Of course, the standard model is based on the Newtonian system of theory, which, based on vector motion, constitutes his program of research to find the fewest number of particle interactions among the fewest number of particles, in the structure of the physical universe.

Though the LST community has successfully modified the concept of vector motion considerably, substituting the description of a particle’s state, by means of its “probability amplitude,” in space and time, for the description of a particle’s state, by means of its change in position, in space and time, they have done so with an algebra Sir William Rowan Hamilton regarded as non-intuitive, unscientific and even self-contradictory, in contrast to Euclidean geometry (see here.)

Larson excluded the RST community from using this defective algebra, when he formulated the second fundamental postulate and included in it the assumptions of Euclidean geomety and the “relations of ordinary commutative mathematics.”

Some have argued that the word “ordinary” is too vague a term, that he should have been more explicit, but it’s clear that he was referring to the unfamiliar rule for multiplication that had emerged in connection with quantum physics, during his formative years. It was this rule, the rule of non-commutative multiplication, where p times q is not equal to q times p, which was known to mathematicians familiar with matrix multiplication, but not to most physicists, which enabled the science of quantum  mechanics from that point on.

It was based on multiplying the amplitudes of two oscillations in order to calculate the energy involved. If the rule of ordinary commutative mathematics were used in these calculations, the result was an infinite sum in a Fourier series. The solution was to use non-commutative multiplication, something that disconcerted physicists, but not mathematicians.

Larson was good at mathematics, but he was not interested in the philosophy of the subject. He was interested in getting away from the necessity of using partial differential equations (PDEs), and the attendant “playing with variables” to find solutions to physical questions. He sincerely believed that there had to be a better way and he sought to find it in the observable world of Euclidean geometry, absolute magnitudes and ordinary commutative mathematics. 

However, as we have studied the consequences of the RST postulates at the LRC, and in working to develop a new theory based on them, we have discovered the work of Hamilton in mathematics, the very man who coined the terms “vector” and “quaternion.” In the process, we discovered the work of Hestenes, who laid out the development of algebra, from the days of Hamilton, in terms of key concepts of Grassmann and Clifford, which has led to our ideas of unit time and space displacements, which have two interpretations, the quantitative (-1, 0, +1), and operational (1/2, 1/1, 2/1) interpretations.

All of this was studied in terms of the tetraktys, which ended up allowing us to unify the mathematical concepts of magnitude, dimension and “direction,” with the corresponding geometrical concepts of point, line, area and volume. The effort ever since then has been to understand our RST-based physical theory in terms of this unified view of mathematics and geometry.

The result has been gratifying to say the least. We have even received a couple of substantial private awards, from those who have followed our efforts and wanted to encourage the work, but, now, we are at the point of having to understand how to add, subtract, multiply and divide with the numbers of the tetraktys that we have developed. We need an “ordinary commutative mathematics,” a multi-dimensional, scalar, algebra, if you will, to enable us to combine units of scalar motion and calculate the relations between them. In short, we need an algebra of multi-dimensional scalar motion.

With an algebra of multi-dimensional scalar motion, we can combine our preons into the bosons of photons, and then combine these into fermions of quarks and leptons, and then combine these into the elements of matter and anti-matter. Such an algebra will enable us to calculate the properties of each of these entities with a precision equal to, if not greater than, the precision obtained in quantum mechanics, but with unprecedented conceptual clarity.

We have made a lot of progress in establishing the fundamentals of the scalar algebra. One important milestone is the identification of the multi-dimensional numbers of the tetraktys with the geometric structure of Larson’s Cube (LC), which has enabled us to describe the S|T units in terms of discrete and continuous magnitudes. This way, we can do calculations, using the discrete numbers of the LC’s lines, squares and cubes, and then we can transform these into the corresponding continuous numbers that we have found in the ratios of its nested radii, surfaces and volumes.

An important indication that we are not wasting our time is the recent derivation of Einstein’s famous mass —> energy equation from the fundamental components of the S|T units. This was done by simply writing the two reciprocal units of the S|T unit as follows:

Figure 1. Einstein’s Equation in Terms of the S|T Unit of Scalar Motion

Since the photon in the LRC’s theory consists of one or more of these S|T units, we now have to calculate the different frequencies possible and their associated energy and how these bosons relate to the fermions, but it’s interesting to see that the properties of the preon of the fermion, photons in our theory, conform to the fundamental energy equation of mass and energy, as shown here.

Hopefully, this new ratio will prove useful, but we can extend it to the two balls that are defined by the LC. The diagonal of a 1x1x1 cube is the square root of 12 + 12 + 12 = 3, and since the radius of the outer ball of the unit LC is the diagonal of the unit cube, then we can use this radius and its inverse as the basis of the 3D number system, and the corresponding tetraktys.

To extend the inverse numbers, defined by the inverse LC to infinity, we simply calculate the reduction of the length of the sides of the current cube by the 45 degree rotation of its side to obtain the side length of the new cube, and then multiply this number by the square root of 3 to get the reduced radius and plug it into the appropriate formula for the diameter, surface area and volume of the ball.

This gives us a 3D number line, analogous to the 1D and 2D number lines we have already compared:

Figure 1. The the Right Lines and Circles of the 3D Number Line. 

It’s important to note that the projected radius of the 3D outer ball is larger than the radius of its 2D cross section counterpart, since its actual third dimension must be large enough to just contain the eight corners of the unit LC. However, if the larger radius (31/2) were actually rotated out of the page, into the third dimension, it’s clear that it would appear to shrink to the 2D size of the cross section (21/2). Therefore, the 2D projection, or representation, of the 3D outer radius is actually larger than the boundary of the unit cross section. 

The inverse is true for the inner ball: Its 2D representation must be smaller than the 2D boundary because its unit length is rotated out of the page and projected onto the cross section from above it. Hence, to show the true length on paper, the outer 3D radius is rotated onto the page, increasing the length of its projection onto the cross section, while the inner 3D radius is rotated out of the page, decreasing its projection onto the cross section.

It was this nuance that escaped my notice and introduced errors into the calculations that I was pursuing earlier. 

More later.

Given Sir Hamilton’s complaint that the science of algebra pales in comparison to the science of geometry, which we have referred to often, it is gratifying that we have discovered that the tetraktys is a mathematical map of the LC. That is to say, the numbers of the tetraktys correspond to the lengths of the right lines in the LC’s 2x2x2 stack of unit cubes:

1. 20 = 1 = 0D unit expansion of LC (point)
2. 21 = 2 = 1D unit expansion of LC (Line)
3. 22 = 4 = 2D unit expansion of LC (Area)
4. 23 = 8 = 3D unit expansion of LC (Volume)

This correspondence of the numbers of the tetraktys with the geometry of Larson’s cube is highly significant, since the LC not only contains the discrete magnitudes of the geometric cubes, but also the continuous magnitudes of the geometric balls.

Indeed, the LC contains the new number line, in the form of nested right lines and circles, which we have been investigating in light of the 4n2 numerical patterns of the wheel of motion, especially in light of the Le Cornec findings.

In the previous post, we discussed the two operational interpretations of the rational numbers, the quotient interpretation and the difference interpretation and how there are two units involved: One unit, 1/2, is the inverse of the other, 2/1, which are the units of the LC, and its inverse, but also the corresponding units of the SUDR and TUDR, respectively.

Since the SUDR is the 3D oscillation of space/time, while the TUDR is the 3D oscillation of time/space, this means that the magnitude, or the speed, of the TUDR is four times greater than the magnitude, or speed, of the SUDR. This was troublesome actually, because the number of preons in the S|T triplets depends upon a 1:1 relative weight between the two.

However, we recognize that, from the perspective of unit speed, the two entities are equal, because each is a unit displacement from unit speed, albeit in opposite “directions.” This equality works out beautifully for identifying the various entities of the standard model as combinations of preons, but not so much for the energy properties of the wheel of motion.

In the latter case, the number four, the quotient relation of the relative number of S|T units in the preons appears to be more important, because we are dealing with the relative energies of the SUDRs and TUDRs in our investigation of the atomic spectra.

But now that we have the quantitative number four (i.e. T|S = (2/1)/(1/2) = 4) , what we need is to understand the n2 part of the equation. What physical property corresponds to the n and why is it squared? We have sought an answer to this question in the mathematics and geometry of the LC and tetraktys for years, but have only been teased with intriguing hints. 

In the quantum mechanics of the LST, the n in their 2n2 equation corresponds to the energy shells of the nuclear atom, and the shells host the orbits of the two electrons allowed by the Pauli Exclusion principle in each orbit, which all fits so nicely into the classical idea of angular and orbital momentum and the four quantum numbers of QM. However, truth be known, you can’t look too closely, or some serious flaws appear in the model.

In the RST based model we are building, consisting of combinations of 3D space oscillations (SUDRs) and 3D time oscillations (TUDRs), the number of electrons is associated with the number of protons, but the electrons are not modeled as residing in concentric shells orbiting a nucleus, but oscillating in connection with the associated proton, which, again, leaves us with the question, “If the n term in the 4n2 periods of the wheel does not correspond to shells, as does the principle quantum number, N, of the QM atomic model, what does it correspond to?”

Whatever the answer is, it has to have a square relation, not a cubic one, which is puzzling given that the the volumes of the atoms would seem to determine their order, not their cross sections. Yet, the 22, 42, 62, 82 of the periods correspond to the increasing areas of the expanding LC, not the increasing volumes. 

Well now it appears that the square relation might reside in the relation of the inverses, at least at the unit level. To see this, we merely need to recognize that the inverse of the tetraktys is the double of the binomial, just as the inverse of the LC is it’s double. Remember, this follows from the equation of inversive geometry, where

r’2 = r * r”

When r is 1, then r’ is the square root of 2 and r” is 2, which is the next set of right lines in the expanding LC. In other words, the 2x2x2 stack of 8 unit cubes expands to a 4x4x4 stack of 64 unit cubes in the discrete expansion of two units of time, which just contains the continuous outer circle expansion of the LC. So, if we relate a 2D slice of the LC to the number line, the nested right lines and circles correspond to the numbers on the 0D time line (radii of the circles), the 1D diameters, the 2D areas and the corresponding 3D volumes (that are implied from the 2D slice.)

However, recall that we found that the number 1, the radius of the inner ball of the LC, is troublesome, since 1n is always equal to 1, regardless of the magnitude of n, so our new number line drops down one level, so-to-speak, to the next smaller ball, with radius r equal to the inverse of the square root of two, which is the inverse of the radius of its associated outer ball, with radius equal to the square root of 2. This is shown in figure 1 below.

Figure 1. The Right Lines and Circles of Larson’s Cube Fitted to the New Number Line

The relation of inversive geometry is still the same, but now r = 1/(21/2) (green radius), r’ = 1 (red radius) and r” = 21/2 (blue radius) and the equation, r’2 = r * r” still holds true: 12 = 1/21/2 * 21/2.

Generating a new unit LC on this basis gives us a tetraktys of

1. 20 * (1/21/2) = 1/21/2 
2. 21 * (1/21/2) = 21/2  
3. 22 * (1/21/2) = 81/2  
4. 23 * (1/21/2) = 321/2 

Therefore, instead of the corresponding inverse tetraktys being doubled to a quadnomial expansion, 40 = 1, 41 = 4, 42 = 16, 43 = 64, it is doubled to the square root of 2, since 2 * (1/21/2) = 21/2, giving us:

1. 20 * (21/2) = 21/2 
2. 21 * (21/2) = 81/2  
3. 22 * (21/2) = 321/2  
4. 23 * (21/2) = 1281/2 

for the inverse tetraktys. 

Hence, whereas the ratio of the unit expansion of the tetraktys to the unit expansion of its inverse tetraktys, at each level of the tetraktys, using the unit of the traditional number line, is,

1. 20:40 = 1:1 = 1:1 (point to point ratio (duration ratio))
2. 21:41 = 2:22 = 1:2 (line to line ratio)
3. 22:42 = 4:42 = 1:4 (area to area ratio)
4. 23:43 = 8:82 = 1:8 (volume to volume ratio),

now, the ratio of the expansion of the tetraktys to the expansion of its inverse, which corresponds to the unit of the new number line, is a constant ratio, 1:2, at all four levels:

1. 1/21/2:21/2 = 1:2 (point to point ratio (duration ratio))
2. 21/2:81/2 = 1:2 (line to line ratio)
3. 81/2:321/2 = 1:2 (area to area ratio)
4. 321/2:1281/2 = 1:2 (volume to volume ratio)

This is very interesting, using the new number line like this, because, since the TUDR tetraktys is the inverse of the SUDR tetraktys, their product is 1/2, at each level, while the inverse of this ratio is, of course 2/1.

Could it be that we are on to something here? More later.

Update: I don’t know why I wrote “product” in that last statement, when it obviously should read “quotient.” However, the product too is interesting, since it gives us:

1. S0 * T0 = 12  
2. S1 * T1 = 22 
3. S2 * T2 = 42 
4. S3 * T3 = 82   

Notice that the exponents of these factors are not summed in the product, because they indicate the geometric dimensions of S and T, not the number of factors in a number. The number of factors is contained in the tetraktys itself, but when we multiply these by the unit, 1/21/2, or the inverse unit, 21/2, it’s as if we are counting these as the sides of the square (4), and the edges of the cube (8).

The exception is the 0D components, because they are mathematical inverses, while the others are not. None of this may matter in the final analysis, since the product of space and time normally doesn’t make sense conceptually. 

In the next post, we will discuss the analog magnitudes of the tetraktys and LC.

I mentioned one of the effects the contest has had on me in the New Physics blog: It forced me to recognize that the number line is sensitive to perspective. With respect to the unit progression, or the physical datum of the physical system, the RST, there is only a difference in “direction” between less than unity and greater than unity speeds, while from the perspective of one or the other, the inverse is alway greater. 

In other words, from the perspective of 0 (i.e. 0 displacement from unity), a unit space/time displacement of 1/2 is no different than a unit time/space displacement of 2/1, except in “direction.” They are separated by two units, one in one “direction” and the other in the opposite “direction.” However, from the perspective of 1/2, 2/1 is four times as great, or it is one-fourth as big. On the other hand, from the perspective of 2/1, the same perception holds. An observer in the t/s sector of the universe would regard his time (our space) and his space (our time) exactly the same way we do.

But, from the perspective of a unit speed, a slower speed than unit speed is not the same as a higher speed than unit speed, just as .5 is not the same as 2, though they both are one unit of displacement removed from unity, in opposite “directions.” There is a quantitative difference as well as a qualitative difference, in the latter case.

Hence, in considering the mathematics of the new number line, there are these two aspects of the same relationship to wrestle with. How do we add, subtract, multiply and divide with these 3D numbers? If we add two s/t units, is the sum greater or less than one t/s unit? If less, then four s/t units are equivalent to one t/s unit. If greater, then one s/t unit is equivalent to one t/s unit. Since the universe of motion deals with speeds, I have always thought that the unequal relation held, but when I realized that the 3D inverse of space is required for 3D oscillation, then the equal relation is required.

This leads me to think harder about rational numbers. When a rational number is equated with the infinite parts of a whole, a fraction of the whole, then these fractions and multiples of the whole reside entirely within the realm of positive real numbers: 0 —> infinity. But when a rational number is equated with two, reciprocal, aspects of one component, such as two orthogonal dimensions of space, then both magnitudes are multiples of the whole, residing entirely within the realm of 0 —> infinity, because they are completely independent variables.

Of course, we can add fractions of the whole to the accumulated total of units, in each orthogonal dimension, in order to obtain greater precision in specifying these positive magnitudes of space, but we can clearly see that the meaning of the rational number, as a fraction of a positive magnitude, and its meaning as the ratio of the magnitudes of the two orthogonal dimensions, are quite distinct.

In the context of the space/time ratios, where space is taken to be the inverse of time, we need to make the same type of distinction between the two meanings of rational number. Larson’s conclusion was that the discrete unit postulate prevents fractions of units, in all but the effective sense. In other words, when the limit of a discrete unit is reached in the relations between motions, then motion, s/t, limited by the discrete unit of space, can revert to motion, t/s, which is to say, motion in time, something Larson called “equivalent space.” He writes in “New Light on Space and Time”:

Let us consider an atom A in motion toward another atom B through free space…. According to accepted ideas, atom A will continue to move in the direction AB until the atoms, or the force fields surrounding them, if such fields exist, are in contact. The postulates of the Reciprocal System specify, however, that space exists only in units, hence when atom A reaches point x, one unit of space distant from B. it cannot move any closer to B in space. It is, however, free to change its position in time relative to the time location occupied by atom B. The reciprocal relation between space and time makes an increase in time separation equivalent to a decrease in space separation, and while atom A cannot move any closer to atom B in space, it can move to the equivalent of a spatial position that is closer to B by moving outward in coordinate time. When the time separation between the two atoms has increased to n units, space remaining unchanged, the equivalent space separation, the quantity that will be determined by the usual methods of measurement, is then 1/n units. In this way the measured distance, area, or volume may be a fraction of a natural unit, even though the actual one, two, or three-dimensional space cannot be less than one unit in any case.

This is an astounding, but perfectly consistent concept. It means that the only way a unit radius ball of space can contract to zero is for an inverse ball of time to increase to unit radius and vice-versa, but Larson never envisions this idea of equivalent space (time) in any other sense than that of relative positions, the non-progressing locations of space and time occupied by atoms. Clearly, however, the consequences of this concept ought to manifest themselves much earlier in the development of his RSt. The reason they don’t, I suspect, is that Larson’s initial progression reversals are 1D not 3D, as are ours, and the requirement for the contraction of 1D units to zero, needing to be accompanied by the expansion of 1D units of the reciprocal aspect, is not as apparent in the 1D case as it is in the 3D case.

Regardless, the idea that 3D time, or 1/s3, must increase from 0 to 1, if 3D space, or s3/1, is to decrease from 1 to 0, is a fundamental consequence of the RST postulates. The fact that it is mathematically consistent is shown by the 2D analogy of rotation, when we describe rotation by the changing angle of the radius, together with the changing angle of its inverse, or the two changing angles of the rotating diameter of the unit circle. As one end of the diameter rotates the last degree, say inward from 179 degrees toward 180 degrees (or 1), the inverse end MUST rotate inward from 359 toward 360 degrees (or 0), and as the rotation of the diameter reverses “direction” at 180 degrees, heading away from 180 degrees outward toward 181 degrees, the inverse end must also reverse “direction” heading outward from 360 degrees (0) toward 1 degree. There is no other way.

So this is a major distinction between the rational numbers of true inverses, and the rational numbers of orthogonal variables. In the latter case, we can change the magnitude of one, without affecting the other, but not so in the case of the former. At least in the case of the space/time progression, where it serves as the datum of the physical universe, an increase in space has the same affect on the magnitude of the motion, as an increase in time, just as the magnitude of an area is affected equally, regardless of which of its two, orthogonal, dimensions is increased or decreased.

The difference is that the magnitude of an area is not normally required to be held constant, while the magnitude of the natural progression of the RST is. Therefore, we cannot always treat the numbers in the space/time ratios that pertain to the order of progression, in the same manner that we treat the numbers of the x/y dimensions that pertain to bounded magnitudes.

For instance, we cannot just add (subtract) quantities of space (s/1), or quantities of time (t/1), to/from existing units of motion, changing their magnitudes. In order to change the magnitude of motion (s/t or t/s), we have to add (subtract) units of motion to/from units of motion.

So this difference requires a different algebra than the one we use with the notion of bounded magnitudes. It is an algebra restricted to rational numbers, where the two units that form the numbers of the number system that constitute the unit ratio cannot be sub-divided, as with a knife. The range of sub-divisions of the bounded magnitudes of traditional algebra is unlimited, but no such concept is possible in the new algebra.

This has many consequences, some of which we will try to explore here soon.

This has worked out fairly well for the research program of physics based on the vectorial motion in the LST, but it is totally unsuitable for the physics based on the scalar motion of the RST. We need a more complex, complete and consistent view of the scalar number line in order to use scalar mathematics in the development of the RST’s scalar theory.

However, the first thing we notice is that the RST’s scalar progression is 3D and therefore non-linear. Fortunately, though, we can use the combination of Larson’s 2x2x2 cube and its associated inner and outer spheres to construct a new, mult-dimensional, scalar number line that is linear. There are several aspects to this approach and, to understand it, we will have to take them one at a time.

The first thing we want to note is that the multi-dimensional magnitudes of the cube are integer indexes to the non-integer multi-dimensional magnitudes of the associated spheres. This is important to understand, since it enables us to unify the integer and non-integer magnitudes the way nature does, and, hopefully, it provides the key to understanding the mysterious connection between mathematics and physics.

To demonstrate what is meant by indexing the continuous magnitudes of physical variables with the discrete variables of numbers, we need to begin by analyzing the dimensions of Larson’s cube, as shown in figure 1 below.

Figure 1. Multi-Dimensional Number Line from Expansion of Larson’s 2x2x2 3D Cube.

In a 3D numerical progression, n3, all three dimensions (four counting 0) - the dimensional resolutes we might say - expand with the cube simultaneously. The magnitude of the 0 dimensional expansion, n(20), increases as a function of one-half of any given axis; the magnitude of the 1 dimensional expansion, 3(2n), increases as a function of the six 1D “directions” of the three axes; the magnitude of the 2 dimensional expansion, 3(2n)2, increases as a function of the 12 2D “directions” of the three axes, and the magnitude of the 3 dimensional expansion, (2n)3, increases as a function of the eight 3D “directions” of the three axes.

Figure 1 shows only one quadrant of the expanding cube, and the inner row/column is labeled with the 0 dimensional numbers, while the corresponding 1D, 2D and 3D numbers are labeled as successive outer layers of the quadrant (the factors of 3 in the 1D and 2D numbers comes from the 3 axes of expansion.)

By selecting just one quadrant of the expanding cube and labeling the magnitudes of all four dimensions in this manner, we get a scalar number line, where the vertical line is independent of the horizontal line, which will eventually allow us to include the reciprocal property that at this point is not apparent. In addition to assuming the presence of the other quadrants in the expansion, the figure does not show the third dimension graphically, but assumes its presence (the z axis with magnitudes in front of and behind the page.)

By accommodating all the magnitudes of the four dimensions this way, we can simplify the required graphics considerably, while maintaining the 3D scalar concept. Next, we can add the non-integer complement magnitudes of the associated inner and outer spheres to figure 1, as shown in figure 2 below.

Figure 2. Multi-Dimensional Number Line from Larson’s Cube with Inner and Outer Spheres

Of course, the magnitude of the radius of the inner sphere is always an integer and that of the outer sphere is always a non-integer, when n >= 1. Multiplying the multi-dimensional values of the outer radii by factors of pi, we obtain 1D (circumference), 2D (surface) and 3D (volume) continuous multi-dimensional magnitudes, indexed by the corresponding integers.

However, while the outer radii are multiples of the square root of 2, when the 0D magnitudes are greater than 1, the inner radii are inverse multiples of the inverse of the square root of 2, when the 0D magnitudes are less than 1, as shown in figures 3 and 4 below.

Figure 3. Outer Radii are Multiples of the Square Root of 2 at Indexes Greater Than 1.

Figure 4. Inner Radii are the Inverse Multiples of the Inverse of the Square Root of 2 at Indexes Less Than 1.

Hence, we can plot the radii linearly on a line, what we are want to call the new scalar number line:

…1/3(1/21/2), 1/2(1/21/2), 1/1(1/21/2), 1/1(2/11/2), 2/1(2/11/2), 3/1(2/11/2)…

comparing this to the traditional scalar number line:

…1/3, 1/2, 1/1, 2/1, 3/1…,

we see several differences. First, there is a distinct difference between the counting multiple and the unit. In the traditional line they are one and the same: 1(1), 2(1), 3(1), …, but in the new line the counting multiple, successive increments of 1, is very different from the unit, which is the square root of 2.

Proceeding in the opposite “direction,” the counting multiple of the traditional line is the inverse of the positive multiple, while the unit is the inverse of 1: …1/3(1/1), 1/2(1/1), 1/1(1/1), but because the inverse of 1/1 is indistinguishable from 1/1, it is not recognized that there are TWO units involved, where one is the inverse of the other.

In the new line, the unit of the outer sphere is the square root of 2, while the unit of the inner sphere is the inverse of the square root of 2, as can be clearly seen by comparing figures 3 and 4, so this requires two instances of the mathematical value of 1, if you will.

Interestingly enough, one of the confusing issues of working with the scalar concepts of the RST, is that while 1/1 is equal to 1/1 mathematically, s/t is not equal to t/s physically. The new scalar number line should be a great help in this regard.

Update: I just noticed that the graphic in figure 4 is the wrong one. I’m making a new graphic for it now and will update the figure soon.

 Update: Replaced graphic in figure 4 (please pardon the distortions.)

However, there’s no doubt where this topic goes. I want to take the new math from the top, and lay out the new concepts from the beginning. I will be referring to them as I develop the physics theory on the other blog.

The first concept that must be clearly understood from the start is that the reason for calling it the new math is that there are two interpretations of number. the first interpretation of number is the usual quantitative one that is a measure of how much or how many of something there is. In the second, the operational number represents a relation between two quantities. 

We begin by viewing the familiar quantitative number line below in light of these two interpretations of number.

Figure 1. The Quantitative Interpretation of Number Line

In the quantitative interpretation of number, the whole numbers and proper fraction, rational, numbers lie to the right of 0 on the number line, in all cases. For instance, the number 1 occupies the first place to the right from 0, and 1/2 lies half way between 0 and 1 on the quantitative number line. The negative numbers and negative proper fractions to the left of 0 are somewhat problematic and were only accepted by mathematicians gradually and grudgingly. Wikipedia defines them as follows:

Negative integers can be regarded as an extension of the natural numbers, such that the expression x – y has a well-defined value for all values of x and y. Other number systems, such as the rational numbers, are then derived as progressively more elaborate extensions and generalizations from the integers.

On the other hand, in an operational interpretation of a rational number, we can take the relation of the numerator and denominator, say the difference between them, instead of the quotient, and it permits us to replace all the positive numbers on the number line with the reciprocal of proper fractions that replace all the negative numbers on the line, none of which are less than 1.

This way, we get a new number line,

1/n, …1/3, 1/2, 1/1, 2/1, 3/1, …n/1,

which is an operational equivalent of the quantitative number line in figure 1, above, but which is not based on integers, but constitutes a new generalization from which integers themselves are derived. In this case, however, instead of positive and negative numbers, we have a rational number and its inverse. To be sure, while the rational numbers are not the same as the quantitative numbers on the quantitative number line, their operational interpretation is; That is, 

1/n = 1-n; …1/3 = -2; 1/2 = -1; 1/1 = 0; 2/1 = 1, 3/1 = 2, …n/1 = n-1;

In Larson’s new system of physical theory (RST), as opposed to the legacy system of physical theory (LST), there are two, reciprocal, sectors of the physical universe, the sector where motion is above unity (the cosmic sector), and the sector where motion is below unity (the material sector.) Within each of these two sectors, there is an important sub-sector, the interior of unit distance, which Larson refers to as the time region (inside unit space) and the space region (inside unit time).

A complete mathematical analogy of this space-time structure can be reproduced by considering the quantitative and operational interpretations of number together. The operational interpretation extends outward from 0 (1/1) to infinity, in both “directions,” while the quantitative interpretation extends inward from 1 and -1 (i.e. 2/1 and 1/2 respectively) to 0 (i.e. 1/1), in both “directions.”

However, there is another important distinction between these two interpretations of number, besides their respective data of 0 and 1, and it must be understood as well. In the operational interpretation, we must pick a perspective; that is, we must view the reciprocal side of the datum from its inverse perspective, just as we must view a see-saw from one side or the other. We cannot view the operational interpretated number line from both sides at the same time, any more than we can view the see-saw profile from both sides of the fulcrum at the same time. In terms of motion, this means we must choose to interpret both views as above and below unit motion, or above and below unit inverse-motion (s/t or t/s, but not both together.)

On the other hand, in the quantitative interpretation of number, we must view the reciprocal side of the datum from its own perspective, where one side is motion, while the other side is inverse-motion (s/t and t/s, at the same time.) This difference is illustrated in figure 2 below.

Figure 2. Operationally and Quantitatively Interpreted Number Lines

The division operation of the quantitative (how much or how many) interpretation of number requires us to differentiate the positive and negative quantities, as if they were real, even though there is no such thing as a negative quantity. As Sir Rowland Hamilton observed:

it requires no peculiar scepticism to doubt, or even to disbelieve, the doctrine of Negatives
and Imaginaries, when set forth (as it has commonly been) with principles like these: that a
greater magnitude may be subtracted from a less, and that the remainder is less than nothing; that two negative numbers, or numbers denoting magnitudes each less than nothing, may be multiplied the one by the other, and that the product will be a positive number, or a number denoting a magnitude greater than nothing; and that although the square of a number, or the product obtained by multiplying that number by itself, is therefore always positive, whether the number be positive or negative, yet that numbers, called imaginary, can be found or conceived or determined, and operated on by all the rules of positive and negative numbers, as if they were subject to those rules, although they have negative squares, and must therefore be supposed to be themselves neither positive nor negative, nor yet null numbers, so that the magnitudes which they are supposed to denote can neither be greater than nothing, nor less than nothing, nor even equal to nothing.

Contemplating the arbitrary nature of the quantitative number line, Hamilton sought a better approach using the dynamic concept of order in progression, rather than the static concept of bounded magnitude. This was a good idea, as far as it went, but it requires two orders of progression to make it work, not just one. Hamilton’s idea was to use the flow of time to give algebra an intuitional foundation, but Larson’s idea was to use the flow of time, together with the flow of space, to put physics on an intuitional foundation.

At first Larson’s idea seems absurd, and it would never had ocurred to Hamilton, but today the flow of space has actually been observed. The logical conclusion is that the two should be considered together. The difficulty is recognizing that they are not separate quantities, but actually two aspects of the same quantity, motion. We start with unit motion and go in both “directions,” toward greater or less than unit motion, when the flow of one aspect is less than the flow of the other.

Larson’s conclusion was that the only possiblility of introducing a difference between the two flows, is to assume that one or the other of them periodically reverses its “direction.” He called this simple harmonic motion, and he pointed out that it was just as reasonable to believe that the flow of space, or of time, could oscillate as not, and that this is the basis of all physics.

Were this the summum and bonum of the subject, we would be home free, but it is not complete at this point, because the two inverse aspects of the universal motion, the two flowing quantities, if you will, do not have the same dimensions. The flow of space exists in three dimensions, while the flow of time has no dimensions. Mathematically, then, the natural progression is not linear. If:

s/t = 23/20

then it does not give us a natural progression of 0, 1, 2, 3, … but rather it gives us a progresssion of 0, 8, 216, 512, …, and, at first glance, it’s totally impractical to construct a number line from such a non-linear progression.

However, it turns out that, within this 3D progression, there is an associated 0D, 1D and 2D progression as well, and by recognizing that the natural progression contains all four numerical progressions, we can construct a new, composite, number line and with it a new number system to use in our investigations of the RST. We will take a look at it next time.

The crucial analysis of the fundamentals that seems to provide us with the clues that this might be so starts with Larson’s idea of scalar motion. As regular visitors of the LRC site know, scalar motion is a definition of motion without reference to moving objects. The equation of motion, v =ds/dt, simply involves a change in space over time, and a changing location of an object is not required to produce the equation’s change of space, just as it is not required to produce its change of time.

In Larson’s system, the initial condition of the universe assumes a natural space clock as well as a natural time clock, the one being the inverse, or reciprocal, of the other. Hence, this assumption defines a universal motion, as the physical datum of the system.  There are several important differences between the new natural type of motion, with no motion of an object involved, and the motion of objects with which we are familiar. One of the most basic differences is that the familiar motion of an object Y, from point X to point Z, increases the distance XY and decreases the distance YZ. On the other hand, the new natural type of motion changes distance itself; that is, both the distance XY and YZ are increased, or decreased, at the same time, making it impossible to define the motion of the object Y, in terms of the changing distance relative to X and Z, with one increasing and the other decreasing. It’s as if the size scale of the system were changing.

This expansion/contraction motion, though easily observed in nature, is quite unlike the motion of an object from one point to another, specified in some specific direction that can be defined in terms of three dimensions. In a 3D system, scalar motion would change the size of a spatial location in all three dimensions simultaneously. This makes scalar motion more difficult to work with in some respects, because the system’s locations (x, y, z), regardless of size, must continuously expand. While at first this is very disconcerting, it turns out that there are ways to cope with it that are straightforward.

Consider a 1D scalar expansion for instance, disregarding the expansion of the points themselves momentarily, the distance between points A and B increases over time. We can choose location A as a reference and measure the expansion in terms of B’s motion away from A, or we can choose B as the reference and measure the expansion in terms of A’s motion away from B, in the opposite direction. Either way, we can conclude that each dimension of scalar motion has two, opposed, directions. In a 1D system there are two scalar directions, in a 2D system there are four scalar directions, and in a 3D system there are eight scalar directions. 

Assigning numbers to the binary directions in each dimension, we get 2⁰ = 1 direction, in the zero-dimensional system (more on this exception below), 2¹ = 2 directions, in the one-dimensional system, 2² = 4 directions, in the two-dimensional system, and 2³ = 8 directions, in the three-dimensional system. Substituting these numbers in the equation of motion, we would get:

ds/dt = d2⁰/d2⁰, for zero-dimensional motion,

ds/dt = d2¹/d2¹, for one-dimensional motion,

ds/dt = d2²/d2², for two-dimensional motion,

ds/dt = d2³/d2³, for three-dimensional motion,

However, as we observe time, it’s clear that it has only one direction, called the “arrow of time,” which is increasing magnitude only; that is, a point in time has no direction, and therefore no extent, in space. On this basis, we can consider time as a zero-dimensional scalar, something that can be counted, but not expanded. Meanwhile, it’s clear that the space that we occupy is three-dimensional; that is, it extends into three dimensions, and, since scalar motion has no specifiable direction, by definition (i.e. it is motion with magnitude only), the expansion of space must be effective in all of the dimensions of the system (i.e. space is a pseudoscalar). Modifying the equation of scalar motion accordingly, we get

ds/dt = d2³/d2⁰,

where space, s, has 2³ = 8 directions, and time, t, has 2⁰ = 1 direction, the scalar “direction” of increasing magnitude only. By defining space and time this way, as the reciprocals of each other, in the equation of motion, the quantity space is differentiated from the quantity distance, which becomes the product of motion and time, as in the ordinary vectorial motion (i.e. motion with direction defined by locations with three dimensions). However, in this case, using the scalar motion equation, distance, d, is a three-dimensional quantity, not a one-dimensional quantity:

 d = Δs³/Δt⁰ * t⁰
    = (n2)³/(n2⁰) * n2⁰ 
    = (8*1³)/(1*1)
    = 8*1³

for each unit of change, n. For example, for two n, we get

d = ((2*2)³/(2*2⁰) * (2*2⁰) = (64*1³/2) * 2 = 64*1³,

or 64 cubic units of volume expansion in two units of time. The expansion series, or “distance” d, as time, t, marches on then is not the familiar linear series of lengths 1¹, 2¹, 3¹, 4¹, …n¹, but the less familiar, non-linear, series of volumes, 8³, 64³, 216³, 512³, …n³.

Geometrically, the first term in this expansion series corresponds to the initial 2x2x2 stack of one-unit cubes, dubbed Larson’s Cube, at the LRC. It is shown in figure 1 below.

Figure 1. Larson’s Cube as the 8 Unit Stack of One-Unit Cubes. 

The red dot in the center corresponds to the 2⁰ = 1, dimensionless, time magnitude, while the stack of eight 3D cubes corresponds to the 2³ = 8 * 1³ space magnitudes, at t₁ - t₀ = 1. Expanding in the next unit of time, at t₂ - t₀ = 2, to two units of space in all directions, it’s easy to see that the stack of one unit cubes, consisting of of 2x2x2 = 8, one-unit, cubes, in figure 1, expands to a 4x4x4 = 64 stack of one-unit cubes. In the third unit of time, the stack expands to a 6x6x6 = 216 units, then to a 8x8x8 = 512 units, and so on, ad infinitum. Meanwhile, the 2⁰ point at the intersection of the cubes, does not expand.

However, this mathematical expansion of the pseudoscalar does not correspond to a physical expansion, because a physical expansion of the pseudoscalar must expand in all directions, defined by three dimensions, not just the three orthogonal directions that constitute its three dimensions. Thus, the physical expansion is manifested as an expanding sphere, not as an expanding cube, and this presents us with the fundamental challenge faced by the Greeks: “How do we calculate the volume of the sphere that corresponds to the volume of the stack of one-unit cubes?” In other words, we need a geometric algebra of quantities that includes the areas of circles and the volume of spheres, as well as the linear extent of right lines, an algebra, which corresponds to a fully functional, non-pathological, numeric algebra, for doing physical calculations in a scalar/pseudoscalar system. In other words, it’s back to the old conundrum of squaring the circle.

Unlike the Greeks, however, we now know that multiplying the sides of a polygon inside the sphere will always result in an approximation, and thus it can’t be represented by a rational number. Since in our universe of discrete motion, as in the Pythagorean universe of discrete numbers, all is number, this is hardly welcome news.

Nevertheless, as we consider the problem, we see that there are two spheres that can be related to the stack of one-unit cubes. One sphere that can be drawn to fit just inside the stack, and the other that can be drawn to just contain the stack. A two-dimensional view of the one-unit instance of these three figures is shown below.

Figure 2. Two-Dimensional View of 2x2x2 Stack of One-Unit Cubes with Inner and Outer Spheres  

In figure 2, the radius, c, of the outer sphere, S₁, is the square root of 2, by the Pythagorean theorem, while the radius, d, of the inner sphere, S₂, is 1, since the radius is r = a = b = 1. By the formula for the area of the surface of a sphere,

A = 4π * r²,

the area of the surface of the sphere S₁ is 8π, while the area of the surface of the sphere S₂ is 4π. Also, by the formula for the volume of a sphere,

V = 4/3π * r³,

the volume of the sphere S₁ is the square root of 2, cubed, times the volume of S₂, which is just 4/3π, since its radius is 1.

Table 1 shows the tabulated circumferences (2*r*π), areas and volumes for spheres S₁ and S₂, and their ratios, for units 1, 2, 3 and 4.

Table 1. Circumferences, Areas and Volumes for Units, 1, 2, 3 and 4

Notice that the S₁/S₂ ratio is just a power of the radius of S₁, or a power of the square root of 2, in each case, denoted “rⁿ” in the last column of the table. The ratio of the surface areas of the spheres is the square of r, or 2, while the ratio of the volumes of the spheres is twice the radius of S₁, which is equivalent to the square root of 2, or r, cubed.

This is an amazing fact that we should be able to exploit in order to replace the 20, 2¹, 2², 2³, numerical units that are so hard to reconcile in a non-pathological, multi-dimensional, algebra.

Recall that, currently, for one-dimensional units, we resort to complex numbers (z = a+bi), the algebra of which is not ordered; For two-dimensional units, we resort to quaternions, the algebra of which is not ordered or commutative, and, for three-dimensional units, we resort to octonions, the algebra of which is not ordered, commutative, or associative!

All of these traditional units depend on one or more imaginary numbers to define their dimensionality, arbitrarily defined as the square root of -1, in the different dimensions of the respective algebras. Of course, in reality, there is no unit that can be physically identified that, when multiplied by itself, is equal to -1, in any dimension.

However, we should remember that the purpose of using the plus and minus signs is only to differentiate between a given dimension’s two “directions.” There’s nothing meaningful about them otherwise. As already noted above, in scalar motion, the choice of a fixed reference (point A or B), with which to measure scalar change, is completely arbitrary.

The same thing is true with numbers. Each number has its inverse and the designation as to which is the number and which is the inverse number is completely arbitrary. Nevertheless, with the number 1, we say that it is its own inverse, and we use this convention to build group theory, where 1 is the identity element.

However, if we could change our number system, from one based on multi-dimensional numbers, using imaginary numbers to define their dimensions, and plus and minus labels to define the two directions of each of their dimensions, to one based on the properties of spheres (i.e. 1D circumferences, 2D surfaces and 3D volumes), the inverse of 1 would no longer have to be itself, but would now be 2, the inverse of 2 would be 4, etc, by the formula for inverse geometry, r’² = r * r’’.

In this way, negative numbers are eliminated conceptually, although the change is actually only one of perspective. It’s like saying that the inverse of -1 is 2 units above it; the inverse of -2 is 4 units above it; the inverse of -3 is 6 units above it, etc. In this case, however, the unit referred to is the square root of 2, r, which is not imaginary, but is the relation between unit dimensions, defining the radius of a sphere.

Just like in the traditional mathematics, the new unit, r, defines the identity element of a group. Figure 3 shows the number 1 of the group, P, the group identity element, P’ (equal to the square root of 2), and the inverse of number 1, p’’ (equal to P’ squared, or 2).

Figure 3. The number 1 of the group (P), the identity element (P’), and the inverse of number 1 (P’’).

In figure 3, P is the radius (1) of the inner sphere, the generator of one 1d (circumference) quantity, one 2d (surface) quantity and one 3d (volume) quantity. Radius P’ generates the 1d, 2d and 3d quantities of the identity element (square root of 2), while P” is the radius (2), or the inverse of radius P (by P’² = P * P’’), the generator of its 1d, 2d and 3d quantities.

This is no different than the number line, where -1 is one unit removed from 0 and two units removed from +1. The difference is huge, though, because we can represent all three of the dimensional numbers with one radius, and do away for the need of imaginary numbers (C = circumference; A = area; V = volume for the given dimension’s P, P’ and P” quantities):

1) 1D: C_Sp = 2π (i.e. -1); C_Sp’ = 2π*r (i.e. 0); C_Sp” = 4π (i.e. +1)

2) 2D: A_Sp = 4π (i.e. -1²); A_Sp’ = 4π*r²  (i.e. 0); A_Sp” = 16π (i.e. +1²)

3) 3D: V_Sp = (4/3)π (i.e. -1³); V_Sp’ = (4/3) π*r³ (i.e. 0); V_Sp” = (32/3)π (i.e. +1³)

The fact that each successive dimension has it’s own “zero” quantity, or identity element, might take some getting used to, but it would be well worth it, if it enables us to get rid of imaginary numbers and the pathology of higher-dimensional algebras.

In that case, we would have an algebra of 0D scalars, an algebra of 1D pseudoscalars, an algebra of 2D pseudoscalars, and an algebra of 3D pseudoscalars, each one with all three algebraic properties of order, commutativity and associativity.

We’ll see.

However, the ancient Hebrews envisioned our days and characterized them, not as the pinnacle of civilization’s progress, but rather as the deterioration of civilization’s worth, inferior to the quality of previous civilizations. The image of the relatively inferior status of modern nations was explained by the Hebrew Daniel, when he saw and interpreted the king of Babylon’s dream, portraying the progressive degradation of the quality of earth’s civilizations, from that time to this.

According to this vision, the ancient Babylonian kingdom was the highest quality civilization in the world, followed by the inferior, but stronger, Persians, who were followed by the still more inferior, but stronger, Greeks, then by the vastly more inferior and stronger Romans, and finally by the remnants of the Romans, mixed in with the conquering Barbarians, the most inferior, totally fragmented, uncivilized nations of all, who were as clay mixed in with the metal of the Romans. The Romans were as iron compared to the more highly prized bronze of the Greeks and to the sliver of the Persians and to the gold of the Babylonians.

Of course, in the end, all of this is irrelevant, as the vision portrayed all of these old kingdoms as being replaced by a new Hebrew kingdom, which would come rolling forth like a stone down a mountain, smashing the image of Western civilization’s heritage, in all these old kingdoms, to dust. Consequently, the dust of the pulverized image simply blows away, like chaff in the wind, and disappears!

But what does this have to do with modern mathematics and science? We don’t know much about the mathematics and science of the Persians and Babylonians, and what we do know comes to us primarily from the Greeks, who learned from the Babylonians, the Persians, and the Egyptians (who, like the Asians, were never a world dominating nation, but nevertheless were sometimes significant players in mathematics and science).

Clearly, however, the strength of the Persians, relative to the Babylonians, and the Greeks, relative to the Persians, and the Romans, relative to the Greeks, and, in general, the modern nations relative to the ancient ones, is based, in part at least, on the progress of technology. Whether it is based on advanced strategic technology, such as provides greater sustenance, infrastructure and internal strength for the nation as a whole, or on advanced tactical technology, providing for improved weapons, communications and mobility to the nation’s armies, navies, and air forces, technology has always played a crucial role in the strength of civilizations.

The interesting aspect of this in the present context is that, while it shows us how understanding the simple fundamentals of mathematics and science makes a profound difference in the power and technological capabilities of nations, it also shows us that there may be nothing particularly enduring about it either. Civilizations come and go, and the particular aspect of their understanding of math and science that made them capable of great feats of organization, engineering and technological exploitation, comes and goes with them. 

From the smallest of means, proceeds that which is great, the ancients said. For example, who could have guessed that the ability of a few Renaissance scientists to deal with the esoteric concepts of irrational and negative numbers would eventually lead to the modern ability to transcend the technology of the ancients so dramatically? But so it is. Without the ability to abstract the square root of 2 and -1, the whole of modern technology would be impossible.

However, knowing this, we are soon lead to ask what other, simple, fundamentals might we be missing? The fundamentals that some future civilization (perhaps the triumphant kingdom of the Hebrews foreseen by Daniel) might discover, might enable them to transcend our technology as much as we have transcended that of the ancients (or even more).

In thinking about this, one might be tempted to revisit the whole notion of irrational and imaginary numbers, the foundation of modern technology, and seek to understand what it is about this whole approach that makes it so powerful. If there is one way to do this, might there be another, maybe even better way to do it?

Of course, readers of these blogs know that here at the LRC we believe there is, and that we are taking our clues on how to proceed from the works of Hamilton, Grassmann, Clifford, Hestenes, and Larson. Hamilton showed us how defining numbers, as traditionally taken for granted, leaves algebra without a suitable scientific basis. Grassmann showed us that there is an underlying connection between geometry and algebra that the Greeks couldn’t make, and Clifford showed us how the two directions of each dimension forms an algebra. Thanks to the work of Hestenes, which brought the works of Grassmann and Clifford to light, we are provided with tremendous insight into the underlying nature of complex and quaternion numbers, and the imaginary numbers that they are built with.

Finally, none of it would have even caught our attention had it not been for the transcendent work of Larson. It is his brilliant recognition and intriguing development of the new and unfamiliar notions of scalar motion that provides us with the motivation for digging into all these ancient mysteries, driving us to uncover the old foundations, in search of new insight into what makes modern math and physics tick.

What we have found astounds us. Could it be that, as Thales and Pythagoras apparently learned from their predecessors, “Everything is number,” after all? The fact that this faith seemed horribly contradicted by the theorem of triangular squares, that squaring the circle could only be approximated, and that the hare could never catch up with the tortoise, and that today, after centuries of effort, we can now name irrational numbers, use them in the calculus to send robots to explore particular parcels of Martian terrain, use computers to calculate π out to a gazillion decimal places, and work with infinite sets, as easily as the Greeks worked with integers, appears to make the whole issue moot.

“Who cares, if the Greeks thought all was number,” one might think. “Our technology, our math and our science reach so far beyond anything ever dreamed of by the Greeks, that it’s patently clear that we have overcome their intellectual obstacles. Let’s just move on.” Ironically, however, that’s just what we can’t do, and the reason that we can’t do it is that the essence of these very same obstacles stands in our way. We now know that nature is both discrete, definitely measured, like numbers, and, at the same time, continuous, infinitely divisible, like distance.

Yet, in spite of the vaunted “work arounds” of our modern mathematics, which have served us so magnificently, irrational, transcendental, and imaginary numbers, finite and infinite sets, etc, we still cannot do as nature does and seamlessly combine the continuous with the discrete. It is frustrating in the extreme. It appears that, if the ancients taught Thales and Pythagoras that all was number, then they were probably just hopelessly naive and the Greeks were simply beguiled by their priestly robes and their high social status. If we can’t do it today, surely the ancient Babylonians and Persians couldn’t do it either.

That may be so, but it doesn’t mean that they didn’t have a valuable insight into numbers and geometry, which has since been lost, one that might prove to be the key to doing what we so desperately want to do. For instance, even though their approximation of π might have been very rough, compared to our very refined approximation, how do we know that it doesn’t matter, in the end? Of course, barring some unexpected archeological find, we are not likely to ever know more about how the ancients thought than the ancient Greeks did, who were in direct contact with them. The point is not, however, that the ancients had the answers we seek. They probably didn’t, but they may have thought about the fundamentals in a way that hasn’t occurred to us, which could prove to be the key for finding the answers.

As it turns out, there are many intriguing clues that the way the ancients thought about numbers, is close to the new way we are thinking about them here at the LRC. In the next post, we will get into some of the details of this.

1⁰, 2⁰, 3⁰ …

must be actually

1*2⁰/2⁰, 2*2⁰/2⁰, 3*2⁰/2⁰, …

because, when, starting with space and time only, there are no “things” to count, which implies that the natural series,

1¹, 2¹, 3¹ …

is mathematically incorrect, as an initial condition in a space|time progression, since

1*2¹/2⁰, 2*2¹/2⁰, 3*2¹/2⁰…,

is a natural progression of double magnitudes (one in each “direction”) not single magnitudes. Therefore, as a space|time progression, the natural 1D mathematical series necessarily begins with 2, not 1, and increases by 2, 1D, magnitudes, not 1:

2*1¹, 4*1¹, 6*1¹…,

while the natural series,

1², 2², 3² …,

is also incorrect, because

1*2²/2⁰, 2*2²/2⁰, 3*2²/2⁰, … 

is the natural mathematical progression of area, which begins with 2² = 4, 2D, magnitudes, not 1, or 2, increasing the base of the series by a factor of 2:

4*1², 16*1², 36*1²….

Finally, the natural 3D series:

1³, 2³, 3³ …

is also incorrect, as a space|time progression, because it is actually,

1*2³/2⁰, 2*2³/2⁰, 3*2³/2⁰, …, 

which is the natural progression of volume, its magnitudes beginning with 2³ = 8, 3D, magnitudes, not 1, not 2, not 4, again increasing the base of the previous series by a factor of 2:

8*1³, 64*1³, 216*1³ …

All of this means, among other things, that the algebra of these numbers begins with the pseudoscalar value of an n-dimensional progression (2ⁿ), not its scalar value (2⁰); that is, each series begins with the corresponding right side of the tetraktys, not the left side. This is because one line has two directions, and one area has four directions, not two, and it is therefore incorrect to write the progression of 1D magnitudes beginning with the scalar magnitude 1 (2⁰), or to write the progression of area beginning with the 2¹, or 1D, pseudoscalar magnitude. Likewise, one volume has eight directions, not two, and not four, and therefore the natural volumetric series must begin with eight cubic scalars, not one. To accurately denote this, we need to rewrite the 1D progression as

(1*2)¹/(1*2)⁰, (2*2)¹/(2*2)⁰, (3*2)¹/(3*2)⁰, …,

the 2D progression as

(1*2)²/(1*2)⁰, (2*2)²/(2*2)⁰, (3*2)²/(3*2)⁰, …,

and the 3D progression as 

(1*2)³/(1*2)⁰, (2*2)³/(2*2)⁰, (3*2)³/(3*2)⁰, …. 

It’s important to recognize that, when the uniform 3D progression is measured from a given point (2⁰ = 1), at t_n - t₀, the apparent one-dimensional interval characterizes the expanding volume by its 1D radius. However, to calculate the true 1D interval, which is the diameter of the volume, the radius must be doubled; to calculate the true 2D interval, the doubled radius, the diameter, must be squared, and to calculate the 3D interval, it must be cubed:

2*1¹ = 2r = d,

4*1² = d²,

8*1³ = d³

However, this brings us face to face with the age old problem of the quadrature, or of “squaring the circle,” because the 2D space component of the 3D space|time expansion must expand geometrically over time, or circularly, and the 3D component must expand spherically, while the algebraic square and the algebraic cube are necessarily rectilinear, and therefore an issue of 2D and 3D numerical integration, or quadrature, and cubature, as it’s sometimes referred to, arises. 

That this problem is related to the foundations of quantum mechanics is indicated, when it’s recognized that only one point on the surface of an expanding circle, or sphere, can be measured at any given time. Special relativity makes it impossible to simultaneously specify t_n at any more than one point on the 2D, or 3D, surface of the expansion, because points on these surfaces are always moving apart. Therefore, we are brought back to consider the physical enigma of point/wave duality, and the mathematical dilemma of quadrature, and the logical challenge of unifying the concept of the discrete numbers of algebra with the concept of the smooth functions of geometry.

In the next post, we will discuss how the ancient way of dealing with these fundamental issues turns out to be remarkably congruent with our ideas of the space|time progression; that is, what has been called the “mediato/duplatio” (halving/doubling) method of ancient reckoning, intimately associated with the notion of the tetraktys, turns out to be our “factor of 2,” playing in the space|time progression series, as described above, and we will discuss the correspondence between them next time.

This topic is very interesting as it relates the modern concept of rotation, implemented with complex numbers, to our new concept of 3D expansion, implemented with scalars and pseudoscalars, which is a crucial point to understand, I believe.

In the previous post, we showed how combining the unit magnitudes of the positive and negative “directions” defines a two unit “distance,” or interval, analogous to a spatial distance, with two algebraic “directions,” one negative and one positive:

1|2 + 2|1 = 3|3 = 0
(1|2) = -(2|1)
(2|1) = -(1|2)

However, the fact that the pipe symbol indicates that the reciprocal relation is to be interpreted as the value of the difference (sum of opposite signs) between the numerator and the denominator means that any number n|n can be used as the identity element, and since the quantities in the numerator and denominator are defined in terms of order in progression, rather than as increased, or diminished, quantities, it is necessary to recognize how those quantities differ; that is, how does 1 become 2 and 2 become 3, or 1?

With non-reciprocal numbers defined as magnitudes, 1 becomes 2, when two independent magnitudes are summed:

1 + 1 = 2, 2 + 1 = 3,

which represents an arbitrary action of addition. 

However, with non-reciprocal numbers defined as order in progression, 1 becomes 2 and 2 becomes 3, as the progression proceeds:

1, 2, 3, … 

but what are the dimensions of these steps of progression? Ordinarily, the absence of a superscript with a number indicates that it is 1-dimensional, and we have seen that in ordinary mathematics, any number raised to the zero power is defined by the law of exponents, as the number 1, since all such numbers

n/n = n¹/n¹ = 1^1-1 = 1⁰ = 1.

However, as we saw in the last post, this definition is problematic, theoretically, because it means that the unit cube, 1³, must be defined as

n⁴/n¹ = 1^{4 -1} = 1³ = 1

and since, in a 3D system, we can’t define the four-dimensional unit required to do this, confusion results. Fortunately, we avoid this problem in the mathematics of reciprocal numbers, because the dimensions of the numbers express their inherent dual “directions” (positive and negative), which gives meaning to 1⁰, as a number with no degree of freedom.  So, we simply start with dimension 0, at the top of the tetraktys, meaning there is no, dual, degree of freedom in the initial number of the tetraktys. It simply corresponds to a geometric point. 

Ordinarily, we would regard a progression of reciprocal numbers, with two, reciprocal, aspects, as an ordered series of 0-dimensional units, or scalars, which would constitute a series of points, not 1D lines. Yet, an unexpressed exponent is assumed to equal 1. So, writing the series,

1|1, 2|2, 3|3, … 

implies an exponent of 1 in the numerator and the denominator, but in this case we can’t subtract the exponent of the denominator from the exponent of the numerator, because the pipe symbol indicates that the reciprocal operation of the reciprocal number is not multiplication (division), but addition (subtraction).  Therefore, the exponents must be the same in both cases, because the subtraction operation (actually sum of opposite signs) wouldn’t be valid otherwise, since we can’t subtract (add) two numbers with different exponents, or dimensions. 

Yet, from our knowledge of the tetraktys, we know that the reciprocal of the scalar (dimension 0) is the pseudoscalar (dimension 3, at the 3D level, or bottom of the tetraktys). So, if one of the terms in a reciprocal number is a 0D scalar, the meaning of the pipe symbol, “|”, requires the other term to be the reciprocal of the scalar, the pseudoscalar! 

We can see that this makes sense, because the series of reciprocal numbers

1⁰|1⁰, 2⁰|2⁰, 3⁰|3⁰, …  =  0⁰, 0⁰, 0⁰, …

is meaningless. A point is only its own reciprocal, when no degree of freedom is present (the n⁰:n⁰ numbers at the top of the tetraktys). Its reciprocal, with any non-zero degree of freedom, is always the pseudoscalar. Hence, the 3D pseudoscalar is the appropriate reciprocal of the 0D scalar in the Euclidean space (i.e. the 2³ numbers at the bottom of the tetraktys).

Consequently, in the three-dimensional system of numbers (the Grassmann algebra), the progression of reciprocal numbers must take the form 

1³|1⁰, 2³|2⁰, 3³|3⁰, …

which is a series of reciprocal numbers expressing a numerical progression of cubes, combined with reciprocally related points, corresponding to the geometric structure of Larson’s cube, with the 0D scalar at the intersection of the stack of 2x2x2 cubes.

However, because the difference between the numerator and the denominator is the difference between reciprocal quantities of different dimensions, we can express its value, as some mathematically meaningful result, only if the denominator is always the 0D scalar, while the numerator is the, reciprocal, pseudoscalar, since subtracting 0 from anything is essentially meaningless, and the breaking of the rule of exponents has no consequences in this case. As they say in the gym, no harm, no foul. 

However, if we change from the pipe operation to the slash operation, then, according to the same mathematical rules, it’s possible to express the operational result as a meaningful quantity. That is to say,

1³/1⁰, 2³/2⁰, 3³/3⁰, … = 1^3-0, 1^3-0, 1^3-0, … = 1³, 1³, 1³, …

Why is this? I submit that it’s because, in the slash operation, the ratio of reciprocals, as a quotient, defines the unit of a function. So, 1³ is a cubic unit of the function, which equates to a cubic pseudoscalar unit per scalar unit. On the other hand, in the pipe operation, 1³|1⁰, the ratio of reciprocals defines the unit of volume, as a 3D interval, with eight directions, between the 0D point and the 3D cube. 

This difference between the two operations enables us to distinguish, in an important manner, the difference between scalar magnitudes of motion, with two “directions,” and vector magnitudes of motion, in two directions. The difference in the magnitudes is the difference in the point of reference. We represent the opposite direction of a vector, by placing the arrow head at the opposite end of the line:

—————————> or <——————————

However, we represent the opposite “directions” of scalars, by placing the arrow head at both ends of a line, pointing in opposite “directions,” like this:

<————————>

This is because motion, as a 1D scalar magnitude, is an expansion from the center outward, in opposite directions, while motion, as a 1D vector magnitude, is a transference from one end of a line to the other. Thus, a scalar line always has a middle point associated with it, which is not part of a vector line. Therefore, the reciprocal number,

1¹|1⁰ = 1 

is a numerical expression of the double headed arrow

<————-0————->

or the result, or interval, we might say, of a 1D scalar expansion outward from a point.

By the same token, the reciprocal number,

1²|1⁰ = 1²

is a numerical expression of the four headed arrow

or the result of a 2D expansion from a point.

Finally, the reciprocal number

1³|1⁰ = 1³

is a numerical expression of the six headed arrow

or the result of a 3D expansion from a point.

The important difference in scalar motion versus vector motion is that the two “directions” in one dimension of scalar motion produce two 1D scalar magnitudes (one in each “direction”), in one unit of time, the four “directions” in two dimensions of scalar motion produce four 2D scalar magnitudes, in one unit of time, while the six “directions” in three dimensions of scalar motion produce eight 3D scalar magnitudes, in one unit of time.

This means that to represent the unit progression of the RST, with reciprocal numbers, we write the series

1³:1⁰, 2³:2⁰, 3³:3⁰, …

where the colon symbol for ratio is used as a general symbol for reciprocity, which can be interpreted as either of the two operations we have defined. Consequently, this gives us two representations of the reciprocal operation: One is a geometric interval, and the other is a function, which produces that interval; that is, one is a representation of a scalar “distance” with two fixed, reciprocal, aspects, the scalar and pseudoscalar, while the other is a representation of a function, with two changing, reciprocal, aspects, the scalar and pseudoscalar.

On this basis, the 0D scalar progression, or scalar expansion of a point, is

1⁰:1⁰, 2⁰:2⁰, 3⁰:3⁰, … = 1⁰, 2⁰, 3⁰, …

where the expanded scalar intervals, i_n, are

i_n = 1⁰|1⁰, 2⁰|2⁰, 3⁰|3⁰, … = 1⁰, 2⁰, 3⁰… (0, 0, 0,…)

And the function of the scalar progression, f(p⁰), which produces them, is

f(p⁰) = Δ1⁰/Δ1⁰.

The 1D scalar progession, or scalar expansion, of a line, is 

1¹:1⁰, 2¹:2⁰, 3¹:3⁰, … = 1¹, 2¹, 3¹, …

where the expanded scalar intervals are

i_n = 1¹|1⁰, 2¹|2⁰, 3¹|3⁰, … = 1¹, 2¹, 3¹ (<-0->, <—0—>, <—-0—->, …)

And the scalar function, which produces them, is

f(p¹) = Δ1¹/Δ1⁰

However, notice that this time, due to the fact that there are TWO directions in the ONE dimension, the progression of the 1D units, as opposed to the progression of the 0D units, is an increase in multiples of two 1D units, one “positive” unit, relative to zero, and one negative unit, relative to zero: 2, 4, 6, …, or the 1D progression, P¹, is P¹ = (2*1¹), (2*2¹), (2*3¹), …

Now, the 2D scalar progession, or scalar expansion, of an area, is 

1²:1⁰, 2²:2⁰, 3²:3⁰, … = 1², 2², 3², … 

where the expanded scalar intervals are

i_n = 1²|1⁰, 2²|2⁰, 3²|3⁰, … = 1², 2², 3², …

And the scalar function, which produces them, is

f(p²) = Δ1²/Δ1⁰

Again, however, due to the fact that there are TWO directions in each of the TWO dimensions, the progression of the 2D units, as opposed to the progression of the 1D units, is an increase in multiples of four 2D units, two polarized units in two independent directions, relative to zero, and two oppositely polarized units in two opposite independent directions, relative to zero: 4, 16, 36, …, or the 2D progression, P², is P²  = (4*1²), (4*2²), (4*3²), …

Finally, the 3D scalar progession, or scalar expansion, of a volume, is 

1³:1⁰, 2³:2⁰, 3³:3⁰, … = 1³, 2³, 3³, …    

where the expanded scalar intervals are

i_n = 1³|1⁰, 2³|2⁰, 3³|3⁰, … = 1³, 2³, 3³, …  

And the scalar function, which produces them, is

f(p³) = Δ1³/Δ1⁰

Now, due to the fact that there are TWO directions in each of the THREE dimensions, the progression of the 3D units, as opposed to the progression of the 2D units, is an increase in multiples of eight 3D units, four polarized units in three independent “positive” directions, relative to zero, and four polarized units in three independent “negative” directions, relative to zero: 8, 64, 216, …, or the 3D progression, P³, is P³ = (8*1³), (8*2³), (8*3³), …

Of course, in the context of the RST, this immediately raises the possibility of the inverse of these intervals, and the functions, which produce them; that is, it is the progression of the temporal tetraktys, in the form of the temporal 2x2x2 stack of cubes. Would this take the form of

f(p^-n) = Δ1^-n/Δ1⁰? 

This is heavy stuff!