The Larson Research Center

In discussing multi-dimensional numbers, we are breaking new ground, since this is the first time anyone has thought about these numbers in terms of reciprocal numbers (RNs). Of course, the only reason we have is because the existence of discrete units of motion have now been postulated in terms of space/time reciprocity, in the RST. Hence, the RSM is only now beginning to emerge, because the RST requires it. In the LST community, the one-dimensional nature of motion, forming the foundation of legacy physics, has directed, and continues to direct, the application and interpretation of dimension in legacy mathematics. Without the concept of n-dimensional motion, n-dimensional numbers don’t hold much interest for physicists. Apparently, this illustrates the justification for Hestenes’ position that theoretical physics drives mathematics.

Increasing Complex Roots and Physics

Another good illustration of this is the invention of the imaginary number. While mathematicians think of these as needed to explain negative roots, or solutions, to quadratic equations, physicists think of them in terms of rotations; that is, without the complex plane, we could not describe one-dimensional rotational motion, or vectors, or functions. The complex numbers that the invention of the imaginary number made possible encode the coordinate positions in the fixed spatial reference system and make it possible to formulate the function x(t), which is the basis of classical mechanics, but they also make it possible to formulate the probability amplitude of the wave equation as a rotation, without which it would be impossible to describe quantum mechanics. This fact is amazing to mathematicians who wonder in awe at the power of this invention of the human mind, the imaginary number.

However, when we consider n-dimensional RNs, we find that imaginary numbers are not necessary, and that the non-commutativity of the fixed spatial reference system, when described by quaternions and octonions, disappears when these numbers are interpreted as units of scalar motion instead. Of course, this is because we have no need for the one-dimensional function, x(t), to define motion, in the RST. In this system, motion is scalar and is defined by the number (s/t)ⁿ, an n-dimensional number. Therefore, what we see happening is that the advanced mathematics in quantum field theory and in string theory, which LST physicists have been investigating for the last half century, that involves Lorentz groups, representations of Lie groups, and gauge groups, is immensely complicated by the lack of recognition of scalar motion and the commensurate mathematics of multi-dimensional RNs. It is the view of one-dimensional vectorial motion in n-dimensional space, instead of n-dimensional scalar motion, that is confusing the otherwise straightforward picture that emerges from the n-dimensional mathematics of scalars.

For example, while John Baez and others can see the relation between the octonions and advanced LST research, the connection is blurred and unclear. Baez writes:

As Corinne Manogue explained to me, this relation between the octonions and Lorentz transformations in 10 dimensions suggests some interesting ways to use octonions in 10-dimensional physics. As we all know, the 10th dimension is where string theorists live. There is also a nice relation to Geoffrey Dixon’s theory. This theory relates the electromagnetic force to the complex numbers, the weak force to the quaternions, and the strong force to octonions. How? Well, the gauge group of electromagnetism is U(1), the unit complex numbers. The gauge group of the weak force is SU(2), the unit quaternions. The gauge group of the strong force is SU(3)….

Alas, the group SU(3) is *not* the unit octonions. The unit octonions do not form a group since they aren’t associative. SU(3) is related to the octonions more indirectly. The group of symmetries (or technically, “automorphisms”) of the octonions is the exceptional group G2, which contains SU(3). To get SU(3), we can take the subgroup of G2 that preserves a given unit imaginary octonion… say e1. This is how Dixon relates SU(3) to the octonions.

However, why should one unit imaginary octonion be different from the rest? Some sort of “symmetry breaking”, presumably? It seems a bit ad hoc. However, as Manogue explained, there is a nice way to kill two birds with one stone. If we pick a particular unit imaginary octonion, we get a copy of the complex numbers sitting inside the octonions, so we get a copy of sl(2,C) sitting inside sl(2,O), so we get a copy of so(3,1) sitting inside so(9,1)! In other words, we get a particular copy of the good old 4-dimensional Lorentz group sitting inside the 10-dimensional Lorentz group. So fixing a unit imaginary octonion not only breaks the octonion symmetry group G2 down to the strong force symmetry group SU(3), it might also get us from 10-dimensional physics down to 4-dimensional physics.

Cool, no? There are obviously a lot of major issues involved in turning this into a full-fledged theory, and they might not work out. The whole idea could be completely misguided! But it takes guts to do physics, so it’s good that Tevian Dray and Corinne Manogue are bravely pursuing this idea.

Multiple Dimensions and The Standard Model

Even if we don’t understand the intricacies of group theory, we can still see that there is a deep connection between RNs and the Lorentz group in 10 dimensions, which means a possible connection between the RST and string theory. This leads us to suspect that RNs could eventually help us to understand what string theory actually is, something which right now is eluding the whole world. Baez makes an interesting comment on the standard model and string theory in this connection. He writes:

I think there are still hopes for understanding the details of the Standard Model with the help of a deeper understanding of Clifford algebras, division algebras and related algebraic structures. Garrett Lisi mentioned Greg Trayling’s work. I’ve never taken the time to properly understand that, in part because of some nonstandard terminology that it uses, but I’m encouraged by the fact that Garrett thinks it’s good. I’ve spent a lot more time on Geoffrey Dixon’s work. The SU(5) and SO(10) grand unified theories, and Jogesh Pati’s work on left-right symmetric theories, also have something very beautiful about them. I suspect that they’re all grasping various parts of some truth that we’re not yet able to fathom, perhaps because we don’t have the right language.

If none of these theories are currently fashionable, well, that’s in part because none of them quite hit the nail on the head - but also because the most influential particle theorists seem to have given up hope on the idea of staring at the Standard Model until something clicks and a beautiful theory takes form which explains all its baroque peculiarities. Most of the physicists who are really good at math seem willing to let the internal logic of string theory guide them where it will: either to a triumphant victory in physics, or to a journey through beautiful mathematics increasingly distant from the physical world. It’s a pity that pondering the Standard Model is being left to mere “phenomenologists”.

Octonions and Bott Periodicity

Perhaps, eventually, the RST and the RSM are going to clear up the truth that the LST physicists and LSM mathematicians are “not yet able to fathom” without them. The key is to clear up the confusion that miscasting the simple symmetry of RNs into terms of “imaginary” roots has caused everyone. Take Bott periodicty for example. They know that this basic pattern of 8 repeats itself over and over again in mathematics and physics, but they don’t know how or why, except that the octonions are dimension 8 and the Bott periodicity seems to be related to these “crazy uncles” of the math world. Baez explains it like this:

Bott periodicity has period 8, and the octonions have dimension 8. And as we’ve seen, both have a lot to do with Clifford algebras. So maybe there is a deep relation between the octonions and Bott periodicity. Could this be true? If so, it would be good news, because while octonions are often seen as weird exceptional creatures, Bott periodicity is bigtime, mainstream stuff!

Baez then goes on to explain that there is indeed a connection between the two that can be explained in terms of something called the homotopy groups of an infinite set of rotation groups that sit inside one another like those series of egg-like Russian dolls inside dolls, which you sometimes see as novelty gifts. When you take the top off of one, you see another inside it. This infinite set of groups inside groups is really just the binomial expansion carried out to infinity and taken as one, huge, whole. The homotopy groups are the number of ways an n-dimensional sphere can be fit into this huge group. Baez explains:

So, what are the homotopy groups of O(∞)? Well, they start out looking like this:

n πn(O(∞))

0 Z/2

1 Z/2

2 0

3 Z

4 0

5 0

6 0

7 Z

And then they repeat, modulo 8. Bott periodicity strikes again! But what do they mean?

He answers the question for each number in the list, which we will see below, but this is what he says about number 7, the last one:

The seventh entry is probably the most mysterious of all. From one point of view it is the subtlest, but from another point of view it is perfectly trivial. If we think of it as being about π7 it’s very subtle: it says that the ways to stick a 7-sphere into O(∞) are classified by the integers. (Actually this is true for O(n) whenever n is 7 or more.) But if we keep Bott periodicity in mind, there is another way to think of it: we can think of it as being about π-1, since 7 = -1 mod 8.

But wait a minute! Since when can we talk about πn when n is negative?! What’s a -1-dimensional sphere, for example?

Well, the idea here is to use a trick. There is a space very related to O(∞), called kO. As with O(∞), the homotopy groups of this space repeat modulo 8. Moreover we have:

πn(O(∞)) = πn+1(kO)

Combining these facts, we see that the very subtle π7 of O(∞) is nothing but the very unsubtle π0 of kO, which just keeps track of how many connected components kO has.

But what is kO?

He defers the detailed explanation of kO and states the answer in one sentence:

Let me just say that when we work it all out, we wind up seeing that the seventh entry in the table is all about ‘dimension’.

Baez finds this startling, because it indicates that these homotopy groups map into the n-dimensional numbers in a very non-trival way, especially the seventh group. He can kind of understand the lower numbered periods, but the dimension aspect of the seventh blows him away. He explains it this way:

To summarize:

π0(O(∞)) = Z/2 is about REFLECTING

π1(O(∞)) = Z/2 is about ROTATING 360 DEGREES

π3(O(∞)) = Z is about TOPOLOGICAL FIELD THEORY IN 4 DIMENSIONS

π7(O(∞)) = Z is about DIMENSION

But wait! What do those numbers 0, 1, 3, and 7 remind you of?

Of course, they remind us of the monad, dyad, triad, and tetrad of the Greek tetraktys, or, in modern terms, the 1, 2, 4, and 8 of the binomial expansion, the first four lines of the Pascal triangle, or, in algebraic terms, the reals, complexes, quaternions, and octonions. Whatever it reminds us of, mapping them to the kOs in the rotation groups reveals something very interesting and intriguing. As Baez summarizes the bottom line:

π0(O(∞)) is about REFLECTING and the REAL NUMBERS

π1(O(∞)) is about ROTATING 360 DEGREES and the COMPLEX NUMBERS

π3(O(∞)) is about TOPOLOGICAL FIELD THEORY IN 4 DIMENSIONS and the QUATERNIONS

π7(O(∞)) is about DIMENSION and the OCTONIONS

Now this is pretty weird. It’s not so surprising that reflections and the real numbers are related: after all, the only “rotations” in the real line are the reflections. That’s sort of what 1 and -1 are all about. It’s also not so surprising that rotations by 360 degrees are related to the complex numbers. That’s sort of what the unit circle is all about. While far more subtle, it’s also not so surprising that topological field theory in 4 dimensions is related to the quaternions. The shocking part is that something so basic-sounding as “dimension” should be related to something so erudite-sounding as the “octonions”!

But this is what Bott periodicity does, somehow: it wraps things around so the most complicated thing is also the least complicated.

In short, this is what Baez and others have to say about Bott periodicity. Actually, it’s very complicated and the details in terms of topology and group theory are probably way beyond all but the professional topologists and algebraists themselves. However, to read about it in its unabbreviated form go to Week 105 of Baez’s site: (http://math.ucr.edu/home/baez/week105.html)

The Periodicity of Reciprocal Numbers

Meanwhile, to understand how octonions relate to Bott periodicity in the simple terms of RNs, read on. In the Reciprocal System of Mathematics (RSM), Bott periodicity is related to octonions simply because, as the dimensions of the triad RNs are increased beyond three (four counting zero), the poles in the higher dimensional RN consist of the multipoles of the lower dimensional RN. The dimensions of the monopole are (3^n-1), where n is the number of the dimensional generation. Thus, we have the first generation:

0) 2⁰ = 1/1 = 1/1 ~ 1mp = 1 = 3⁰

1) 2¹= 1/1 1/1 = 2/2 ~ 1mp + 1dp = 3p = 3¹

2) 2² = 1/1 2/2 1/1 = 4/4 ~ 1mp + 2dp + 1qp = 9p = 3²

3) 2³ = 1/1 3/3 3/3 1/1 = 8/8 ~ 1mp + 3dp + 3qp + 1op = 27p = 3³

where p, mp, dp, qp, and op are poles, monopoles, dipoles, quadrupoles, and octopoles respectively. The dimensions of the poles in this initial sequence are 3^1-1 = 3⁰ = 1. In the next sequence, the octonion pseudoscalar, consisting of 8 multipoles that are equivalent to 3³ = 27 1D poles, becomes the basis of the next higher “monopole,” which follows in the second generation of three dimensions. In other words, the pseudoscalar of the previous series becomes the basis of the scalar of the subsequent series, and the dimensional expansion continues for another four increments, increasing the value of the scalars, as higher dimensional points, lines, planes, and volumes:

0) 2⁴ = 1/1 4/4 6/6 4/4 1/1 = 16/16 ~ 1mp = 1p = ((3⁰)x(3⁴)) = 81mp = 3⁴

1) 2⁵ = 1/1 5/5 10/10 10/10 5/5 1/1 = 32/32 ~ 1mp + 1mp = 3p = ((3¹)x(3⁴)) = 243mp = 3⁵

2) 2⁶ = 1/1 6/6 15/15 20/20 15/15 6/6 1/1 = 64/64 ~ 1mp + 2dp+ 1qp = 9p = ((3²)x(3⁴)) = 729mp = 3⁶

3) 2⁷ = 1/1 7/7 21/21 35/35 35/35 21/21 7/7 1/1 = 128/128 ~ 1mp + 3dp + 3qp 1op = 27p = ((3³)x(3⁴)) = 2187mp = 3⁷

where the p in p, mp, dp, qp, and op is now equivalent to three pseudoscalars of the previous generation, which in this case is the 2³ octonion. Thus, each “monopole” in the second generation consists of three 2³ octonion pseudoscalars, 3^2-1 = 3¹ = 3, or one 3⁴ tripseudoscalar “monopole.” In the next sequence, the 2⁷ pseudoscalar, consisting of 27 3⁴ tripseudoscalars, which is equivalent to 3⁷ = 2187 3D “monopoles,” becomes the next higher “monopole,” which constitutes the next generation of tripseudoscalar “monopoles,” the 3⁸ tripseudoscalar “monopole”:

0) 2⁸ = 256/256 ~ 1mp = 1p = ((3⁰)x(3⁸)) = 6561mp = 3⁸

1) 2⁹ = 512/512 ~ 1mp + 1dp = 3p = ((3¹)x(3⁸)) = 19683mp = 3⁹

2) 2¹⁰ = 1024/1024 ~ 1mp + 2dp + 1qp = 9p = ((3²)x(3⁸)) = 59049mp = 3¹⁰

3) 2¹¹ = 2048/2048 ~ 1mp + 3dp + 3qp + 1op = 27p = ((3³)x(3⁸)) = 177147mp = 3¹¹

where, again, the p in p, mp, dp, qp, and op is the 3⁸ tripseudoscalar “monopole.” Of course, this pattern of three dimensional progression can be continued indefinitely, but this is enough to show why the octonions are related to the meaning of dimension, which is so puzzling to modern mathematicians: It’s proof that the bidirectional integers we have named RNs are limited to three geometric dimensions that are the eight dimensions of the mathematicians! That is, Bott’s proof of the base 8 periodicity in the RNs of the RSM, proves that they, like the units of scalar motion in the RST, must exist in three dimensions, which, in terms of polarities, is limited to 2³ = 8 “directions,” or polarities. The combo of scalar|pseudoscalar, besides wrapping things around “so the most complicated thing is also the least complicated,” is just an RN scaling in three dimensions!

In terms of the RST magnitudes of scalar motion, the time-displaced, unit displacement, progression ratio, s/t = 1/2, is a pseudoscalar and the space-displaced, unit displacement, progression ratio, s/t = 2/1, is a scalar. Therefore, when these two magnitudes are combined into one composite magnitude, they form the basis of an n-dimensional RN. The geometric representation of the RN³ octonion pseudoscalar, the Larson cube, contains the scalar and the psuedoscalar, but also “a copy” (actually three copies) of the 1D and the 2D pseudoscalars as well, as shown in figure 1 below:

Figure 1. Larson’s Cube, the Geometric Representation of the Octonion.

In figure 1, the red point is the 2⁰ scalar, which has no “direction” and is thus located at the center of everything. The three, orthogonal, blue lines are the three ways that a copy of the 2¹, or 1D, pseudoscalars can be oriented in the cube, the three, orthogonal, green planes are the three ways that a copy of the 2², or 2D, pseudoscalars can be oriented in the cube, and the orange cube itself is the 2³ pseudoscalar. The entire assembly is the octonion, a 3D scalar number with [1 + (3x2) + (3x4) + 8] = 27, n-dimensional, “directions,” or polarities.

The Wisdom of the Greeks Returns

It’s easy to see the nesting here that Baez and other LST mathematicians describe in the complex terms of topology, but we can see that, in the simple terms of the multipoles of the RSM, these nested values are the three-dimensional iteration of the pseudoscalars; that is, the 0D, 1D, 2D, and 3D pseudoscalars, or what we will call the pseudopoint, pseudoline, pseudoplane, and pseudovolume, the 1s that run down the right side of the first four lines of the Pascal triangle that the Greeks called the monad, dyad, triad, and tetrad of the tetraktys:

1) (monad - pseudopoint) 2⁰ = 1 = 1

2) (dyad - pseudoline) 2¹ = 1+1 = 2

3) (triad - pseudoplane)2² = 1+2+1 = 4

4) (tetrad - pseudovolume) 2³ = 1+3+3+1 = 8

However, the first point is not actually a pseudopoint, but a true, 3⁰ = 1, point (1), with a potential for the unfolding of many more points. The pseudoline, or dyad (1+1), contains 3¹ = 3 points (1mp, 1dp), while the pseudoplane, or triad (1+2+1), contains 3² = 9 points (1mp, 2dp, 1qp), and the pseudovolume, or tetrad (1+3+3+1), contains 3³ = 27 points (1mp, 3dp, 3qp, 1op). We will designate this first pseudovolume, the octopseudovolume, short for octonion pseudovolume.

Now, in the next generation of the tetraktys, the number of mathematical dimensions increases by four, from four to eight, so this increases each of the pseudoscalars by a factor of three:

1) The 2⁴ monad is a pseudopoint that is equivalent to three octopseudovolumes of the first tetraktys, which we will designate the 3⁴ pseudopoint, or the trioctopseudopoint, because it contains the three octopseudovolumes of the previous tetrad. Thus, line five of the triangle, the monad of the second tetraktys, is equivalent to 3x27 = 81 pseudopoints that are equivalent to three octopseudovolumes, or 1 trioctopseudopoint.

2) The 2⁵ dyad is a pseudoline that contains three trioctopseudopoints, and is therefore a trioctopseudoline.

3) The 2⁶ triad is a pseudoplane that is equivalent to nine trioctopseudopoints, and is therefore a trioctopseudoplane.

4) The 2⁷ tetrad is a pseudovolume that is equivalent to 27 trioctopseudopoints and is therefore a trioctopseudovolume.

At this point, a third tetraktys follows, and the pattern of its respective monad, dyad, triad, and tetrad repeats ad infinitum. It is important to understand that these four dimensions of the tetraktys that repeat in the Bott pattern are four mathematical dimensions, not four spacetime dimensions, as LST physicists are want to interpret them, due to the erroneous concept of Einstein’s 4D spacetime. The n-dimensional tetrakti are the power series of the four RSM terms of monopole, dipole, quadrupole, and octopole.

In other words, we count the third dimension of unity, 2⁸, the third monad, as four dimensions above the second dimension of unity, 2⁴, or the second monad, which is four dimensions above the first dimension of unity, 2⁰, or the first monad. This relates to a power series of geometry, as powers of points, lines, planes, and volumes, or as powers of monads, dyads, triads, and tetrads!

For instance,

1) 2⁴ is the second power of the monopole, equivalent to the second power of a 2⁰ geometric point (Monad²).

2) 2⁵ is the second power of the dipole, equivalent to the second power of a 2¹ geometric line (Dyad²).

3) 2⁶ is the second power of the quadrupole, equivalent to the second power of a 2² geometric plane (Triad²).

4) 2⁷ is the second power of the octopole, equivalent to the second power of a 2³ geometric volume (Tetrad²).

Hence, the geometric expansion of 2, the binomial expansion of the dual dyad, also includes the expansion of 3, the trinomial expansion of the triad, consisting of the dual dyad plus the monad, as reflected in the three component RN, a triad number, which is the first number of the RSM. The number ones on the left side of the triangle are 0-dimensional monads, but in a sense they are each equivalent to the number ones on the right side of the triangle, which are the n-dimensional triads; that is, the value of the monads on the left are the source of the value of the triads on the right, given the specified dimension.

This can best be understood in terms of the trinomial RNs, however, because, when the dimensional property of the monad in the RN is increased, it is limited to the three dimensions of geometry, which is tantamount to saying that it is limited to the three dimensions of scalar motion. Once this limit is reached, it can only be expanded by a larger pseudoscalar. In other words Pascal’s triangle is the mathematical equivalent to a scalar expansion, the unit progression of the integers of the RST.

Nevertheless, if we write the equation of the octopseudovolume as OPV = 2³ = 27, people will think that we are nuts! So, we should probably write it as OPV = 2³ = 8 ~ 3³ = 27, which means that 2³ is equal to 8 linear units, or multipoles, which all together contain 27 points, or the 27 pole sum of the eight multipoles of the octonion pseudoscalar, the octopseudovolume:

OPV = 2³ = 1mp + 3dp + 3qp + 1op = 8xp ~ 27p,

Where p, mp, dp, qp, op, and xp are poles, monopoles, dipoles, quadrapoles, octopoles, and multipoles respectively.

On this basis, the next four dimensions, constitute the second power of the expansion, the second tetraktys, or the Hexa Series, and the second power of their geometric equivalents. The next set of four dimensions constitute the third power of the expansion, the third tetraktys, or the Duohexa Series:

1) Octo Series (1st Tetraktys)

a) 2⁰ = 1 ~ 3⁰ = 1p = p = (point (pseudopoint))

b) 2¹ = 2 ~ 3¹ = 3p = L = (line of three points (pseudoline))

c) 2² = 4 ~ 3² = 9p = P = (plane of nine points (pseudoplane)

d) 2³ = 8 ~ 3³ = 27p = V = (volume of 27 points (pseudovolume)) = 1 octopseudovolume

2) Hexa Series (2nd Tetraktys)

a) 2⁴ = 16 ~ (3⁰)x(3⁴) = (3⁴) = 81p = p = (trioctopseudopoint)

b) 2⁵ = 32 ~ (3¹)x(3⁴) = (3⁵) = 243p = L= (trioctopseudoline)

c) 2⁶ = 64 ~ (3²)x(3⁴) = (3⁶) = 729p = P = (trioctopseudoplane)

d) 2⁷ = 128 ~ (3³)x(3⁴) = (3⁷) = 2187p = V = (trioctopseudovolume) = 1 hexapseudovolume

3) Duohexa Series (3rd Tetraktys)

a) 2⁸ = 256 ~ (3⁰)x(3⁸) = (3⁸) = 6561p = p = (trihexapseudopoint)

b) 2⁹ = 512 ~ (3¹)x(3⁸) = (3⁹) = 19683p = L= (trihexapseudoline)

c) 2¹⁰ = 1024 ~ (3²)x(3⁸) = (3¹⁰) = 59049p = P = (trihexapseudoplane)

d) 2¹¹ = 2048 ~ (3³)x(3⁸) = (3¹¹) = 177147p = V = (trihexapseudovolume) = 1 duohexapseudovolume

With this much understood, we can designate the octopseudovolume, hexapseudovolume, and the duohexapseudovolume as the tetrad, tetrad₂, and the tetrad₃, respectively. Indeed, we can designate each pseudoscalar in this way:

1) Monad₁, Monad₂, Monad₃…Monad_n

2) Dyad₁, Dyad₂, Dyad₃…Dyad_n

3) Triad₁, Triad₂, Triad₃…Triad_n

4) Tetrad₁, Tetrad₂, Tetrad₃…Tetrad_n

where n designates the Tetraktys that the number belongs to, and this will enable us to explore the various combinations and label them. The first three Tetrakti are shown in table 1 below.

Table 1. Values of First Three Tetraktys Calculations

Since a tetraktys means a set of four, we can clarify what is happening if, instead of incrementing the dimensions of the polarities of these numbers linearly, we increment them modulo four. This gives us a clearer picture of what is actually happening: Each tetraktys is a 3D structure of two “directions” (2³) consisting of 3ⁿ quantities:

Tetraktys₁:

1) 2⁰ = 1 = 1 ~ 3⁰ = 1

2) 2¹ = 1+1 = 2 ~ 3¹ = 3

3) 2² = 1+2+1 = 4 ~ 3² = 9

4) 2³ = 1+3+3+1 = 8 ~ 3³ = 27

Tetraktys₂:

1) 2⁰ = 1 = 1 ~ 3⁴ = 81

2) 2¹ = 1+1 = 2 ~ 3⁵ = 243

3) 2² = 1+2+1 = 4 ~ 3⁶ = 729

4) 2³ = 1+3+3+1 = 8 ~ 3⁷ = 2187

Tetraktys₃:

1) 2⁰ = 1 = 1 ~ 3⁸ = 6561

2) 2¹ = 1+1 = 2 ~ 3⁹ = 19683

3) 2² = 1+2+1 = 4 ~ 3¹⁰ = 59049

4) 2³ = 1+3+3+1 = 8 ~ 3¹¹ = 177147

Thus, we see that this series of infinite geometric groups are an ascending series of the multipoles of the octonion: 1 monopole, 3 dipoles, 3 quadrupoles, and 1 octopole, where the associated quantities of the poles in these multipoles are based on the previous octonion. Finally, given this numerical pattern of multipoles, we can systematically calculate the total units of motion and the units of displacement in these RNs.

First, though, in the next article, Reciprocal System of Mathematics - Reciprocal Numbers, we will see how the algebra of RNs must be ordered, commutative, and associative.

See also: Reciprocal System of Mathematics - Background — Reciprocal System of Mathematics - Fundamentals