the laws of motion, where the unit space/time progression produces two, inverse, sectors of the universe, constantly engaged in an eternal feedback loop.

The details of this development, as far as Larson could develop them are contained in his works, and they are amazingly consistent and compelling, much more so than the standard hot big bang cosmology, especially as more and more observations such as galaxies older than they should be in a serial type evolutionary process, and observations of gravitational anomalies that are attributed to dark matter and dark energy, etc. are made. However, there is a fly in the ointment: Larson's concept of "scalar rotation," upon which the whole development rests, is an oxymoron!

I'll explain this in the next post.

Just a quick note to say I'm still reading. Thanks.

__________________
Forming opinions as we speak

Readers hungry for a more precise presentation of some of these ideas, especially the relation of Geometric Algebra to Larson's ideas, should read this presentation on a recent PhD thesis:
http://www.simplephysics.org/

As explained in the previous post, to construct his theory of the universe of motion, Larson's developed the necessary consequences of the RST, in which space is defined as the inverse of time in the equation of a universal motion. It is then assumed that the physical structure of the universe is composed entirely of discrete units, or instances, of this motion, existing in three dimensions, and it is further assumed that "direction" reversals in the progression of the space or time aspect of this motion produce "speed-displacements," or local instances of the motion, where the space/time ratio is altered from the initial value of ds/dt = 1/1, to 1/n, or n/1.

The first question that usually occurs to those studying Larson's works is, "What causes these 'direction' reversals?" Larson's answer is that no mechanism is required to be identified in this case, because it lies outside the scope of the system (related to Godel's incompleteness theorem I think.) In other words, in order for the system to be applied at all, n units of one aspect must be associated with 1 unit of its inverse aspect, and the only way that this is possible is for the scalar "direction" of the progression of one aspect or the other, to "oscillate;" that is, it must alternately increase/decrease in value, which is the only sense in which a scalar value can "oscillate."

Thus, if, at some given space/time location in the progression, the scalar value of one aspect is alternately increasing/decreasing, while the inverse aspect at that location continues to increase normally, the space/time progression ratio at that location will not be 1/1, but 1/2 or 2/1. This result can be easily plotted on a world line chart as shown in figure 1 below, where the scalar increase of time is plotted on the vertical axis and the scalar increase of space is plotted on the horizontal axis.



Figure 1. The Space/Time Progression World Line Chart

Clearly, if the unit, 1/1, space/time progression is plotted on this chart, it will fall along the green line, where ds/dt progresses as:

1/1, 2/2, 3/3, ..., n/n.

However, if the space aspect of the progression, at a given location, say at 3/3, is alternating between 2 and 3 continuously, the uniform increase in the progression of space at that location will effectively stop, while the uniform increase of the time aspect at that location will continue to increase normally.

Notice that the space progression doesn't actually stop, as this would be impossible, but since each increasing step is offset by a decreasing step, the forward progress effectively ceases, as it would if a marching soldier took a step backward every other step. Thus, the spatial location at this point in the progression is fixed by the space "oscillation," while the space/time progression ratio of the location is ds/dt = 1/2, because the space progression is now confined to one unit (3-2 = 1), and within that unit only half of the total units of progression are increases, the other half are decreases. This location is indicated on the chart by the circled red S, and its inverse, caused by an equivalent "oscillation" in the time aspect, is indicated by the circled blue T.

Consequently, the world line for the red S is plotted as a vertical red line, since only its vertical component (time) uniformly increases. Likewise, the world line for the Blue T is plotted as a horizontal blue line, since only its horizontal component (space) uniformly increases. In other words, discrete units of space motion, albeit stationary in space, are created by space "direction" reversals, while discrete units of time motion, albeit stationary in time, are created by time "direction" reversals.

So far, the logical consequences of the system are clearly evident, or what more formally is referred to as apodictic; that is, they are demonstrably true, or incontrovertible. However, when Larson arrived at this point in his development, he had to find way in which n units of one aspect of the progression are associated with 1 unit of its inverse aspect, where n is not fixed at 2. To do this he concluded that another "direction" reversal pattern is possible, where the alternating increase/decrease in one aspect of the space/time progression only continues for a period of time (or space). When this period is up, then the alternating increase/decrease pattern reverts to a non-alternating increase/increase pattern for two units, after which the alternating increase/decrease pattern begins once again.

This establishes a periodic pattern of "direction" reversals, as opposed to the continuous "direction" reversals, at a given location, and changes the space/time progression ratio from the ds/dt = 1/2 of the continuous pattern, to a progression ratio of ds/dt = 2/3. The advantage of this is clear, when it is understood that combining units of 1/2 ratios with other units of 1/2 ratios, or combining units of 2/1 ratios with other units of 2/1 ratios, produced by the continuous reversal pattern, the space/time progression ratio of the combined unit remains constant. For examble,

1/2 + 1/2 = 2/4 = 1/2,
2/4 + 6/12 = 8/16 = 1/2,
2/1 + 2/1 = 4/2 = 2/1,
4/2 + 12/6 = 16/8 = 2/1

where we are employing the operational view of number in summing the ratios. However, combining units with continuous reversals, with units of periodic reversals, produces a different result. For example,

1/2 + 2/3 = 3/5,
2/4 + 3/5 = 5/9,
2/1 + 3/2 = 5/3,
4/2 + 5/3 = 9/5.

Here, the progression ratio changes as the displacement changes by adding units of continuous to units of periodic reversals. In fact, we get an infinite series of integer displacements:

2/3 = 1 unit of displacement
1/2+2/3 = 3/5 = 2
1/2+3/5 = 4/7 = 3
1/2+4/7 = 5/9 = 4
.
.
.
x(1/2) + 2/3 = x+2/2x+3 = n,

where x is a multiplier of continuous units. In other words, without the periodic pattern of "direction" reversals, which Larson calls "another possibility," there can only be three space/time progression ratios:

ds/dt = 1/1, or 1/2, or 2/1,

Obviously, without the periodic reversals, the development must stop at this point, because no other space/time ratio could be formed. However, while the initial pattern of continuous "direction" reversals is something that must be assumed philosophically, this is harder to do for the subsequent periodic pattern. Yet, as far as I can determine, no one ever raised this issue with Larson, before he passed away. Indeed, as far as I know, no one ever raised the issue until I did a few years ago, more than a decade after his decease.

So, the question is, then, "How can this periodic pattern of reversals arise?" If we assume that they just do, as apparently Larson did, perhaps we can also develop a theory of photons as he did, where the integer displacement in the m/n space/time ratios accounts for the frequencies above and below unity, which otherwise couldn't be accounted for.

However, now we can see that the world lines of these periodic displacements, unlike the world lines of the continuous displacements, are not vertical, or horizontal, but somewhere in between the two, on a diagonal less than, or greater than unity, which means that they possess a scalar motion (increase of space and time) that is greater than zero space motion and less than unit space motion, or greater than zero time motion and less than unit time motion. This is contrary to observation, because only two types of physical entities are observed, those with mass (zero-speed matter) and those with unit speed (c-speed radiation).

The recent observations of what appears to be massive neutrinos as seen in the "neutrino oscillation" phenomenon seem to indicate an exception along these same lines, but things are far from clear at this point. The challenge that we are faced with in the development of the RST universe of motion is explaining the existence of the periodic pattern of "direction" reversals, and, given the periodic pattern, explaining how it produces a massless photon, as Larson concluded that it does, but possessing all the properties of photons, including frequency, propagation, and chirality.

Larson's approach was to assume that the "period" of the continuous portion of the periodic pattern (its length in the periodic PA), representing an integer value of speed-displacement, is the value of its frequency, but that it propagated, relative to matter, because the reversals are effective in only one dimension of space, or time, effectively stopping its uniform progression in that one dimension only, while it remains fixed in the remaining two dimensions. Some claim that this would make the photon expand outward at unit speed, two-dimensionally, like an expanding circle, but Larson insisted that it was carried outward by the unit expansion in only one of the two remaining dimensions, and so propagated outward in a straight line.

Needless to say, this concept of radiation has been challenged by students of Larson's system from the beginning, but Larson pretty much left it at that point, in order to move on to develop his concepts of matter, even though his concepts of matter depend on his concept of radiation. We discussed the reason for this in the previous post.

Fundamentally, Larson's matter concepts begin with the scalar "oscillation" of "direction" reversals, interpreted as a one-dimensional "oscillation," that can then be "rotated." The scalar "direction" of this "rotation" is opposite the scalar "direction" of the 1D "oscillation," and therefore, the net scalar value of the combination of the "oscillation" and the "rotation" is zero. Larson calls this combo the rotational base, and proceeds to add units of rotation to this base, units of "rotation" in both scalar "directions." This leads to theoretical entities with net time or space displacements, relative to the rotational base, that are identified with corresponding physical entities that have the observed properties of negative, positive or no charge, various values of mass, etc.

Larson never considers the property of spin, pretty much regarding it as a value conjured up from the imagination to make quantum mechanics workable. Clearly, in the early stages of his development, its many problems, now so obvious, were not all that obvious, and he was convinced that his development was not inconsistent with the known facts, only with the accepted theories of quantum mechanics and the interpretation of science built upon quantum mechanics.

However, while today we know better, it is nevertheless apparent that his work points to a deep underlying reality that, if properly developed, potentially could prove to be the way out of the fundamental crisis now besetting theoretical physics. The purpose of the Larson Research Center, is to investigate this potential.

At the LRC, we start by recognizing that Larson's new system is a scalar system and that, as such, it depends on new concepts of scalar values, as opposed to existing concepts of vectorial values. For example, we view Larson's assumption, that scalar "direction" reversals can be interpreted as 1D oscillations, as an incorrect conclusion. Scalar values cannot have the directions of vectorial values; that is, they cannot be differentiated by the dimensions of geometry.

However, what we have discovered, to our delight, is that scalar values can be differentiated without converting them into vectors first; that is, instead of developing scalar magnitudes from the left, proceeding to the right, in the linear spaces of GA, as Larson attempted to do, we start from the right. In this way, the circled red S and the circled blue T in figure 1 above, are the multi-dimensional pseudoscalars in the right most space of GA.

When we combine them, we get a combination that "oscillates" in all three dimensions of space and time, and also propagates in all three dimensions. The resulting world line of the combo is shown below in figure 2.



Figure 2. Combining S and T

As can be seen from the chart in figure 2, since the S unit's time progresses, and the T unit's space progresses, there is a non-zero possibility that the two units can coincide at some point in the space/time progression. If this happens, and they combine, the resulting S|T unit's space and time both progress, at the unit rate. Thus, the world line of the S|T unit is plotted parallel to the green diagonal line of unity.

Moreover, as indicated by the text below the chart, there are three, orthogonal "dimensions" to this combo:

1) Outward space motion (ds/dt = 1/2)
2) Outward time motion (ds/dt = 2/1)
3) Inward space/time motion (ds/dt = 1/1)

Hence, with the new concept of scalar motion, comes a new concept of scalar math, and a new concept of scalar geometry, providing the basis of a new concept of scalar science.

In the new scalar science, rotation, in the sense of changing direction, is a meaningless concept, because scalars don't have directions. Therefore, the concept of "scalar rotation" is an oxymoron, and Larson's entire development is based on the concept of "scalar rotation." However, if we can manage to replace his inconsistent concept of scalar rotation, as a concept that depends upon the orthogonality of three vectorial directions, with a consistent concept of scalar expansion, as a concept that depends upon the orthogonality of three scalar "directions," we are confident that great things can come of it.

Already, we can see results beginning to emerge from the shadows. For instance, the notion of integer and half integer spin of bosons and fermions appears to be reflected in the difference between balanced RNs and unbalanced RNs. The intimation that magnitudes of electrical charge correspond to space and time "directions" in unbalanced RNs, while they manifest as chirality in balanced RNs, also seems promising, and so on.

The work along these lines is just beginning, but Larson's work stands as a beacon, lighting the way to what promises to be a whole new world of physics.

In Larson's development of the RST, what we call the RSt, his results are both qualitative and quantitative, but mostly qualitative, because, in part, while he had a new scalar system to work with, he didn't have the scalar mathematics to go along with the scalar motion concepts that are necessary to develop the theory in a scientifically rigorous manner.

However, as the principles of scalar mathematics emerge, a true scalar science becomes possible in which scalar theory can be developed, exploiting the new principles of the system more systematically.

Larson's qualitative theoretical results, including concepts of radiation, charge, magnetic moment, mass, energy, etc, enabled him to develop many quantitative results, including specifics of global interactions, such as the dynamics of inward, or gravitational motion of mass, combined with the outward, or expansion motion of galaxies, in a new cosmology.

This lead to extended concepts of the macrocosmic physical structure of the universe, such as the evolutionary sequences of star and galaxy formation, and the overall structure of the universe, as two interacting, inverse sectors, linking the isotropic emergence of high-energy cosmic rays, in the material sector, with the high-speed ejecta of material concentrated by time gravity, in the cosmic sector.

The incoming matter is then concentrated by space gravity, where it aggregates, concentrates, and eventually explodes and ejects matter at such high speeds that it re-enters the cosmic sector, as cosmic sector cosmic rays, and the same process begins anew, as gravity in that sector once again begins the process of aggregation and concentration of matter in that sector.

Consequently, we see a new cosmology, based on the reciprocity of space and time, forming the universe into an eternal cycle of radiation, matter, and energy, where the cosmic sector, based on 3D time and scalar space, "feeds" the material sector, based on 3D space and scalar time, and the material sector, in turn, "feeds" the cosmic sector, in one eternal round.

Not only is there no need for a "hot big bang" in this cosmology, but it comes with many features that match our observations, such as the flat geometry of the universe, the expansion of space and time, the properties of gravity, the ages of distant galaxies, the energy of quasars, etc.

Many people, when they read Larson's works, are compelled by the development's logical symmetry, and its consistency with observations, but they wonder about the quantitative side of the theory. Fortunately, as the vectorial motion of high-speed astronomical events is the primary motion under consideration, at this scale, the current LST methods of most calculations are not affected, except in the regime of greater than light speeds, which in the RST take the form of time motion.

It's this concept of time motion, and the concept of the two interacting, inverse, sectors of the universe, which provide the basis for Larson's new results, and they are primarily qualitative in nature, and he concentrates on their qualitative effects.

However, when it comes to the quantitative details of specific interactions, pertinent to the topic at hand, Larson uses the best data available at the time to deduce his conclusions. Of course, the scope of what one man can accomplish in such a vast field is very limited. In Volume III of his work, The Universe of Motion, Larson writes:

Obviously, given a new system of physical theory, much of the early work is bound to be immature and preliminary. No one can expect anything else, yet the results Larson achieves, by applying the new system in many areas, are remarkable. Nevertheless, the value of these results lies primarially in serving as indications of the validity of the new system, and illuminating the general nature of the new processes involved.

To say that much remains to be done is an astronomically sized understatement. Certainly, however, enough has been accomplished to warrant further investigation of this scalar system, where Larson has been able to blaze the trail so to speak, and, by so doing, to deliver the outlines of exciting developments to come.

However, a significant part of that outline depends on the new definition of motion and the changes it introduces to the physical picture, but while Larson's conclusions, in the field of physical cosmology, viz-a-viz the standard, or hot big bang cosmology, are fairly easy to compare and contrast with current theory, the same is not true at smaller scales.

Yet, much of what forms the basis of physical cosmology, the theories of the large-scale structure of the physical universe, depends upon the theories of the small-scale universe, and the comparison and contrasting of the two systems is not that easy at the microscale, because the terminology and language of the standard model of particle physics is much more daunting than that of cosmology.

Indeed, the world of particle physics has become so esoteric that it is almost impossible for anyone, at an undergraduate level, to master the key concepts, and the language used to describe them, well enough to get a clear picture of how it all works to even venture an opinion, regarding the issues arising from the processes involved.

This is because the experiments, the phenomenology, are described in terms of the theory, not in terms of observations. Therefore, instead of a universe of observed galaxies, clustered together, yet receeding away from one another and composed of various types of physical enties with a range of observable properties, describable in terms relatively easy to define, we have a universe of observed debris, created by manmade collisions, reacting to an artificial environment in which they move out at near light speed in the flash of nano-explosions, describable only in terms of esoteric equations.

These equations, describing the observed phenomena, are equations ultimately based on the principles of vectorial motion, and the language of mathematics that has been developed to express the laws of these principles. Clearly, what this does, in effect, is cloak the processes involved, because they can only be perceived through the equations that describe them.

While it's true that the picture of reality that emerges from this study, called the standard model of particle physics, can be graphically depicted in a schematic diagram, it contains only a fraction of the information used to construct it. For outsiders, trying to understand the results of the experiments, it's as if we had to study the stars and galaxies from pictures drawn in the sand.

The bottom line is that in order to develop the new scalar science of astrophysics, planetary physics, and cosmology, in many cases, we have to build a scalar science of particle physics, atomic physics, and nuclear physics first. To do that, we need to understand what the observers at the microscale are seeing so that we can compare our theoretical entities with the observed entities, but this task is complicated by the successes of the current theories, which tends to drive the descriptions of the observations in terms of the theories.

Thus, we have leptons and hadrons, which are all fermions, the basic building blocks of matter, but not bosons, which are the particles of radiation. Yet, these are all described in terms of fields, which for the physicist, are as real as the chair in which he sits. However, fields are not enough to totally describe the hadrons, because hadrons are not elementary as once thought, but are now considered to be composed of unobservable quarks.

There are three families of fermions, each of which is composed of different sets of two different leptons and two different quarks. The leptons and quarks of the first family, the members of which form the relatively stable matter of most of the elements in an earth environment, are the electron, and the electron neutrino, and the up and down quark, respectively.

It takes three quarks to form a hadron fermion in the first family. These hadron fermions each have three quarks, two ups and one down, or two downs and one up, the first forming the proton hadron fermion, and the second forming the neutron hadron fermion.

The atoms of successive elements of matter are composed of a successive number of hadron fermions and lepton fermions, in a fixed proportion: there is one lepton fermion for every hadron fermion that is a proton hadron fermion, and, generally, there is neutron hadron fermion for every proton hadron fermion in each atom, except the first, although this number may vary somewhat.

Now, what do we have in the RST? Larson's RSt has subatoms of matter that consist of a system of motions: A linear vibration, which rotates in two of three dimensions, as one 2D rotation, and can also rotate in the third, orthogonal, dimension, as one 1D rotation, forms the basic system of rotations, in his system.

Atoms consist of two of these subatomic systems combined as one. Subatoms consist of one of these systems and are distinguished from one another by the number of discrete units of rotation in their system of rotations. The first system of rotations to emerge is identified with the electron, or the positron, depending upon the scalar "direction" of the rotation of its unit of motion. Successive subatomic entities emerge by adding units of rotation to the system of the previous entity. Hence, an electron becomes a proton, or a positron becomes an anti-proton, through the addition of the appropriate unit of rotation. These systems can take on different charges as well as different masses, depending upon the nature of the added unit of rotation.

So how does this RSt model compare with the standard model? Well, it predicts the periodic table of elements much better than does the standard model, and it explains the inter-atomic distances of elements better than does the standard model, but in other ways it can't compare to the standard model.

The most important of these shortcomings is the inability to calculate the atomic spectra from it, which is the corner stone of the standard model. However, at the LRC, we think that we have identified a fundamental error in Larson's development that will enable us to correct this deficiency, but at the cost of having to redo a lot of the theoretical development.

This would have never been possible to contemplate, if it weren't for the discovery of the scalar mathematics that enables us to develop the theory mathematically. Though this discovery has only emerged in the last few years, it seems promising in many ways, but perhaps the most promising aspect of all is found in the shadows of the standard model that its light is casting.

We have discovered three scalar "dimensions" of motion that are analogous to the three vectorial dimensions of geometry, quite unexpectedly. These three dimensions come in several configurations similar to the configurations of quarks in the standard model (i.e. uud or ddu) that constitute units of motion that cannot be separated, or exist apart (one of the major mysteries of the standard model now considered solved by Gross and company's concept of asymptotic freedom in QCD).

We have also discovered that the scale of these configurations forms three discrete families (another of the major mysteries of the standard model, still unsolved), and we have hints of how the electron and neutrino and photons of radiation fit into the picture as well, albeit perceived only dimly at this point.

With all this, many of Larson's substantial contributions remain intact, especially the understanding of special relativity, the ubiquitous force of gravity, as the property of the intrinsic motion of mass, and the space/time dimensions of physical constants. In addition, we now have the new found understanding of the octonions and the Bott periodicity theorem, so we can see shadows of string theory's Supersymmetry lurking about.

In short, antoniseb, there are lots of exciting qualitative results coming from the system, and we are working hard to extract the requisite quantitative results, which, if and when we do, we'll post on our site.

Stay tuned.

Excal

Last night I watched a PBS special on E=mc², called "Einstein's Big Idea: The Ancestors of E=mc²." I learned several things from the show. For instance, I learned that the letter c, as a symbol for the speed of light, comes from the latin word for swiftness, "celeritas." I never knew that, I don't think.

However, I learned something else as well, something much more significant. I've written before that I can easily understand how energy and matter are related in the RST, since it's so straightforward in the new system, but I really have trouble understanding how Einstein knew this. Now, after seeing the show, the mystery has been cleared up for me. Einstein accepted, as most others did by then, that energy could be defined by the mass of an object times its velocity squared, because Emilie du Châtelet, following Leibniz lead, had done it, and she demonstrated the veracity of the definition with the help of Willem 'sGravesande's clay penetration experiments.

So, when Einstein realized that the velocity of mass was limited to the value of c, he realized, at the same time, that its energy must also be limited by c. In other words, what was being asserted was that the maximum energy that a moving object could obtain, given the accepted definition of energy, was

E = mv²,

which, given the newly discovered speed limit of v at c, naturally leads to

E = mc².

However, it's not clear to me that Einstein understood that this means that the mass of matter could be converted into energy by some process other than putting the energy into it by accelerating it to c-speed (or as close as possible). In other words, it wasn't necessarily understood right away that mass and energy are equated by the equation, since the dimensions of mass, kg, are obviously not the same as the dimensions of energy. Energy is measured in units of joules in the SI system of units (probably ergs in the cgs system of his day). Therefore, the dimensions of the equation must have initially been interpreted as

ergs = mass x velocity x velocity,

but then what is velocity times velocity? I don't think Leibniz, or du Châtelet, or Einstein knew the answer to this question. I wish I could find out WHY Leibniz thought it was necessary to square the velocity in the energy equation, but all I can find so far is that he did think so, even though he couldn't prove it. Therefore, as far as I can tell, there is no theoretical explanation of the definition of kinetic energy as mv²/2.

On the other hand, using the RST dimensions of motion, energy clearly has space/time dimensions,

t¹/s¹,

and momentum has space/time dimensions,

t²/s²,

while velocity has space/time dimensions,

s¹/t¹,

However, the dimensions of the energy equation balance, only if mass has dimensions,

t³/s³,

as can be seen in Einstein's equation,

t¹/s¹ = (t³/s³) * (s²/t²).

But Einstein did not suspect that mass has these space/time dimensions. Indeed, LST scientists do not understand it this way, even today. Kilograms, the units of mass, like the units of temperature. or the units of apples, or bananas, are understood as scalar units, as units of an amount, or a quantity, of a substance. Energy is also a scalar quantity; that is, it has magnitude only, with no direction in space, but the dimensions of scalar units are zero, not three, as required by the dimensional analysis of Einstein's energy equation show above.

Yet, as we have discussed previously, while the dimensions of a scalar quantity are zero, the dimensions of the pseudoscalar octonion, are three, so this admits the possibility that a substance can also have three dimensions and still be a "scalar." In this case, the units of the amount of substance would be units of volume.

However, as it turns out, even though energy is a scalar quantity, its definition in LST physics, where its divided into two categories, potential and kinetic, has the dimensions of work. In Wikipedia's article on energy, we read:

Hence, in LST physics, the scalar value of energy is transformed, by virtue of its definition, into the scalar value of "work done by a force," which is the scalar product of force and a so-called "displacement vector," and, therefore, can have one of three values: positive, negative, or zero, depending upon the degree of orthogonality of the force vector. In other words, energy is a scalar defined as work, but work is a scalar that is defined as something only manifest as a vector product, the inner vector product that can be either positive, negative, or zero. From the Wikipedia article on work, we read:

Thus, energy, in LST physics, is defined as the projection of one vector upon another. When the vectors are perpendicular, no shadow is cast by the vertical upon the horizontal, but, if the angle between them is greater than, or less than 90 degrees, the shadow (projection) takes on a definite positive or negative value - just like a scalar. Maybe we could call this a pseudoscalar too, but not in the same sense. It's a physical pseudoscalar, we might say, not a mathematical one.

The interesting aspect of all this, in the present context, is the insight that it gives us into the relation of numbers and magnitudes. LST physics deals only with vectorial motion magnitudes, as we have been explaining, so to define scalar motion magnitudes (in this case the motion of inverse scalar velocity, or energy), a way has been found to describe it in terms of vector magnitudes. It would be considered ingenious were it the work of one individual, but really it's a concept that has evolved unconciously, as it were, over centuries. The driving evolutionary force, success in physics, like biological success in living and reproducing, is powerful, if not all that efficient.

Nevertheless, with this much understood, we are left to ponder the meaning of scalar. If energy is a scalar quantity, and a scalar has no direction, then how can it have the mathematical dimensions of 1 in Einstein's energy equation? As scalars in a three-dimensional physical system, one would think that mass and energy should have the mathematical dimensions of the octonion scalars (2⁰), or pseudoscalars (2³), but, while we can see that, in the RST at least, mass has the three dimensions of the fourth line in the tetraktys, the dimensions of the pseudoscalars, energy has the dimensions of the second line in the tetraktys, the 2¹, or one-dimensional, line, not the first line, the 2⁰ line. What's up with that?

I'll try to answer that question in the next post.

Excal

We can gain a useful understanding of the conflict in the view of the dimensions of scalars, discussed above in terms of the definitions of mass and energy, and see how the existence of non-zero scalar dimensions actually clarifies how a physical scalar value such as energy can have non-zero mathematical dimensions, by studying the dimensional properties of the Greek tetraktys and comparing/contrasting the meaning of these dimensions in terms of vectors, Clifford algebas, and proportions, using the operational interpretation of number.

In the vector view of the tetraktys, the 2⁰ points are scalar multipliers of 2¹ vectors, and a vector times a vector is another vector. So we have a resultant vector as the diagonal between two orthogonal vectors, or two non-parallel vectors, times the magnitude of a scalar, or this product times another vector times a scalar, etc. All the possible combinations and the mathematics for these vectors, in the tetraktys, are described by the vector algebra, using the numbers in its hypercomplex number system, the set of reals, complexes, quaternions and octonions.

In the Clifford algebra view of the tetraktys, used to formulate Geometric Algebra, the 2⁰ points are again scalars, but vectors are directed, one-dimensional, lines, multiplied by the scalars, while the product of vectors is not another 1D vector, but a directed 2D bivector, or 3D trivector, again, multiplied by the scalars. All the possible combinations and the mathematics for these multivectors in the tetraktys are described in the Geometric Algebra, using the multi-dimensional number system, the set of zero, one, two, and three-grade blades.

By contrast, in the proportional view of the tetraktys, the "points" are also scalars, but, unlike in the previous views of 1D vectors and nD multivectors, the scalar in the proportional view is the source of the higher dimensional numbers, in the sense that all the higher-dimensional numbers in the tetraktys are expanded scalars, rather than rotated vectors, or multivectors. There are no vectors in this view, no vectors, no bivectors, and no trivectors, only "n-dimensional" scalars.

To illustrate how this works, we can use scalar values such as colors and walk them through the tetraktys. Each scalar value represents a relative proportion, which is either equal to, greater than, or less than, the reference proportion. We begin with the first element, at the top of the tetraktys (1), the 2⁰ = 1 scalar, or the void. We assign the color black to it, a scalar value corresponding to a black "point," if you will.

At the next higher dimension (11), 2¹ = 2 scalar, we can expand the "point" scalar value in two "directions" to form a 1D value corresponding to a geometric "line," with three scalar values, representing the expansion of the black scalar, expanded to a scalar value, or "point," on either side of black, to a red value, or "point," on the left, and to a blue value, or "point," on the right. The blue "point" is a scalar value of greater proportion than the black "point," while the red "point" is a scalar value of less proportion than the black "point." We will give the set of these three values, corresponding to a 1D geometric "line," defined between the three scalar values, or "points," the color green, representing the one-dimensional equilibrium established by its three scalar numbers.

At the next higher dimension, 2² = 4 scalar (121), again we have the zero-dimensional, black, "point," but now we can expand it into two 1D scalar values, or "lines," the new one of which we will color red. However, the difference in the color of the 1D values, represents a scalar difference in the symmetry of the two lines; one is symmetrical and one is not, the difference in symmetry defining two scalar "dimensions."

There is a scalar difference of dimension between the red value and the green value, and this difference is manifest as the difference in the symmetry of the two 1D values; that is, the green 1D value is symmetrical, or balanced, but the red 1D value is unbalanced. Its symmetry is broken, we might say, in the red "direction," representing the new, or second, dimension at this level. The product of these two 1D values, the green 1D "line" ^ red 1D "line", is a yellow, "two-dimensional," scalar value, corresponding to a geometrical "area."

At the next higher dimension of the tetraktys, the 2³ scalar on the fourth line (1331), we again have the 0D, black, "point," but now we can expand it three ways, corresponding to the three vectors of the Clifford algebra tetraktys. One of these is the balanced scalar value, or the symmetrical 1D expansion, and the other two are the unbalanced scalar values, or two, 1D, non-symmetrical, expansions of the 0D black "point."

The first, balanced, 1D value is again colored green, while the second 1D value, unbalanced in the red "direction," is again colored red. The third 1D value, unbalanced in the blue "direction," is colored "blue." Now, we can combine each of the three, 1D, scalar values, with each of the others, so we have three combinations of two, 1D, scalar values, forming a 2D scalar value, and these three combinations correspond to the three, 2D, bivectors of the Clifford algebra tetraktys:

1. green 1D "line" ^ red 1D "line" = yellow 2D "area"
2. green 1D "line" ^ blue 1D "line" = cyan 2D "area"
3. red 1D "line" ^ blue 1D "line" = magenta 2D "area"

Notice that, because these are scalar values, they are commutative; that is, the order of combining them makes no difference in the result. Now, at this, the bottom level of the tetraktys, there are also three more combinations, where we combine one of the three 1D values, with one of the three 2D values. However, there is only one result, regardless of the combinations, and it corresponds to the Clifford algebra, 3D, trivector, a "volume:"

1. blue 1D "line" ^ yellow 2D "area" = white 3D "volume"
2. red 1D "line" ^ cyan 2D "area" = white 3D "volume"
3. green 1D "line" ^ magenta 2D "area" = white 3D "volume"

Figure 1 below illustrates the scalar tetraktys.



Figure 1. Scalar Tetraktys

Again, since these values are scalar values, their algebra is associative; that is, it doesn't matter how the three, 1D, scalar values and the three, 2D, scalar values are grouped to form the one, 3D, scalar value, the result is always a white, 3D, volume.

Of course, the point is that the scalar combinations of the scalar tetraktys correspond to the combinations of the scalar values of the red SUDR, and the blue TUDR in the development of the physical theory that we are working on. The SUDR and TUDR, are initially joined together to form the green SUDR|TUDR combo. This combo (green 1D value) represents the one-dimensional, balanced, RN, the symmetry of which can be "broken" in two "directions," by the addition of red SUDRs, and/or blue TUDRS, to the green symmetrical combo. Thus, we see that the units of scalar motion have three "dimensions," and though these scalar "dimensions" are not the vectorial dimensions of Euclidean geometry, they are nevertheless consistent with three-dimensional mathematics. Not an insignificant result.

Once we understand this, we can see that the 1s running down the right side of the tetraktys in figure 1, have n "dimensions" (multicolors), while the 1s running down the left side of the tetraktys have 0 "dimensions" (black color), but they are all scalar values nonetheless.

Therefore, we see that the zero-dimensional units of mass, which we measure in kilograms, can also consistently be expressed as the three-dimensional units of scalar motion. Hence, all the physical dimensions reduce to consistent multi-dimensional units of space/time in two, reciprocal, scalar groups, when we provide the correct dimensions of the scalar values involved:

The energy group:

1. mass = t³/s³
2. momentum = t²/s²
3. energy = t¹/s¹

The velocity group:

1. inverse mass = s³/t³
2. inverse momentum = s²/t²
3. velocity = s¹/t¹,

where I'm explicitly indicating the one-dimensional values in the superscripts, for greater clarity. If 3D inverse mass is the mass of antimatter, then 2D inverse momentum is the momentum of antimatter, but it is also 1D velocity squared. So, multiplying 2D inverse momentum by 3D mass, yields 1D energy, as shown above, but, by the same token, multiplying 2D inverse mass (antimatter) by 2D momentum, yields 1D velocity,

v = s³/t³ * t²/s² = s¹/t¹

So, then, what is 2D momentum and 2D inverse momentum? We know 2D momentum is a product of 3D mass and 1D velocity, so 2D inverse momentum must be the product of 3D inverse mass and 1D inverse velocity, but 1D inverse velocity is energy, therefore, 2D inverse momentum is the product of 3D inverse mass and energy, or

(p) = (m)*E = s³/t³ * t/s = s²/t²,

where the parentheses indicate inverse. So, though we don't know what it means at this point, at least we have a consistent and fundamental definition of velocity times velocity, or velocity squared.

More to come on that later, but in the meantime, since force (a quantity of acceleration) is required to produce a 1D velocity of mass (2D momentum), an inverse force (a quantity of inverse acceleration) should be required to produce a 1D inverse velocity of 3D inverse mass (2D inverse momentum).

So, the next question is, then, what are force and acceleration? In the LST vectorial system, force and acceleration must be defined apart from mass and momentum, but in the RST this is not necessary.

In the new scalar system, force, is energy per unit space:

f = t/s * 1/s = t/s²,

while acceleration is velocity per unit time:

a = s/t * 1/t = s/t².

So, force extended over space is energy,

E = t/s² * s/1 = t¹/s¹,

while acceleration extended over time is velocity,

v = s/t² * t/1 = s¹/t¹.

Further, force extended over time is momentum.

p = t/s² * t/1 = t²/s²,

while acceleration extended over space is inverse momentum,

(p) = s/t² * s/1 = s²/t².

The only thing I want to emphasize at this point, is that these space/time dimensions are entirely consistent. Larson used this fact to great advantage, as you can see in Chapter 12 of "Nothing But Motion," but he did so without knowledge of the scalar tetraktys. Now that we understand it better, we expect to be able to exploit these scalar equations of motion, both on the velocity side and on the inverse (energy side) of unity, to great advantage.

Stay tuned.

Excal

Reading the new policy in ATM, I was wondering if this thread is now closed? If not, I would like to continue with it. Is anyone else out there interested in learning more of the RST?