## Reciprocal System of Mathematics - Reciprocal Numbers

The Greek tetraktys contained important cosmological significance for the ancients. Now, we see that it also contains the foundation of the RST cosmology as well. For the Greeks, the first principle was 1 (1/1), because from 1 all else proceeds. However, the first *number* for the Greeks was not 1, but 3, where the dicotomy of 2 was unified by combining it with 1. We do the same thing in the RSM. Recall that we interpret rational numbers operationally, so that they are equivalent to the signed integers:

- n/m, …, 1/2, 1/1, 2/1, …m/n = -x, …, -1, 0 , 1, …x.

Thus, in the RSM, the monopole is equivalent to zero, or

- 1/1 ~ 0,

but this unity becomes unbalanced, or split, by the duality of the dipole, where:

- 1/2 ~ -1, and

- 2/1 ~ +1.

Combining the two poles of the dipole with the monopole, however, to form a 3 pole, or triad number, forms the first number of the RSM, the reciprocal number (RN), and restores the balance of unity,

- (1/2 + 1/1 + 2/1) = (4/4) = (1/1)

So, just as the ancient Greeks did, we see great significance in the numbers 1, 2, 3, and 4. For them, these numbers formed a harmonious whole, where all things proceed from 1 (1/1), which, becoming divided in 2 (1/2 & 2/1), is restored in 3 (1/2 + 1/1 + 2/1), and then completed in 4 ((1/2 + 1/1 + 2/1) = 4/4).

Thus, the RN is the first, complete, number of the RSM, containing the four numbers of the tetraktys, 1, 2, 3, and 4 in a coherent, harmonious way. But the magic is that when we square the RN, since it is equivalent to a line, it truly is a square, and when we cube the RN, it truly is a cube! That is to say, the 1D RN is a number with quantity, direction, and dimension, equivalent in every way to its geometric counterpart, magnitude.

For instance,

- RN
^{2}= (4/4)^{2}= 16/16, or

- (1/2 + 1/1 + 2/1)
^{2}= (4/8 + 4/4 + 8/4) = 16/16,

but this is equal to:

- RN
^{2}= 4 * (1/2 + 1/1 + 2/1) = [(1/2 + 1/1 + 2/1) + (1/2 + 1/1 + 2/1) + (1/2 + 1/1 + 2/1) + (1/2 + 1/1 + 2/1)],

which are four lines:

- (1/2 + 1/1 + 2/1) = - +
- (1/2 + 1/1 + 2/1) = - +
- (1/2 + 1/1 + 2/1) = - +
- (1/2 + 1/1 + 2/1) = - +

Arranged in a square, these become:

Likewise, the cube of the RN is:

- RN
^{3}= (4/4)^{3}= 64/64, or

- RN
^{3}= (1/2 + 1/1 + 2/1)^{3}= (16/32 + 16/16 + 32/16) = 64/64,

but this is equal to:

- RN
^{3}= 16 * (1/2 + 1/1 + 2/1) = [(1/2 + 1/1 + 2/1) + (1/2 + 1/1 + 2/1) + (1/2 + 1/1 + 2/1) + …(1/2 + 1/1 + 2/1)],

which are 16 lines that we can arrange in a cube (6 horizontal, 6 lateral, and 4 vertical):

So now we can see that the 1D RN is equivalent to a line, the 2D RN is equivalent to a square, and the 3D RN is equivalent to a cube, and

- 1) The 1D RN is 4 units of scalar motion with one “direction,” interpreted as the two poles of one dimension.
- 2) The 2D RN is 16 units of scalar motion with two “directions,” interpreted as four poles in two dimensions.
- 3) The 3D RN is 64 units of scalar motion with three “directions,” interpreted as eight poles in three dimensions.

Also,

- 1) The 2D RN = 4 1D RNs.
- 2) The 3D RN = 4 2D RNs or 4 x 4 = 16 1D RNs.

Thus, it takes 64 1D units of scalar motion to form a 3D RN, or 4 2D units, or to put it another way:

- RN
^{3}= RN*RN^{2}= RN*RN*RN = (4/4)*(4/4)*(4/4) = 64/64, or

- RN
^{3}= (16/32 + 16/16 + 32/16) = (1/2 + 1/1 + 2/1)(4/8 + 4/4 + 8/4) = (1/2 + 1/1 + 2/1)(1/2 + 1/1 + 2/1)(1/2 + 1/1 + 2/1) = 64/64

and since RNs are scalar numbers they have all the properties of scalars and therefore their algebra is ordered, commutative, and associative, meaning that the algebra of RNs is a normed division algebra for all dimensions.

Now that we can see this, we can also see why the legacy system of mathematics (LSM) loses these properties as the dimensions of scalars is increased to form vectors: The importance of distinguishing “direction” from direction is not recognized in the LSM. Therefore, when raising the dimensions of real numbers to form complex numbers, they are formulated in terms of locations in the plane and these locations are not ordered, as they are when they are on the 1D line; that is, there is an infinite number of locations on a given circle in the complex plane such that the absolute value of (a + ib) located on the circle is not greater or less than the absolute value of (c + id) on the circle (the radius of the circle from the origin is constant, regardless of 2D direction).

Then, when raising the dimensions of the complex numbers to form the quaternions, the numbers lose the commutative property, because (a*b) is not equal to (b*a), which is, again, only due to the failure to distinguish between “direction” and direction (the radius of the sphere from the origin is constant regardless of 3D direction). Finally, when raising the dimensions of the quaternions to the octonions, the numbers lose the associative property, because (a(b*c)) is not equal to (b(a*c)), once again due to the failure to distinguish “direction” from direction (the vector of the radius of the sphere from the origin to the surface is opposite in sign to the radius, as a vector, from the surface to the origin). Hence, the vectorial algebra of the 3D octonions (LSM mathematicians call these 2^{3} = 8D octonions, but, geometrically speaking, they are 3D entities) is non-ordered, non-commutative, and non-associative.

On the other hand, the scalar algebra of RNs is completely ordered, commutative, and associative, regardless of dimension. Moreover, the multipoles of the RSM, unlike the multivectors of Hestenes’ Geometric algebra (GA), have no need of a property such as “orientation,” which in GA must be used to indicate the “direction” property of a multivector. The multipoles, by definition, are inherent “directions” of magnitude equivalent to the n-dimensions of geometric magnitudes; that is, there are two polarized “directions” in a line, four polarized “directions” in a plane, and eight polarized “directions” in a volume.

In the previous article, we saw how that the Bott periodicity theorem is explained by the triform of the RN, resulting in the formation of higher dimensional tetrakti, each with its own monad, dyad, triad, and tetrad. These are actually n-dimensional RNs; that is, the dyad_{1} is the dimensional expansion of the monad_{1} from a single point (2^{0} = 1) to a line of three points, the one in the monad (2^{0} = 1), plus the two in the dyad (2^{1} = 2), forming the triform RN, the first number of the RSM.

Thus, the RN^{2} number is RN^{1} * RN^{1} = RN^{2}, or, in terms of poles (points), 3*3 = 9, and RN^{3} is 3*3*3, or 27 poles. Similarly, (RN^{2})^{2} = RN^{4}, or 9*9 = 81 poles, which is the monad_{2}, and the whole process starts over again.

Since the first number of the RSM, the RN, raised to some power, can be expressed as RN^{n}, the question arises, what about subsequent numbers in the RSM? Can they also be dimensionally expanded? The answer to this question is yes. The RNs correspond to the natural numbers and, so, just as any natural number can be raised to some power n, any RN can be raised to some power n, as well. The RN numbers of the RSM are identified by subscript in the form RN_{n}, where n = 1 to ∞.

On this basis, the reciprocal numbers RN_{1}, RN_{2}, RN_{3}, …RN_{n} correspond to the natural, or counting numbers, and

- RN
_{1}^{2}= RN_{1}^{1}* RN_{1}^{1}

just as

- 1
^{2}= 1^{1}* 1^{1}

means the unit square, and

- 1
^{3}= 1^{1}* 1^{1}* 1^{1}

means the unit cube. However,

- RN
_{2}^{2}= RN_{2}^{1}* RN_{2}^{1}, and since RN_{2}is

- RN
_{1}^{1}+ RN_{1}^{1}= (4/4) + (4/4) = (8/8), or

- [(1/2 + 1/1 + 2/1)] + [(1/2 + 1/1 + 2/1)] = (2/4 + 2/2 + 4/2) = 8/8, then

- RN
_{2}^{2}= [(2/4 + 2/2 + 4/2)]^{2}, or

- RN
_{2}^{2}= (8/8)^{2}= 64/64, and

- RN
_{2}^{3}= (8/8)^{3}= 512/512.

With this much understood, its easy to see that a bewildering array of possibilities can be formed, and that the square or cube root of one RN is equal to another RN. For instance,

- RN
_{2}^{2}= RN_{1}^{3}, or

- (8/8)
^{2}= (4/4)^{3}= (64/64).

Likewise,

- (RN
_{2}^{2})^{1/2}= 2*(RN_{1}^{3})^{1/3}, or

- (64/64)
^{1/2}= 2*((64/64)^{1/3}) = 8/8.

Considering the complete correspondence between the RNs of the RSM and the magnitudes of geometrical entities, this ability to algebraically manipulate the n-dimensional numbers of the RSM is tantamount to algebraically manipulating the n-dimensional magnitudes of geometry.

### Bott Periodicity of Magnitude

In discussing Bott Periodicty theorem in the Reciprocal System of Mathematics (RSM), used in the scalar science of the LRC, we have shown how the theorem limits geometry to three dimensions in terms of *poles*. However, as we also have discussed, the expanded poles of RNs are scalar equivalents of rotated, or outer, products of vectorial poles, due to the three pole structure of the RN, which is manifest in the powers of 3. Nevertheless, we can demonstrate the theorem more clearly in the *magnitudes* of the RN, which are based on the powers of 4 (4/4), by focusing on the theorem’s role in current legacy system (LST) research, and contrasting the use of the math of the two systems.

We can start with the Wikipedia article on Bott Periodicty, recalling that what the theorem says in short is that there is no new phenomena beyond three dimensions. This implies then that the concept of 10 dimensions, which is the foundation of string theory, has to cope in some manner with the theorem. The Wikipedia article states:

In mathematics, the

Bott periodicity theoremis a result from homotopy theory discovered by Raoul Bott during the latter part of the 1950s, which proved to be of foundational significance for much further research, in particular in K-theory of stable complex vector bundles, as well as the stable homotopy groups of spheres. Bott periodicity can be formulated in numerous ways, with the periodicity in question always appearing as a period 2 phenomenon, with respect to dimension, for the theory associated to the unitary group. See for example topological K-theory.

Now, homotopy theory has to do with functions in topology, and K-theory has to do with topology, as a branch of algebraic K-theory, which is used to study topological spaces. Michael Atiyah, the originator of the Index theorem, states that it is used to find the number of solutions to a system of differential equations in a given algebraic space, which can be useful in finding the actual solutions themselves, which is something like knowing that there are 180 degrees in the angles of a triangle is helpful in finding the value of a given angle that otherwise would be much more difficult to do.

With the index theorem, the shape, or the topology, of the space under study provides the clues needed to get at the solutions to differential equations that the vectorial system of physical theory requires. What Atiyah (and Singer) did, was to use K-theory to demonstrate that the index could be described topologically. This was huge and led to the popular connection between topology and theoretical physics. Indeed, from what I understand, it was the mathematical methods derived from the index theorem that enabled Witten to make his significant contribution to string theory.

Anyway, the Wikipedia article on algebraic topology states:

The goal is to take topological spaces and further categorize or classify them. An older name for the subject was combinatorial topology, implying an emphasis on how a space X was constructed from simpler ones. The basic method now applied in algebraic topology is to investigate spaces via algebraic invariants, by mapping them, for example, to groups which have a great deal of manageable structure in a way that respects the relation of homeomorphism of spaces. This allows one to recast statements about topological spaces into statements about groups, which are often easier to prove.

Of course, it is the study of groups, in group theory, that connects all this to modern physics, especially to string theory. I’ve written several times about the connection of the octonions to modern physics and their connection through the periodicity theorem that John Baez finds so mysterious (Period Eight, or Bott Periodicity, Strikes Again!), but back in May of 2000, Dr. Rick Ye, gave a talk at what is now the KITP in Santa Barbara entitled “The Bott periodicity theorem,” where he explains that there are various ways to state the theorem in terms of the topological language of groups, and later he goes through proofs of these, but, after he explains three versions of this period eight theorem, he talks about why it is important, and gives four reasons:

- It is the fundamental structure theorem for K-theory, which enables computations to be made in homotopy structure.
- Homotopy structure of the unitary groups is the cornerstone for everything based on these groups (standard model).
- It leads very quickly to Thom isomorphism and then to the Atiyah-Singer Index theorem.
- It’s needed for D-branes!

The relation between the period eight of Bott periodicity, K-theory, mathematical groups, Index theorem, and D-branes in string theory, seems to be simply that it maps the D-branes of different dimension, as seen through the eyes of vectorial physics, using vectorial spaces, and the system of differential equations clothed in the language of the shapes of spaces, or topology.

It’s all very complicated and involved. However, what we have discovered in scalar physics is that vectorial physics appears to be the wrong system for describing the internal degrees of freedom of physical entities, and the vectorial mathematics of differential equations is the wrong language for investigation of these properties, and the vectorial geometry of vectorial motion is the wrong geometry. Thus, the science of using complex concepts of topology, connected to complex concepts of groups, connected to complex systems of differential equations, are simply inappropriate for calculating the properties of these entities. Physicists may be able to get there from here, eventually, or not, but why beat our heads against the wall? It appears that scalar science offers a much simpler route to understanding these properties.

Take Bott periodicty for instance. When viewed from the standpoint of scalar mathematics, this phenomenon becomes trivially simple to comprehend. Recall that the first reciprocal number (RN) is

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

where nm stands for natural units of motion. As explained previously, this equation exploits the operational interpretation of numbers of the RSM to express the motion combination of the two unit speed-displacements of the RST, which we have called the SUDR and TUDR, into one composite unit, designated as S|T. The RN is a three-dimensional number that can express the period eight phenomenon in terms of a power expansion, by first showing that the quadratic expansion of the RN is isomorphic to the binomial expansion, given the operational interpretation (OI) of RSM (Note: In the following illustrations, the number on the left side of the “|” symbol indicates the dimension, or the level, we might say, of the binomial expansion, while the number on the right side indicates the corresponding value of the RN magnitude at that dimension, or level):

•20|40 (1/1) = 1/1 nm

•21|41 (1/2 + 1/1 + 2/1) = 4/4 nm

•22|42 (4/8 + 4/4 + 8/4) = 16/16 nm

•23|43 (16/32 + 16/16 + 32/16) = 64/64 nm

^{11}, covering three generations of 2

^{3}= 8 “directions” in three dimensions, we get:

•20|40 (1/1) = 1/1•21|41 (1/2 + 1/1 + 2/1)1 = 4/4•22|42 (1/2 + 1/1 + 2/1)2 = (4/4)2 = 16/16•23|43 (1/2 + 1/1 + 2/1)3 = (4/4)3 = 64/64•24|44 (1/1) = (4/4)4 = 256/256•25|45 (1/2 + 1/1 + 2/1)5 = (4/4)5 = 1024/1024•26|46 (1/2 + 1/1 + 2/1)6 = (4/4)6 = 4096/4096•27|47 (1/2 + 1/1 + 2/1)7 = (4/4)7 = 16384/16384•28|48 (1/1) = (4/4)8 = 65536/65536•29|49 (1/2 + 1/1 + 2/1)9 = (4/4)9 = 262144/262144•210|410 (1/2 + 1/1 + 2/1)10 = (4/4)10 = 1048576/1048576•211|411 (1/2 + 1/1 + 2/1)11 = (4/4)11 = 4194304/4194304

This shows how each dimension, n, can be expressed as a power of the first RN, or 2^{n} ~ 4^{n} = RN^{n}. Finally,

•20|40 (1/1) = 1/1•21|41 (1/2 + 1/1 + 2/1)1 = (2/2)2•22|42 (1/2 + 1/1 + 2/1)2 = (4/4)2 = 16/16•23|43 (1/2 + 1/1 + 2/1)3 = (8/8)2 = (1*8/1*8)^{2}= 64/64•24|44 (1/1) = (4/4)4 = (16/16)^{2}= (2*8/2*8)^{2}= 256/256•25|45 (1/2 + 1/1 + 2/1)5 = (32/32)2 = (4*8/4*8)^{2}= 1024/1024•26|46 (1/2 + 1/1 + 2/1)6 = (64/64)2 = (8*8/8*8)^{2}= 4096/4096•27|47 (1/2 + 1/1 + 2/1)7 = (128/128)2 = (16*8/16*8)^{2}= 16384/16384•28|48 (1/1) = = (4/4)8 = (256/256)2 = (32*8/32*8)^{2}= 65536/65536•29|49 (1/2 + 1/1 + 2/1)9 = (512/512)2 = (64*8/64*8)^{2}= 262144/262144•210|410 (1/2 + 1/1 + 2/1)10 = (1024/1024)2 = (128*8/128*8)^{2}= 1048576/1048576•211|411 (1/2 + 1/1 + 2/1)11 = (2048/2048)2 = (256*8/256*8)^{2}= 4194304/4194304

shows why the period eight is fundamental to higher dimensional numbers. And we have always thought that the 8 bit byte was a necessary, but rather arbitrary, basis for the binary language of computers!

Anyway, the bottom line here is that because we don’t need the systems of differential equations upon which vectorial physics is based, we obviously don’t need all that complex homotopy stuff either. Evidently, all that we need is some good computer engineers!

We need some really competent mountain climbers at the foot of this mountain, a mountain that the amatuer Larson discovered, but that the professionals like Lee Smolin should prepare to climb (to use his own metaphor).

See also: Reciprocal System of Mathematics - Background — Reciprocal System of Mathematics - Fundamentals — Reciprocal System of Mathematics - Multi-dimensional Numbers