The New Mathematics

Discrete Magnitudes of Right Lines & the Analog Magnitudes of Circles

Posted on Friday, April 8, 2011 at 03:39AM by Registered CommenterDoug | CommentsPost a Comment

Things are moving quite fast in the LRC development of the RST. I hope and pray that the Peter Principle doesn’t overcome us, before we can explain the atomic spectra, which is our immediate research goal. Mathematically and geometrically, we are focused on two things and their relationship: These two things are the geometrical set of right lines and circles constructed with Larson’s Cube (LC), and the algebraic set of numbers in the tetraktys, generated by the binomial expansion.

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.

The Philosophy of Mathematics, Geometry and Physics

Posted on Tuesday, March 1, 2011 at 05:36PM by Registered CommenterDoug | Comments6 Comments

One of the things that the FQXI contest highlights is just how much mathematics, geometry and physics enter into philosophical discussions! There is no way to get a handle on anything other than a small fraction of the discussions the contest generates, and the mix of professionals, semi-professionals and amateurs makes for a unique and stimulating experience. I encourage all ISUS members and interested non-members to participate.

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.

 

The New Scalar Number Line

Posted on Saturday, November 6, 2010 at 08:50AM by Registered CommenterDoug | Comments1 Comment | References3 References

As we’ve seen, the new scalar math requires a new scalar number line. The familiar number line, though simple and straightforward, is philosophically troublesome due to the enigmatic status of zero and negative numbers. Even so, it has been used to define integers and rational numbers, using a concept of 0 as a sort of number and -1 as the foundation of a set of multi-dimensional algebras called Lie algebras.

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.)

 

 

 

 

 

The New Math

Posted on Monday, April 26, 2010 at 08:16AM by Registered CommenterDoug | CommentsPost a Comment

It’s been a long time since my last entry on this blog. Mostly that’s due to time constraints, but also because I’ve written about things on the new physics blog that probably should have gone here. Sometimes, though, it’s hard to separate the math from the physics topics.

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 xy 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.

Toward Non-pathological Algebras

Posted on Wednesday, July 16, 2008 at 05:40AM by Registered CommenterDoug | CommentsPost a Comment

Arguably, the two most challenging mathematical/philosophical problems for the Greeks were manifest in the attempt to square the circle and to accept the existence of irrational numbers. In modern times, we’ve proven that the former is impossible, and the latter is actually quite useful. However, as discussed in the previous post, it is possible that there are other approaches to meeting these formidable challenges, unknown to us, which might even prove more useful than our current method of handling them.

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

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

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