The Larson Research Center

I tried to point out to John Baez, via Peter Woit’s blog, that by not recognizing that the “dimensions” of mathematics do not correspond to physical dimensions, the LST community is tripping up on a small, but very significant, stumbling block.

They equate the four levels of the tetraktys with four different, ad hoc, number systems, based on the ad hoc use of imaginary numbers: At the first level, 0 imaginary numbers are associated with the familiar real number system, but adding 1 imaginary number to the reals enables man to generate the marvelous complex numbers, the second level which provides the foundation of all the science and technology running the world today.

Recently, another number system has been widely incorporated in computer simulations and robotics that was invented in the Nineteenth Century, by Sir Hamilton, which is called the quaternions. Quaternions have found wide application lately, even though their true nature is misunderstood in most cases. This number system, residing at the third level of the tetraktys, incorporates three imaginary numbers.

Finally, at the fourth level, the octonions incorporate no less than seven imaginary numbers and are the subject of Baez’s Scientific American article, which Woit blogged about, because it ties octonions to string theory, and Woit’s purpose in life is to debunk string theory hype, whereever and whenever it appears.

However, Woit had to admit that Baez and his co-author were not actually hyping string theory: They were hyping octonions, declaring that, “if string theory is right, the octonions are not a useless curiosity: on the contrary, they provide the deep reason why the universe must have 10 dimensions: in 10 dimensions, matter and force particles are embodied in the same type of numbers—the octonions.”

This is a reference to the supersymmetry of string theory. It turns out that the only way to describe the elements of the theory without inducing anomalies, is to use the 8 “dimensions” of octonions plus the two extra dimensions of strings and time - a total of ten “dimensions.”

Of course, I tried to point out that the universe doesn’t have ten dimensions, it only has the three observed dimensions of space and the one observed dimension of time - the four dimensions of motion, if you will, and their inverses, but just as the members of the LST community can’t understand that motion doesn’t have to be one-dimensional, they also can’t seem to understand that each physical dimension has two “directions,” and that they should look into the mathematics of ten “directions,” instead of ten “dimensions.”

Unfortunately, however, in our era of political correctness, such views are squelched and Woit refused to allow my comment on his blog to be published. Oh, well. It’s their loss. We will continue to apply our meager brain power to the truth and keep plugging along to see what we can accomplish without their Cadillac brains and resources.

In the next post, I will begin to explain the integration of the geometry of Larson’s Cube, the mathematics of the tetraktys and the numbers of the new number line, which will enable us to desribe the preons of our version of the standard model in terms of more than the initial color combinations we have been using. Now we can put real numbers to the entities in the model, numbers that are related to the energy levels of the atomic spectra.

Proving once again that many times, by small and simple means, great things are brought to pass.

Scientific American recently published an article by John Baez and John Huerta on the use of octonions in string theory. Their motivation was that octonions “may explain why the universe has the number of dimensions it does,” if string theory is right.

The number of dimensions of the universe they refer to is either ten, which includes the eight dimensions associated with the LST tetraktys, one real and one, three or seven imaginary dimensions, plus two more swept out through space as a 1D string propagates, or else eleven dimensions, which includes one more, when the 2D membranes of M theory propagate through space over time.

These ten (string) or eleven (membrane) dimensions are to be understood in terms of the mathematical operations used in describing a unified picture of a physical universe that has two sectors, one sector consisting of the observed matter particles (spin 1/2), the other a mirror image of the first, but consisting of force particles (spin 1), an idea called supersymmetry in string theory.

If it weren’t for the extra dimensions that strings or membranes sweep out over time, say the authors, the interactions of force and matter particles can be described with simple multiplication within the tetraktys (thus providing a unified description of nature), but “[The evolution over time] changes the dimensions in which supersymmetry arises, by adding two—one for the string and one for time. Instead of supersymmetry in dimension one, two, four or eight [of the tetraktys], we get supersymmetry in dimension three, four, six or ten [for strings, or four, five, seven, or eleven for membranes.]”

In other words, they need to keep the mathematics confined to the dimensions of the tetraktys (of course, they don’t use the word tetraktys, but the shortcut is useful in referring to “the standard collection of one, two, four and eight dimensions.”) This is understandable, because the Bott periodicity theorem proves that there are no new phenomena beyond the dimensions of the tetraktys. Yet, instead of accepting this, they spend billions of dollars and decades of time looking for the evidence that the universe can escape the tetraktys!

But it is the eight dimensional octonions of the tetraktys that works out for strings. Using the four-dimensional quaternions, or the two-dimensional complexes, or the one-dimensional reals introduces anamolies, in which string theory breaks down. String theory and M theory (presumably) are only self-consistent and anamoly free, when the system is described using the eight dimensional octonions.

“So,” they conclude, “if string theory is right, the octonions are not a useless curiosity: on the contrary, they provide the deep reason why the universe must have 10 dimensions: in 10 dimensions, matter and force particles are embodied in the same type of numbers—the octonions.”

Regular readers of the LRC’s three blogs will probably be wondering why in the world do these people insist on complicating the algebraic picture, by counting the real and imaginary numbers as mathematical dimensions that correspond with physical dimensions? I cannot for the life of me answer that question. It is a complete mystery to me why they can’t see that the three physical dimensions of space and the one of time are embodied in the tetraktys.

It’s clear that there are two inherent “directions” of dimensions; that, instead of the numbers one, two, four and eight of the tetraktys representing physical dimensions, these numbers represent the physical “directions” of space and time, the 20 = 1, 21 = 2, 22 = 4, and 23 = 8, “directions” of the 4D tetraktys, corresponding to the point, line, area and volume of geometry.

When we construct the right lines and circles of Larson’s Cube, with its two balls (eeew, that’s hard to write!), we get a wonderful picture of the discrete and continuous structure of the physical tetraktys, which corresponds perfectly to the observed space and time of our universe. 

The only thing that remains is to set it in motion; that is, describe how it changes over time. However, it’s not the vectorial motion of the LST we should envision, which is so misleading, but rather the scalar motion of the RST, which does not add an extra two, or three, dimensions to the 4D tetraktys, thus eliminating the vexation of extra dimensions that is so perplexing to the LST community.

The idea of supersymmetry, that there are material and cosmic twins, one the inverse of the other, in all but magnitude, then falls out within the four space/time dimensions of the tetraktys, revealing an inverse tetraktys with four time/space dimensions, if you will, in which the dimensions of space and time are swapped, where time has three dimensions and space has one dimension.

It is just so simple, but don’t look for it to appear in a Scientific American article any time soon. 

Update: I should point out that the expansion/collapse of the 3D oscillation adds two “directions” to the eight “directions” of the tetraktys. If we call the eight diagonals in Larson’s Cube dimensions, which is what the LST would do, then the inward and outward “directions” of these over time would constitute two additional dimensions in that sense, I suppose.

I can see how this thinking evolved from the correlation of 1D vector motion with numbers on the number line, but when it is realized that scalar changes in space and time are legitimate instances of motion, as well, it clarifies the whole picture.

We’ve heard it for so long and in so many ways that we have become inured to it, but I happened across an old BBC documentary on Youtube last night, and decided to watch once more as Brian Cox explained the nature of time in spacetime to his audience.

He tried to explain that even though time is treated as a 4th dimension in Einstein’s spacetime, it’s treated a little bit differently than a magnitude with the three dimensions of space is treated, because its magnitude has a minus sign in the equation.

It reminded me of an earlier video I had watched in which Brian and Max Tegmark discussed the same thing. Tegmark observed that, if it weren’t for that minus sign, there would be no point in having a brain, because we wouldn’t be able to predict anything.

But that’s a double-edged sword, because as Brian explains in his BBC documentary, since Einstein’s time dimension is continuous and one-dimensional, everything is present - past, present and future - and we just travel along the ribbon of time that leads to all the events of our life, which already exist, waiting only for our arrival. The minus sign keeps us from stopping and reversing.

It’s this nonsense that the LST community has bought into that drives thinking people away, to look for something better. When we take Larson’s assumption that time and space don’t exist independently, but are the reciprocal aspects of the one single component of the universe, motion, we can see intuitively that it’s a better course to take.

The thing is, LST defenders of the unification of space and time into 4D spacetime argue that it has been proven and even incorporated into technology like GPS. However, as I watched the documentary, I wondered why they couldn’t see that what they think of as the spacetime warp is simply a result of motion that is not readily apparent.

But just as in Hollywood films, where the film makers straighten out the curved trajectory of tracers from the guns of fighter planes turning, twisting and rolling in aerial combat, for esthetic reasons, scientists seem to prefer the sight of the pretty lines in their rubber bed sheet illustrations, curved by the weight of large masses, to the image of the eternal motion of expanding space and time.

The truth is, however, there is no rubber bed sheet, and there is no fixed set of future events, waiting for our arrival. The 3D expansion of space and time is eternal, and the course of an aggregate of matter in this expansion is more like a fish swimming upstream, than a bowling ball depressing a bed sheet.

The motion of the observed expansion is scalar; that is, it has no specific direction or dimension, but expands in all dimensions and therefore in all directions. The expansion at a given location stops when it oscillates in all dimensions, expanding/contracting in 3D space over 0D time alternately, in the case of our SUDR, and 3D time over 0D space, in the case of our TUDR (see here).

At first thought, we might object to the idea that a given location in the expansion could be oscillating. Why is a given location oscillating as opposed to any other location? and what, or who, caused it to oscillate? Of course, there are no answers to those questions, any more than there are to the questions, Why does a given particle exist, and why does it spin or oscillate the way it does?

No, the only question that makes sense is “What difference does the 3D oscillation make in our development of physical theory?” The answer to this question is just beginning to be explored. It boils down to substituting the vectorial motions of ill-defined points for the scalar motions associated with consistently defined points. Both kinds of motions are simply ways of describing changing space and changing time, but the former has been developed into a highly sophisticated science, from which all our modern technology emanates, while the latter is nothing more than a glimmer of hope, born out of a desire to find a more intuitive way.

As it turns out, the mysteries of some very fundamental issues have been exposed, having to do with some very fundamental equations. Mysteries like the reality of 4 π rotation and the strange mathematical relation of square roots and integers in the periodic energy relations of the elements.

Certainly there is more to come, but whether or not we will be able to reach the goal of explaining the atomic spectra with it or not is anybody’s guess, at this point in time. I hope so, though, because, if we do, we will have a whole new physical model of the atom to work with. One in which scalar motions replace forces, and in which gravity can be explained as a straightforward property of mass. 

If we succeed, the theoretical universe will retain its causality, but without locking in the future. The importance of the minus sign in Einstein’s equations will be greatly diminished I think.

The winning essays in the FQXI Essay contest will be announced in early June, but there’s little chance that my essay will win anything. Right now it’s buried in the ratings, a little more than half-way down in a field of many, many essays.

However, I am pleased to announce that an anonymous group of Russian scientists has honored it with a prize of their own, thanks to my long-time fellow traveler in all things Larsonian, Horace from Poland. I am most grateful for this gesture.

Horace explained that “They were taken by someone attacking an axiomatic concept such as the point,” and were intrigued by the view of space and time as fundamental constituents of the universe instead of the orthodox view of space and time as a container of matter.

Of course, the idea that space and time exist in three dimensions, in discrete units, and are simply the reciprocal aspects of motion, which motion is the single component comprising all things in a physical universe, conforming to Euclidean geometry, the ordinary relations of commutative mathematics, with absolute magnitudes, and that there is no background, no container of matter, was not my idea, but the great insight of the man, Dewey Larson.

Even the idea that the point can and should be defined in a new way, which is the point of my essay, arose from the development of the consequences of Larson’s fundamental postulates. He deserves the credit that I hope will come to him in due time.

It took me several months to complete the FQXI essay, but most of the time was spent trying to compress the main idea into something that would fit into the contest constraints. Now I realize that I should have spent more time on thinking about the idea itself and how it applies to the topic.

If I had done that, I would have no doubt realized the main point of the RST, that space and time are quantized, could have been brought to the forefront in a much more dramatic and convincing manner than the wishy-washy way I ended up presenting it. I posted a comment that hints of something to that effect on my FQXI discussion thread here.

In that post, I wrote:

The point is, that there is no use trying to define a point in space that has any extent, or an instant of time that has some duration. This contradiction at the foundation of our science and mathematics cannot help manifest itself in terrible ways later on. Our concept of the electron is the best example, but there are many others. 

A really advanced alien society would no doubt laugh at our pathetic theories that we take so seriously that we build silly machines like the LHC, going to astronomical expense to look for figments of our imaginations.

Why look for the Higgs, when we can’t even understand the electron? If there is a discrete unit of space, then, by definition, it means that it cannot be subdivided. Yet, we can represent any magnitude with figure 1 of my essay. ANY geometric length magnitude whatsoever, including the so-called Planck length, can be represented by the radius of the unit circle. This means that the radius of the square root of 2 circle can be represented as well. With these two radii and the eight cubes between them, we have both digital and analog 1D, 2D and 3D geometric quantities such as circumference, area and volume, represented. So, how can we say space and time are doomed at some length, as today’s leading theoreticians contend?

I went on to try to explain that, choosing a unit, which we must do, subordinates any subdivisions of that unit to the unit ratio. Since the unit space/time ratio is the unit speed, then the speed (time) of any subdivision thereof is necessarily greater (less). For example, if we use Larson’s space and time unit to form the unit ratio (light speed, c, based on the Rydberg frequency of hydrogen), then there would be about 2.8205 x 1031 subdivisions of the Planck length within the space unit of that unit speed. Hence, the time it takes the last of those Planck lengths to collapse, or the first to expand, in the course of the unit oscillation, is miniscule indeed.

But the point is that even if we took the Planck length as the unit length, along with the corresponding time unit to give us the unit ratio of speed c, the construction of figure 1 of my essay is still valid, and it could be subdivided into even smaller units. Then the ratio of those units could be calculated and taken as standard, and the process repeated ad infinitum.

This is the crux of the problem, when subdividing the continuum: There is no conceivable mathematical limit to the size of divisions. The question is, though, is there a physical limit? Larson’s RST, our new system of physical theory, assumes that there is, and Larson calculated it based on the Rydberg frequency of hydrogen and the speed of light.

The key consequence of this assumption, which Larson described, is that space/time motion limits the distance between two entities to the unit of space, after which time/space motion reduces the distance (to zero if need be).

Now, the question is, how can we express this inverse relation mathematically? The problem we run into is that s/t = 1/1 is equal to t/s = 1/1, mathematically. With Larson’s “direction” reversals, this unit ratio gives us s/t = 1/2 and t/s = 1/2 (assuming space reversals in the former and time reversals in the latter).

In our development at the LRC, we try to treat these two values as negative and positive units of motion, by taking the arithmetic ratio (difference between denominator and numerator) as the operational differential. But how do we do that with figure 1 of the essay, since inverting the radius and diameter is impossible (we can’t have a radius of 2 and a diameter of 1). Therefore, what we have done up until now is switch the labels. We say that space (diameter) is now time (radius) and time (radius) is now space, but what justification do we have to do this? How does space in the material sector become time in the cosmic sector and vice-versa?

The problem is the 1/1 ratio. If we take the ratio of the advanced function of the expansion/contraction (e/c), which is the square root of 2r2, (the outer circle radius of figure 1) and the retarded function of the e/c, which is the inverse of the advanced function (the radius of a third circle, smaller than the inner circle of figure 1 that is not shown), we get two positive units in the two “directions” of the numbers. The number 2 for the greater than unit numbers and the number 1/2 for the less than unit numbers . To see this for yourself, just take the inverse of the square root of two over the square root of 2, and then the inverse of this, the square root of 2 over its inverse. You get the following number line:

…, 2.5, 2.0, 1.5, 1.0, 0.5 | 2, 4, 6, 8, 10, …

This number line has advantages. First of all, it preserves the greater than, less than, relation between the inverses, inherent in the geometric ratio (numerator/denominator quotient relation), while at the same time it incorporates the unit symmetry of the arithmetic ratio (numerator/denominator difference relation).

Second, the numbers are both functions of the unit radius (the inner circle of figure 1): The square root of 2 is the unit advanced function, we might say, while its inverse is the unit retarded function. But why call them advanced and retarded? In reality, mathematically, one is the inverse of the other, just as space is the inverse of time. True, these numbers are not the elements of a group, but since their respective units are inverses, we need only to invert their multiples to form the group under division, given the quotient interpretation:

…, 1/5(1/2), 1/4(1/2), 1/3(1/2), 1/2(1/2), 1/1(1/2) | 1/1(2/1), 2/1(2/1), 3/1(2/1), 4/1(2/1), 5/1(2/1), …

The beauty of this unorthodox group is that the identity element itself is a product of the two units, not an explicit element of the group. It’s a product of the two inverse units; that is, 1/2 / 2/1 = 1/4 and 2/1 / 1/2 = 4/1, but the product of these two, 1/4 *4/1 = 4/4 = 1/1, acts as the identity element of the group. Moreover, taking the difference interpretation of the numbers, they form a group under addition, the integer group, where the identity element is 0.

Now, a very interesting observation is that, with the difference interpretation, we can build the standard model of particle physics, as shown here, and, with the quotient interpretation, we can build the periodic table of elements.

Larson’s mathematical pattern for the periodic table is 4n2, to which he gives a physical meaning in order to obtain the four periods, 22 = 4; 42 = 16; 62 = 36; 82 = 64 of the table. These are the double periods of the Wheel of Motion, as opposed to the 2n2 half-periods of quantum mechanics, obtained using the four quantum numbers, n, l, m and s. 

The trouble is, in each case, the periodic nature of the table is built on the variability of constituent particles of atomic structure (QM), or on the variability of constituent motions of the atom (RSt). Both work out fairly well in accounting for the order of the elements, but neither theory conforms to the new empirical data shown by Le Cornec, which uses a ratio of square roots of atomic ionization potentials to order the elements by their energy levels. Le Cornec shows that in the QM theory of the atom, the “s” and “p” energy level groups are reversed, while in the development of his RSt, Larson altogether abandoned the effort to explain the spectroscopic data of the elements, using scalar rotations.

With the new numbers, however, the quantitative relations of the periods fall out from the physical concept of the 3D oscillation. The sequence of 2, 4, 6, 8 of the new number series under the quotient interpretation, shown above, contains 4, 16, 36 and 64 positive “slots” for the respective, positive, inverses. Since these results come from the advanced/retarded functions of the 3D oscillation, the relation of square roots and their inverses, as described above, it’s easy to conclude that a connection with Le Cornec’s work is plausible.

At least that’s what I’m hoping, but this unorthodox view of a mathematical group may end up derailing the whole thing. I guess we’ll see if it’s valid or not, in the end.