To my dismay, however, I started thinking about it again and, now, I cannot get it out of my mind. Readers of the blog might recall how one of my favorite topics in mathematics is how math, employed in theoretical physics, has developed historically.

I like to start with Sir William Hamilton and his discovery and fascination with quaternions (see here.) I first learned about it from David Hestenes, who more or less resurrected the idea of quaternions, in connection with his work on his Geometric Algebra.

Not being a mathematician, or even one inclined to the subject, my interest was driven by the implications of scalar motion, which seemed to me to be almost, if not entirely neglected by everyone except Dewey Larson and his followers.

Just before the disagreement within ISUS drove me to organize the LRC, I had given some thought to expressing the fundamental ideas of Larson numerically. I had this idea that the numbers 1/2, 1/1, 2/1 could be used to quantify the RST concepts of material and cosmic sector space and time displacements, as defined by Larson.

But soon, I realized that these one-dimensional numbers, if they were to be useful to express scalar motion entities, would have to be extended to multi-dimensional numbers, and this eventually led me to understand the great disconnect in modern number systems, as mathematicians tried to understand how to extend non-scalar motion, or vector motion, to multi-dimensions.

Ironically enough, it was Hamilton’s discovery of quaternions that led the way to the derailment, even though it was he who complained most eloquently about the use of so-called imaginary numbers, which eventually were to enable him to form quaternions, ironically enough. However, this use of imaginary numbers to increase the dimensions of ordinary, or scalar numbers, has led to pathological algebras, which I’ve talked about many times (e.g. see here.)

The fact is, only the algebra of one-dimensional numbers is non-pathological. The two-dimensional complex numbers lose the order property of scalar algebra, while the three-dimensional quaternions lose both the order property and the commutative property, and the octonians, which would be four-dimensional numbers, add associativity to the lost properties of multi-dimensional numbers.

Of course, if we consider scalars as zero-dimensional numbers, then complex numbers would be one-dimensional, quaternions would be two-dimensional and octonions would be three-dimensional, which would help clear up the understanding of non-mathematicians, with a geometric perspective, letting them relate the number systems to the corresponding geometrical concepts of point, line, area and volume.

However, the mathematicians look at numbers as algebraic operators, and they like to designate their dimensions in those terms, so an algebraic expression of complex numbers, with two terms, z = (a + bi), for example, is considered two-dimensional, and an expression of quaternions, with four terms, H = (a+bi+cj+dk), is considered four-dimensional, and one of octonions, with eight terms, a scalar number and seven imaginary numbers, is considered eight-dimensional.

Thus, unfortunately, the dimensions of algebra, lose their direct correlation with the dimensions of geometry, among the mathematicians. Nevertheless, we were able to show how the dimensions of numbers are indeed correlated with the dimensions of geometry, when viewed through the tetraktys and Larson’s Cube (e.g. see here.)

With all this as prelude, we come to the core of this blog entry: What can replace the multi-dimensional vector aglebra of today’s number systems, based on imaginary numbers? The answer is a multi-dimensional scalar algebra, based on ratio, which is the mathematical corollary to motion. In the set of zero-dimensional numbers, correlated with geometric points, the members of the set are the familiar rationals called integers and their inverses:

1/n, …1/3, 1/2, 1/1, 2/1, 3/1…, n/1

We can say that this set is based on multiples of the scalar unit 10 = 1, but what about a higher dimensional set based on multiples of the linear unit 11? What would its members look like? Or a set based on multiples of the square unit 12? What would its members look like? Or a set based on multiples of the cube unit 13?

The problem is that, in algebra, 10 = 11 = 12 = 13= 1 = (1x1) = (1x1x1), while in geometry, a unit point is a zero dimensional magnitude (10), differentiated from a unit line, which is a one-dimensional magnitude (11), differentiated from a unit area, which is a two-dimensional magnitude (12), and differentiated from a unit volume, which is a three-dimensional magnitude (13).

These multi-dimensional, geometric entities are not algebraically inter-operable. How do you multiply or divide a line times an area, or a point times a volume, or how do you add and subtract them from one another? You can’t, because the geometric dimensions of these units differ, while the dimensions, represented by the exponents in the corresponding algebra are not even dimensions in the same sense.

Unlike geometric dimensions, where each dimension is an orthogonal magnitude, the algebraic dimensions are not orthogonal magnitudes, but simply denote a number of factors in the expression of a mathematical operation.

However, if we recognize this difference, we can change the meaning of the exponents, so that:

0D = 10 = unit point

1D = 11/2= unit line

2D = 21/2 = unit area

3D = 31/2 = unit volume

This may appear to be a strange suggestion, but there is a way to employ these multi-dimensional numerical units, as geometric units, by utilizing a geometric theorem: the theorem of Pathagoras. Thus, given

n = 1, 2, 3, …, ∞,

Unit point = (12)1/2;

Unit line = (n2)1/2;

Unit area = (n2 + n2)1/2;

Unit volume = (n2 + n2 + n2)1/2.

On this basis, the members of the 0D scalar set, based on unit point numbers, are noted as before:

1) 1/n, …1/3, 1/2, **1/1**, 2/1, 3/1…, n/1,

But the members of the 1D linear set, based on unit line numbers, are:

2) 1/n(12)1/2, …1/3(12)1/2, 1/2(12)1/2, **(12)1/2/(12)1/2**, 2(12)1/2/1, 3(12)1/2/1…, n(12)1/2/1.

The members of the 2D square set, based on unit area numbers, are:

1) 1/(n2+n2)1/2, …1/(32+32)1/2, 1/(22+22)1/2, **(12+12)1/2/(12+12)1/2**, (22+22)1/2/1, (32+32)1/2/1…, (n2+n2)1/2/1.

The members of the 3D cube set, based on unit cube numbers, are:

1) 1/(n2+n2+n2)1/2, …1/(32+32+32)1/2, 1/(22+22+22)1/2, **(12+12+12)1/2/(12+12+12)1/2**,(22+22+22)1/2/1, (32+32+32)1/2/1…, (n2+n2+n2)1/2/1.

If these multi-dimensional number sets turn out to have the properties required for mathematical groups, then their members should be able to be algebraically manipulated, with no pathology in the higher dimensions, as is now found in the higher dimensions of the legacy number system, based on imaginary numbers.

Some of the group properties are easily seen in the new sets. For instance, the identity element of all of them is equal to 1, which implies that multiplication by it will not change a member of the set. Also, a member of the set, when multiplied by its inverse is equal to 1. I think all the other group properties should hold as well, but I haven’t tested them to see.

For example, in the 2D case, the second element multiplied by its inverse element is:

Given x = sqrt(n), written in the form of the Pathagorean theorem, then:

(x)*(1/(x)) =

sqrt(2)*(1/(sqrt(2))) =

(12+12)1/2 * (1/(12+12)1/2 = ((12+12)1/2)/((12+12)1/2) = 1

Now, when each element is *divided* by its inverse, the 1D elements are squared, giving us the progression series,

1) e/(1/e) = e2 = 1, 2, 3, …n2,

while the 2D elements divided by their inverses produce the progression series:

2) e/(1/e) = e2 = 2, 8, 18, …2(n2),

and then the 3D elements divided by their inverses, produce the progression series

3) e/(1/e) = e2 = 3, 12, 27, …3(n2)

We can visualize these multi-dimensional, geometric numbers graphically, by drawing the unit ball inside the Larson cube, which just contains it, and drawing a second ball that just contains Larson’s cube. Taking a 2D slice of the 3D result, we get the graphic of figure 1 below:

**Figure 1.**

Note that the graphic of figure 1 does not illustrate the 1D linear number system in Larson’s cube, as it does the 2D and 3D systems. This is because I did not understand at the time I first made the graphic that, just as our RSt here at the LRC differs from Larson’s RSt, in that it BEGINS with 3D entities, the 3D oscillations of SUDRs and TUDRs, rather than with the 1D entities, the 1D oscillations of photons, with which Larson’s RSt begins, so too should our new number systems.

We can do this by writing one system equation, with three values of n: n1, n2, n3, in the 3D equation of the Pathagorean theorem, as follows:

e/(1/e) = e2 = ((n12 + n22 + n32)1/2)/(1/((n12 + n22 + n32)1/2),

where n2 and n3 = 0, for the 1D system,

and n3 = 0, for the 2D system.

The result is the three progression series, 1(n2), 2(n2) and 3(n2), shown above in 1), 2) and 3), respectively, for sucessive values of n1, n2 and n3.

While this simplification of the current multi-dimensional number system that is based on imaginary numbers (i.e. the square root of -1), may not appear to be relevant, at this point, to a civilization that has managed to construct the marvels of modern science and mathematics, building upon this imaginary foundation, we must remember that the theoretical physics community cannot complete it’s explanation of reality on this basis, given that quantum mechanics and general relativity are fundamentally incompatible.

It must be recognized that the dimensions of the fundamental energy equation, E = mc2, are the dimensions of inverse motion and that it follows that this fact cannot be ignored in light of our thesis that the universe consists of nothing but motion and its inverse.

This is especially clear, given Xavier Borg’s demonstration that the true physical dimensions of all physical units are the dimensions of motion (sn/tn) and inverse motion (tn/sn), where n = 0, 1, 2 or 3. Thus, the mass energy equation can be written in terms of these dimensions of motion and its inverse,

E=mc2 =

(t/s) = (t3/s3)x(s2/t2),

and the radiation energy equation can also be written in these terms, when the frequency term is recognized as just a convenient measure of vibration, clarifying the dimensions of Planck’s constant, s2/t, the dimensions of the so-called “quantum of action,” which dimensions are simply an ad-hoc compensation, derived to compensate for the use of the non-physical, but mathematical expression of the frequency term, 1/t. Making this correction, the radiative energy equation is simplified to,

E=h*v* =

(t/s) = (t2/s2)x(s/t).

However, treatment of physical equations such as these, and their implications, will be handled in our New Physics blog. For our purposes here in the New Math blog, it is sufficient to reform the number systems and their associated algebras, for application to the new physics.

]]>This changed, however, when I found out about a new essay contest at FQXi, and I decided to enter it, albeit in February, just before the March 4th deadline. It was very stressful, but I managed to write and submit a paper in time.

The theme of the contest is: “**Trick or Truth: the Mysterious Connection Between Physics and Mathematics.**” It was impossible to resist, but almost impossible to imagine that I could do justice to the theme, given the constraints FQXi puts on the essay entries and the little time that was left to write it.

Nevertheless, with much effort and prayer, I managed to meet the deadline with a paper I think made sense, even though there is nearly zero chance that it will be noticed much, with the hundreds of papers submitted, including one by Lee Smolin.

There are also typos and editing errors in it that I wish I could have discovered in time to correct. Yet, I hope that the readers there will overlook them. The effort was worthwhile, I think, because it gets the ideas of the LRC’s RSt, and thus Larson’s work, on the table of the judges, participants and readers of the essay contest, many of whom are professional mathematicians and physicists.

The title of my essay is: “**Trick or Truth: The Mysterious Connection Between Numbers and Motion and Geometry.**” I hope everyone will read it and comment on it at the FQXi site, when it is posted (it hasn’t appeared there yet, but should sometime next week,) but I thought I would post a corrected and expanded version of it here, as well.

And I will do that soon, but in the meantime, as a preface to the paper, I have determined to publish what I wrote nearly two years ago, after I met with Rainer, Bruce and Gopi, because, it is relevant, I think:

Yesterday I enjoyed a discussion of scalar motion fundamentals with Bruce Peret and Gopi, as we met at ISUS HQ, in SLC, UT, as guests of Rainer Huck. It was the first time that I had met Gopi, a young physics PhD student, studying in Texas, but originally from India.

It was actually a fruitful discussion, in large part due to Gopi’s interest in the LRC’s RSt, which gave me a chance to not only explain somewhat the unique development of our RSt and how it differs from Larson’s RSt, and the RS2 re-evaluation of his RSt, but also to introduce Gopi to the

newideas of scalar motion fundamentals, which we have been developing, while they have been experimenting with substituting projective geometry for Euclidean geometry in their work.I explained how the tetraktys incapsulates the concept of two “directions” in three dimensions (four including zero), and how Larson’s 2x2x2 stack of unit cubes is the geometrical equivalent of the tetraktys, which, by adding the concept of magnitude, to the tetraktys concepts of dimension and “direction,” incapsulates the totality of scalar motion fundamentals: magnitude, dimension and “direction.”

I’m not sure how impressed he was with this phenomenal connection we have found between the binomial expansion that we call the tetraktys and the 3D stack of unit cubes we call Larson’s cube, especially in the course of the disjointed give and take of a living room discussion. However, I was so pleased with his careful and thoughtful curiosity that I entertained the idea of extending the discussion into the implications for a scalar algebra fit for use in RST theory development.

In fact, we did talk about imaginary, complex, quaternion and octonion numbers, and the algebraic pathology that the use of them engenders, but I was not able to get much beyond that except to touch briefly on Hestenes’ geometric algebra and Altmann’s paper, “Hamilton, Rodrigues, and the Quaternion Scandal.”

From there we got into quadrantal versus binary rotation, and I tried to explain how that fit into our work with 3D oscillation (“pulsation” as Gopi calls it. I like that.), and its physical explanation of the 720 degree “rotation” of the LST community’s concept of quantum spin.

I was so pumped, for having the opportunity to lay out these vital discoveries we have made here at the LRC to Bruce and Gopi, that I continued the discussion in my mind as I drove home, after our meeting had ended.

It was then I felt a great desire to finish the conversation, maybe in a message to Gopi, or a presentation at ISUS HQ, or something else. I finally settled on writing this blog entry, because it is here that we document the development process of the LRC’s efforts to develop a useful RSt.

Of course, like I said, I never actually got around to writing it then, but now I have written it for the essay contest. In fact, I wrote it three times, trying to fit it into their constraints. So, in the next few days, I will take those three versions and try to combine them into one expanded article, without the constraint of the contest rules.

Wait for it!

**Update: **A few days later and here is the expanded paper!

Horace writes:

We have always been looking for the mathematical link between the magnitudes above and below one unit. Bruce uses the idea of projective geometry and counterspace to show how linear motion gets converted to reciprocal rotational motion below one unit (a.k.a. “crossing the unit boundary”).

It looks like Miles Mathis has found another way:

Miles Mathis wrote:

I developed an equation to find one velocity from the other, using the radius r, and I later showed thatat the size of the photon, a tangential velocity of c was equivalent to an orbital velocity of 1/c.

Source: http://milesmathis.com/charge3.html

This is good news indeed for those who have accepted the departure from the fundamental postulates of Larson’s RST, but for those of us unwilling to accept the argument that requires the abandonment of Euclidean geometry in favor of projective geometry, as the geometry of the universe of motion, and a concommitant change in the wording of the second fundamental postulate, not so much.

Not that the controversial work of Miles Mathis might not be valid, it might be, but it illustrates a fundamental misunderstanding in the mathematics of scalar motion that exists in the “RS2” community.

In their “re-evaluation of the Reciprocal System of Physical Theory,” the followers of the RS2 community have accepted the notion that the inverse of translational scalar motion in the material sector of Larson’s RSt appears to be rotational motion, from the point of view of the material sector and vice-versa: The inverse of translational scalar motion in the cosmic sector appears to be rotational from the point of view of the cosmic sector.

Consequently, crossing the boundary between the two sectors involves a transition from translational c-speed to rotational 1/c-speed, which Mathis has appeared to confirm in their minds with his work differentiating orbital and tangential speeds of photons.

However, in the LRC’s view, this approach confuses the fundamental concepts of scalar motion in the universe of motion, which are based on the principle of reciprocity, easily graphed as shown below:

**Figure 1**. The Unit Boundary Between the Material and Cosmic Sectors of the Universe of Motion.

Just as 1/2 is the mathematical inverse of 2/1, Larson’s “direction” reversals in the unit space and time progression created two sectors of the universe of motion, by effectively stopping the progression of one or the other of the two reciprocal (hence orthogonal) components of the universal progression.

When the space component of an expanding location oscillates, changing the space:time progression ratio at that point from 1:1 to 1:2 (think of the increase of space as alternatingly increasing, decreasing, while the increase of time continues increasing normally), it effectively changes the progression at that point to a progression of time only, as shown in figure 1 above.

Conversly, when the time component of an expanding location oscillates, changing the space:time progression ratio at that point from 1:1 to 2:1 (think of the increase of time as alternatingly increasing, decreasing, while the increase of space continues increasing normally), it effectively changes the progression at that point to a progression of space only, as shown in figure 1 above.

This concept can be formulated mathematically by understanding that there are two interpretations of numbers: One represents quantity (how much or how many), the quantitative interpretation of number, while the other represents a relation between quantities, the operational interpretation of number.

The difference is readily understood when we consider the rational number line:

1/n, …1/3, 1/2, 1/1, 2/1, 3/1, …, n/1

The numbers to the left of unity, 1/1, are the inverse of the numbers to the right of unity. However, they may be interpreted both quantitatively and operationally, with different results, depending on the desired interpretation. Interpreted quantitatively, the numbers to the left of 1/1 are less than 1, or fractions of the whole, while the numbers to the right of 1/1 are multiples of 1.

On the other hand, interpreted operationally, the sets of numbers to the left and right of 1/1 are both multiples of 1: the set to the left are multiples of 1 in the opposite “direction” of that of the set to the right. The set to the right is the inverse of the set to the left, which changes the “direction” of its magnitudes, in the following sense:

1:n, …1:3, 1:2, 1:1, 2:1, 3:1, …, n:1

This is the sense of “direction” we have when comparing two quantities, as, for example, when we weigh quantities in a pan balance. There are three possibilites and only three: The pans are balanced with equal quantities on either side, or the pans are unbalanced with unequal quantities favoring one side or the other.

The oscillation of the space or time component at some given point in the 3D universal progression produces the discrete units of motion postulated in the first fundamental postulate of the RST. These discrete oscillating units of scalar motion may be algebraically combined and the relations between such combinations algebraically analyzed in a manner completely analogous to number systems that are well understood.

For instance, we can combine integer multiples of each:

1x(1:2) = 1:2 1x(2:1) = 2:1

2x(1:2) = 2:4 2x(2:1) = 4:2

3x(1:2) = 3:6 3x(2:1) = 6:3

. .

. .

. .

nx(1:2) = n:2n nx(2:1) = 2n:n

Clearly, in terms of magnitude and “direction,” these two sets of numbers are equivalent to opposites

-1, +1

-2, +2

-3, +3

.

.

.

-n, +n,

when quantitatively interpreted, but when operationally interpreted, they are multiplicative inverses, such that 1/n times n/1 = n/n, or unity.

When used to formulate speeds of the universe of motion, as we have illustrated in figure 1 above, we must make a distinction between material sector speeds and cosmic sector speeds, because one is speed-displacement from c-speed in terms of time, while the other is speed-displacement from c-speed in terms of space:

Material Sector Cosmic Sector

Δs/Δt = 1/1 = c Δt/Δs = 1/1 = c

Δs/Δt = 1/2 = 1Mc Δt/Δs = 1/2 = 1Cc

Δs/Δt = 2/4 = 2Mc Δt/Δs = 2/4 = 2Cc

. .

. .

. .

Δs/Δt = n/2n = nMc Δt/Δs = n/2n = nCc

Where Mc = material speed-displacement = Cc = cosmic speed-displacement = 0.5c.

Of course, because these speeds are inverses, one is four times greater than the other, quantitatively, when considered from the reciprocal point of view (i.e. 2/1 = 4x0.5).

The attempt to quantify the RST concepts has led to different approaches, but the LRC was established to ensure that Larson’s fundamental postulates were not changed in the process. Opponents object to Larson’s idea of “direction” reversals and have sought to find an alternate way to obtain the required reciprocity of the system, by resorting to rotation. This is understandable, but the LRC takes exception to the concept of rotation as scalar motion.

Of course, the rejection of rotation as scalar motion means that the LRC departs from Larson’s development of the consequences of the RST in his RSt, but it does not imply departure from the RST itself, which must be held inviolate.

In its abandonment of the RST, it appears to us that the RS2 community has lost sight of the meaning of true reciprocity and in the attempt to justify its rationale, it now seems willing even to abandon the concept of scalar motion.

]]>However, the integer and rational number systems are actually composed of one dimensional numbers, or lengths. As we have discussed many times before, to deal with higher dimensional numbers, legacy system mathematics adds “imaginary” numbers to these number systems.

In the reciprocal system of mathematics (RSM), we recognize four scalar dimensions, with two “directions” each. At unit magnitude, these are isomorphic to the 4th degree of the binomial expansion, 20, 21, 22 and 23, which we call the tetraktys, because it consists of the first four numbers of the Pythagorean system of numbers: the monad, the dyad, the triad and the tetrad.

As we happily discovered some time ago, the binomial expansion of the tetraktys is the numerical equivalent of the geometry of the 2x2x2 stack of 8 unit cubes that we call Larson’s cube (LC). Its center, located where one corner of all 8 cubes coincide, defines the middle point of three, 1D, orthogonal lines, the intersection of three, 2D, orthogonal planes, as well as the intersection of the diagonals drawn between the eight reciprocal corners of the stack.

**Figure 1.** The Stack of Eight 1-unit Cubes Known as Larson’s Cube

The fact that this geometric figure contains the geometric representation of the four dimensions of the tetraktys and thus the first four binomial expansions of the tetraktys (counting zero), it follows that it also connects geometry with the eight mathematical dimensions of the four normed division algebras, the positive and negative reals, complexes, quaternions and octonions.

Moreover, the geometry of the LC not only represents the numbers of the tetraktys, it also defines a unit volume within its interior, and defines the inverse of this volume with its extent, a fact that connects the integer and rational numbers with irrational numbers, in a fundamental manner.

This connection between integers, rationals and irrationals provides a different approach to numbers and number systems that is not defined by Cantor’s sets nor Dedekind’s cuts, nor imaginary numbers. It is based on scalar expansion, where the three 1D lines, the three 2D planes and the eight 3D cubes expand outward from the 0D point.

Since a unit of space and a unit of time can be defined for the LC expansion, from point to cube, it follows that a unit of motion, v = Δs/Δt, can also be defined for it, and the inverse of the outward expansion, the unit inward motion of collapse. Since such a motion has only two possibilities, outward or inward, we can define them as the two scalar “directions” of motion, in contrast to the possible vector motion directions, which is a set of infinite directions.

Because these two scalar “directions” of motion, outward and inward, manifest themselves in each of the non-zero geometric components of the LC (the 1D line, the 2D area, the 3D volume), it’s easy to confuse them with the geometric “directions,” or poles associated with each of these entities, but we should note that the two poles of the 1D line can expand and contract, the four poles of the 2D area can expand and contract and the eight poles of the 3D volume can expand and contract.

Hence, we can use the base number 2 with non-zero exponents 1, 2 and 3 to represent the total “directions” of each non-zero dimension’s expansion, as indicated by the exponents: 21 = 2; 22 = 4, 23 = 8 “directions” respectively.

Given this isomorphism between the magnitude, dimension and “direction” of geometric expansion of the LC, and the magnitude, dimension and “direction” of the numerical expansion of the tetraktys, the question arises, “Can these geometric and numerical entities be used to form a number system that is closed under addition, subtraction, multiplication and division?” If the answer is “yes,” then it follows that the resulting 3D algebra will also have 2D, 1D, and 0D subalgebras that are closed as well.

While this may seem obvious given Clifford algebras, it isn’t, because the dimensions of the 3D Clifford algebra are used only to define the 1D (vector) space that the Clifford algebra operates in. Thus, it is all about the mathematical operations that translate and rotate vectors in that space.

In our case, we are dealing with an algebra of the spaces themselves. We seek to add and multiply LCs, if you will - the whole LC at once, represented by its geometric properties and the isomorphic numerical properties of the tetraktys.

As described in the previous entry, the numerical expansion of the LC, and thus the tetraktys, can be accomplished by addition or multiplication of its poles. Since the monopole, the three dipoles, the three quadrupoles and the single octopole comprise the unit and its subunits, they can be consistently manipulated algebraically.

To demonstrate this requires only that the geometric coefficients (1331) of the four-part numbers (1(20)+3(21)+3(22)+1(23)) be removed, before the algebraic operation is performed, and then reinserted into the number after the calculation is complete. For example, we can show that this works for addition, subtraction, multiplication and division, by letting the unit LC equal ‘a.’ Then,

a = (1+2+4+8)

b = 2a = 2 x (1+2+4+8) = (2+4+8+16)

c = 3a = 3 x (1+2+4+8) = (3+6+12+24)

d = 4a = 4 x (1+2+4+8) = (4+8+16+32)

Next we perform the following operations:

For multiplication,

1) a/b x c/d = ac/bd

((1+2+4+8)/(2+4+8+16)) x ((3+6+12+24)/(4+8+16+32)) =

((1+2+4+8)(3+6+12+24)/((2+4+8+16)(4+8+16+32)) =

((3+12+48+192)/(8+32+128+512)) =

((3(1+4+16+64))/(8(1+4+16+64)) = 3(1+2+4+8)2/8(1+2+4+8)2 = 3/8

Just as a/2a x 3a/4a = a(3a)/(2a)(4a) = (3a2)/(8a2) = 3/8

For addition,

2) a/b+c/d = ad+bc/bd =

(1+2+4+8)/(2+4+8+16) + (3+6+12+24)/(4+8+16+32) =

((1+2+4+8)(4+8+16+32) + (2+4+8+16)(3+6+12+24))/((2+4+8+16)(4+8+16+32)) =

((4+16+64+256)+(6+24+96+384))/(8+32+128+512) =

(10+40+160+640)/(8+32+128+512) =

10(1+4+16+64)/8(1+4+16+64) =

10(1+2+4+8)2/8(1+2+4+8)2 = 5/4

Just as

(a/2a)+(3a/4a) = ((a x 4a)+(2a x 3a))/(2a x 4a) = 10a2/8a2 = 10/8 = 5/4

And finally, for division,

3) (a/b)/(c/d) = (ad)/(bc) =

((1+2+4+8)/(2+4+8+16))/((3+6+12+24)/(4+8+16+32)) =

((1+2+4+8)(4+8+16+32))/((2+4+8+16)(3+6+12+24)] =

(4+16+64+256)/(6+24+96+384) =

4(1+4+16+64)/6(1+4+16+64) = 4(1+2+4+8)2/6(1+2+4+8)2 = 2/3

Just as

(a/2a)/(3a/4a) = (a(4a))/((2a)(3a)) = (4a2)/(6a2) = 2/3

Of course, since a = 1, 2/3 = 2a/3a = 2(1+2+4+8)/3(1+2+4+8) or, equivalently, 2(20+21+22+23)/3(20+21+22+23)

With the calculation complete, we insert the geometric numbers (1331) back into the final terms, which gives us the correct number of poles. For instance, in this last example,

2(20+3(21)+3(22)+23) = (2+12+24+16) = 54

3(20+3(21)+3(22)+23) = (3+18+36+24) = 81

54/81 = 2/3.

But then, why not just use the number of poles in the LC to begin with?

LCp = (1x(20)+3x(21)+3x(22)+1x(23)) = (1+6+12+8) = 27,

and since 27 = 33, then let

a = 33

b = 2(33)

c = 3(33)

d = 4(33)

and naturally, the algebra is still closed under addition and multiplication.

]]>

where poles are the constituent components of monopoles (mp=1), dipoles (dp=2), quadrupoles (qp=4) and octopoles (op=8). This gives us a means to express the sum of the unlike dimensions of unit numbers: In other words, we can sum the unit points, the unit areas, and the unit volumes of elements of the expanded tetraktys (1331), which is the numerical equivalent of the expanded LC, in terms of these poles.

Consequently, the numerical expression of the scalar expansion of the LC is

33, 53, 73, …(2n+1)3, n = 1, 2, 3, …∞

Now, it’s important to distinguish between this 3D expansion of the tetraktys and the 0D expansion of Pascal’s triangle, or the binomial expansion, which is normally considered as the numerical expansion of geometric dimensions, without including the notion of magnitude and “direction.” The scalar expansion of the triangle is simply,

20, 21, 22, 23, …2n, n = 0, 1, 2, 3, …∞

We can interpret this expansion as the expansion of the countable number of points:

20 = 1 instance of 1 point; 21 = 2 instances of 1 point; 22 = 2 sets of 2 instances of 1 point; 23 = 2 sets of 2 sets of 2 instances of 1 point, etc.

1) 20 = (*) = 1

2) 21 = (*)**+**(*) = 1**+**1 = 2

3) 22 = **[**(*)+(*)**]+[**(*)+(*)**]** = 2**+**2 = 4

4) 23 = **{**[(*)+(*)]+[(*)+(*)]**}****+{**[(*)+(*)]+[(*)+(*)]**}** = 4**+**4 = 8

Hence, unlike the numerical expansion of the tetraktys, which is equivalent to the 3D geometric expansion of the LC, the numerical pattern of the triangle is not equivalent to any geometric expansion, but it is simply a regrouping of 0D terms. Take line four for example:

4) 23 = **{**[(*)+(*)]+[(*)+(*)]**}****+****{**[(*)+(*)]+[(*)+(*)]**}** = 4**+**4 = (*)**+**(***)**+**(***)**+**(*) = 1+3+3+1 = 8

That this sum of points in line 4 of the triangle (4+4 = 8) is equal to the sum of dimensional coefficients in the tetraktys (1+3+3+1 = 8) appears to be the cause of a colossal error in the development of mathematics: The fundamental confusion resulting from an incorrect understanding of magnitude, dimension and “direction” in mathematics and geometry, is tantamount to a detour along the road to comprehending their unity.

As a result of this non-comprehension, mathematics has been developed along a non-geometric line, leading to the confusion in the theory of algebra lamented by Hamilton, wherein he 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. It must be hard to found a Science on such grounds as these, though the forms of logic may build up from them a symmetrical system of expressions, and a practical art may be learned of rightly applying useful rules which seem to depend upon them.

The effect of this detour probably only appears to the non-genuises (Hamilton was truly a genuis) among us, when we seek to understand the properties of scalar motion. After all, who needs to understand n-dimensional scalars in the practical world of vector motion?

In the theoretical world of nothing but motion, we have to combine n-dimensional entities of motion and determine how these combos relate to one another. We can’t do it without first clearing up the confusion in our understanding of these fundamentals.

Hopefully, it’s getting to the point that we can begin to formulate the n-dimensional, scalar algebra we need to move forward, though the very idea must appear preposterous and nonsensical to the uninitiated.

]]>In a letter to Frank Meyer, two years before he passed away. Larson wrote:

As I have tried to emphasize throughout my writings, the conceptual aspects of physical theory, our understanding of what the mathematics of physical events mean, is independent of the mathematical relations. There are usually many possible interpretations of the same mathematics. Consequently, the true meaning cannot be derived from the mathematics. As matters now stand, the accepted physical meaning of each mathematical relation is based on assumptions applicable to that particular case. Conventional physical theory has a general mathematical structure into which each individual conclusion is required to fit, but it has no similar conceptual structure, and it therefore has no way of verifying the conceptual interpretations of the mathematical relations. Our contribution is to provide the conceptual structure that is needed. Since the previous interpretations were based on unconnected assumptions, it was inevitable that some of them would turn out to be wrong, but this does not necessarily mean that the mathematical expressions are incorrect. And where we do find that some modification of the mathematical relations is necessary, we do not need any new kind of mathematics.

However, it is evident that “a new kind of mathematics” is necessary, since the legacy mathematics has taken a development route that follows vector motion, rather than scalar motion. In vector mathematics, as in vector motion, the point’s position, relative to another, is paramount, while in scalar mathematics, as in scalar motion, the magnitude’s value, relative to another, is paramount.

However, because a vector also has magnitude, confusion in terms of the mathematical meaning of scalar, since it has been understood mostly in the context of vector mathematics, results when we speak of scalar dimensions.

This confusion is cleared up when we cast scalar dimensions in the light of the first four levels of the binomial expansion, which yields the tetraktys (2^0 = 1, 2^1 = 2, 2^2 = 4, 2^3 = 8), since we can understand the base number 2 as two “directions” of a given dimension, which produces the doubling expansion 1, 2, 4 and 8 of poles, defining the geometry of Larson’s cube (point, line, plane and cube).

However, the fact that numbers, as factors, can be expanded without limit (n x 2; n—>∞), and the fact that mathematicians regard these factors as dimensions, has led them to regard factors of 2 greater than 3 as geometric dimensions (hyper geometric dimensions), even though a child can see that the logic is bogus.

It is a fact that nature stops at three dimensions, Raul Bott proved it, so trying to ignore that limit only leads to confusion, in the end, as we see it has in LST theory, which has been stopped dead in its tracks by it (see here.) While this was not so evident in Larson’s time, it is clear to almost everyone now: the trouble with physics is that physicists have sought solutions, by following mathematicians into the realm beyond three dimensions.

Here at the LRC, we not only assert that this is a mistake, we contend that there is no need to be tempted to do it, when one understands the science of simple mathematics in terms of the science of simple geometry.

When we expand Larson’s cube, the progression is 8, 64, 216, 512…(nx2)3, where n = 1, 2, 3, …∞. In this way, the base 2 increases not as factor of 2, but as a multiple of 2, and each dimension can be expanded in turn for each successive base:

(1x2)0 = 1; (1x2)1 = 2; (1x2)2 = 4; (1x2)3 = 8,

(2x2)0 = 1; (2x2)1 = 4; (2x2)2 = 16; (2x2)3 = 64,

(3x2)0 = 1; (3x2)1 = 6; (3x2)2 = 36; (3x2)3 = 216,

(4x2)0 = 1; (4x2)1 = 8; (4x2)2 = 64; (4x2)3 = 512.

**Figure 1. **Expansion of Larson’s Cube (LC).

This is scalar magnitude expansion in 0, 1, 2 and 3 dimensions. Each dimension has two “directions,” and the expansion proceeds step by step, as the natural, or counting numbers increase, according to the dimensional number of the term.

Since the expansion is eternal, picking a point in the count is tantamount to starting at the number one and increasing from there.

The unit number, 1, is the number 2 expanded from a point, 10, to a 2x2x2 = 8 stack of unit cubes, but all three dimensions, the line, the plane and the cube expand simultaneously in two “directions”, four “directions,” and eight “directions,” respectively (the point doesn’t expand, of course.) This gives rise to the numbers of the fourth line of Pascal’s triangle, 1331, which, when summed, equal 8.

This composite number, the tetraktys, or its geometric equivalent, the LC, can be understood as the 3D unit scalar and, if it is treated as such, it is clear that the subsequent lines of the expanded triangle are multiples of it, as we have already discussed in the previous post.

However, to treat the subsequent lines of the expanded triangle, line 5 and up, as 2n geometric dimensions, or as some kind of “extra” geometric dimension, is a grave error for physicists, who, we contend, must respect the limits of Euclidean geometry in their study of the structure of the physical universe, if they expect to progress.

Taking the binomial expansion of the tetraktys, the 2x2x2 = 23 = 8 unit cubes, and the corresponding 1mp+3dp+3qp+1op = 27 poles, as the fundamental scalar unit of our new system of mathematics, replacing the number 10, as the fundamental scalar unit, as understood in today’s mathematics, to which mathematicians then add “imaginary” numbers to this “real” number, in order to increase its dimensions, requires much of us to be sure, but it seems to me that it must be done, if we are to progress in our understanding of the physical universe.

All righty then.

Comparing the LC pseudoscalar expansion, 8, 64, 216, 512, …, to the triangle pseudoscalar expansion, 8, 16, 32, 64, …, it’s hard to see much of a correlation between the two. Whereas the sum of the numbers of a given line of the triangle is always equal to the pseudoscalar of that line, which is always a power of 2, and the sum of the number of poles of a given line is always equal to a power of 3, the sum of the number of poles of each successive LC expansion doesn’t follow this pattern, rather it follows a (2n+1)3 sequence: 33, 53, 73, etc.

So, while it’s easy to see that the tetraktys (1331) is a unit of the triangle and can be used to calculate its expansion, either the expansion of its numbers, or the expansion of their associated dimensions, when these are understood in the 3D terms of the tetraktys, the question now is, “Can we relate the tetraktys to the expansion of the LC, which has a different pseudoscalar than that of the triangle?”

Happily, the answer is yes. In fact, it is simple: We just need to join the tetraktys numbers with the dimensions of the LC expansion, as shown in figure 1 above, and voila, it appears:

**Figure 2.** The Expansion of the Tetraktys and Larson’s Cube

There is much more work to do, but understanding how the dimensions of the tetraktys relate to the expansion of Pascal’s triangle **AND** the expansion of the LC and the implications of this for today’s mathematics and physics is a necessary step toward the development of a “new kind of mathematics.”

I think Larson would be happy, cautiously happy, but happy nonetheless.

]]>I posted a new excellent video in LRC Lectures Online by John Baez that explains the LST view of the mathematics that is so important to understand, as background to the development of an RST-based theory of particle physics here at the LRC.

In his video, John talks about the tetraktys, but he doesn’t call it that. He tells the math history of imaginary numbers and their use to devise higher-dimensional algebras from the reals. He explains the important concept of normed division algebras and how they only exist in certain dimensions, and how that fact kept Hamilton frustrated for a long time, as he sought to find a three-dimensional normed division algebra (although he didn’t have a label like that to express it concisely). “Normed,” in this context, means that the absolute value of the products is equal to the product of the absolute values.

The problem is, Baez and company have changed the meaning of the word dimension, without explicitly pointing it out to non-specialists. Their use of the word doesn’t necessarily refer to geometric dimensions, but, like Larson explained, the word dimension, for them, means the number of independent magnitudes in an equation.

Thus, complex numbers (ℂ) are two dimensional numbers for modern mathematicians, because they consist of a real and an imaginary part (a+ib), and they are used in calculations of 2D rotation.

However, in terms of the RSM, line 2 of the tetraktys (11), or 1(2^0), 1(2^1) in its binary expansion, defines a one-dimensional line:

Equation 1: (2^0 + 2^1 + 2^2 + 2^3) = 27

This is exciting because it relates the discrete numbers of the tetraktys to the continuous magnitudes of LC. But to understand it, one has to understand the equation above. It is an equation of what we call the Reciprocal System of Mathematics (RSM).

In the legacy system of mathematics, the sum of the numbers in equation 1 above is 15, not 27. In the RSM, the base number 2 in the binomial expansion is the magnitude’s number of “directions” (think polarities) and its exponent is the number of its dimensions.

This relates to Pascal’s triangle and Clifford algebras, but also to Raul Bott’s periodicity theorem, which limits the number of geometric dimensions to no more than three. The key is to understand that each number represents the number of geometric entities of 2n (n = 0-3), ** contained** in a given dimension, where the dimension is the number of terms in the equation, less 1.

Thus, in equation 1 above, the total number of terms is 4, so the dimension is 3 and there is one 20 term contained in that dimension, three 21 terms, three 22 terms, and one 23 term. These happen to be the four coefficients of the fourth line of Pascal’s triangle, the sum of which is the mysterious number 8 of Bott periodicity:

Line 1) = 1 = 1

Line 2) = 11 = 2

Line 3) = 121 = 4**Line 4) = 1331 = 8**

Since these numbers correspond to the first four dimensions of the binomial expansion and Clifford algebras, and they correspond to the geometry of Larson’s 2x2x2 stack of unit cubes, the philosophy of the ancient Greek tetraktys, and the dimensions of the only known normed division algebras, not to mention the blades of Hestenes’ Geometric Algebra, and the domain of string theory, our interpretation of them as the “directions” in the geometric dimensions of numeric magnitudes is in good company.

Hence, we can rewrite equation 1 above to reconcile the RSM and legacy math difference, by combining the coefficients of Pascal’s triangle with the terms of the binomial expansion:

Equation 2: **1**(2^0) + **3**(2^1) + **3**(2^2) + **1**(2^3) = 27

However, the coefficients of the fifth line of Pascal’s triangle, corresponding to the fourth dimension (5 terms less 1), are

Line 5) 14641 = 16

Obviously, the coefficients of each line of the triangle sum to powers of 2, while the sum of the products of the coefficients and the n dimensional geometric entities are powers of 3. For example , the sum of the products of the fourth dimension is:

Line 5) **1**(2^0) + **4**(2^1) + **6**(2^2) + **4**(2^3) + **1**(2^4) = 3^4 = 81.

The coefficients of the triangle’s fifth dimension are

Line 6) 1 5 10 10 5 1 = 2^5 = 32

and the sum of the products of the coefficients and the n dimensional entities is

Line 6) **1**(2^0) + **5**(2^1) + **10**(2^2) + **10**(2^3) + **5**(2^4) + **1**(2^5)= 3^5 = 243.

But since our RSM interpretation counts the number of n-dimensional entities *contained* in each dimension, and those dimensions cannot exceed 3, what meaning can the triangle’s lines 5 and up have for the RSM?

Well it turns out to be the key to the RSM actually. Because the sum of the products of the triangle’s coefficients and the n-dimensional geometric entities is always a power of 3, we can simply multiply each coefficient of line four (the 3rd dimension) of the tetraktys by 3 to obtain the next higher set of coefficients, thus preserving the four terms of the three-dimensional form:

Line 5) 1331 x 3 = 3+9+9+3 = (8 x 3) = 24

Line 6) 3993 x 3 = 9+27+27+9 = (24 x 3) = 72

Line 7) 9 27 27 9 x 3 = 27+81+81+27 = (72 x 3) = 216

Notice that this is tantamount to multiplying the number of geometric entities of the LC by the number 3.

So much for obtaining the coefficients. Now, what about the sum of their products with the n-dimensional geometric entities?

Well, as we have already seen, these are just powers of 3:

Line 4) 3^3 = 27

Line 5) 3^4 = 81

Line 6) 3^5 = 243

Line 7) 3^6 = 729

So, we have another periodicity. This time it is a period of 27. We will call it the tetraktys periodicity. Therefore, in the RSM, the number 8 of Bott periodicity, which is really the sum of the four coefficients of the triangle’s fourth line, is related to the number 27 of the tetraktys periodicity, which is really the sum of the four coefficients’ products with the corresponding four n-dimensional entities of the tetraktys (LC,) by the number 3, which is the key factor of both periodicities, uniting the discrete periodicity with the continuous periodicity.

This is a wonderful gift, a gift of 3, the trinity we might say, one which we neither understand nor deserve, but which we celebrate this time of year.

]]>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.

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(-) (+)

2^0 = 1 is the magnitude of the point in the middle, 1/1, and 2^1 = 2 is the one dimensional magnitude, with two opposite polarities, or a dipole, stretching out on either side of the point. In other words, it is the numerical equivalent of a geometric line in one dimension.

Hamilton finally realized that he needed four terms to construct what he thought of as a 3D algebra, moving one dimension up from what he thought of as a 2D algebra. He suddenly realized one day that he could accomplish his objective with one real number and

threeimaginary numbers, or what mathematicians would call a 4D number a+ib+jc+kd.Today, these numbers are called quaternions (ℍ), and Baez refers briefly to the battle that ensued as to whether these allegedly “4D” quaternion numbers, or the allegedly “2D” complex numbers were the numbers of choice to use in science and engineering.

The champions of (ℍ) lost, and (ℍ) descended into the dusty bin of academics for a long time, but the important thing to understand is that they are actually 2D numbers, not 4D numbers! Think of them as a scalar, s, and a 3D vector, v (or s,v).

They correspond to the third line of the tetraktys (121), which, in binomial expansion form, is the numerical equivalent of a geometric 2D area: 1(2^0), 2(2^1), 1(2^2), because it consists of one center point, and two, orthogonal, dipoles, extending out from it, forming one quadrupole.

Increasing line 3 of the tetraktys to line four (1331) gives us a true 3D numerical equivalent of a geometric volume, which, as Hamilton’s friend Graves pointed out to him, could be formulated with seven imaginary numbers, but neither Hamilton nor Graves realized that these constructions, now called octonions,

O, were in the true 3D level of the tetraktys, which we at the LRC now know is the numeric equivalent of the 2x2x2 stack of unit cubes, which we call Larson’s cube, containing one 2^0 point in the middle, three orthogonal dipoles, three orthogonal quadrupoles and one octopole, or 1(2^0), 3(2^1), 3(2^2), 1(2^3), in terms of the binomial expansion.This is where the professor stomps his foot, because Baez and company apparently don’t recognize this. Instead they take the 8 corners of a 1x1x1 = 1 unit cube, not a 2x2x2 = 8 stack of 1-unit cubes, and they assign 7 of the 8 corners of the cube (poles) to the 7 imaginary numbers of Graves 9 (omitting the scalar a=1), and from this they devise a way to concoct the last remaining normed division algebra that can be constructed.

It’s complicated and a total distortion of the true nature of the tetraktys (they collapse their cube into a Fano plane.) This is not how we understand the tetraktys in terms of the binary expansion and its numerical equivalents of the geometric point, line, area and volume, forming Larson’s cube.

But what is very useful is the recognition that the four sums of the numbers of each the of the tetraktys, define the first four lines of Pascal’s triangle,

1 = 1

1+1 = 2

1+2+1 = 4

1+3+3+1 = 8

and which they call “dimensions,” limits the number of normed division algebras to these four geometric real dimensions, 0,1,2 and 3. This is the tetraktys and it contains all the geometric entities of our 3D universe, corresponding to the point, the line, the area and the volume. It’s a case of numbers confirming the first postulate of the RST, which states that everything consists of one component, existing in three dimensions (“Hear oh Israel, the Lord our God, the Lord is One.”)

Moreover, because of their mathematical concept of the single unit cube, with its 8 corners, defining the algebra of octonions, they assume that the next line of Pascal’s triangle (14641), leads to a fourth geometric dimension (24) in the form of the first hypercube, and they don’t stop there, but continue to higher dimensional hypercubes, collapsing them into a plane and studying their properties with group theory and Lie algebras.

To us, this is a grievous mistake, or at least unnecessary, not only because it misses the point of the identification of the binomial expansion with the magnitudes, dimensions and polarities of the tetraktys, geometrically expressed in the form of Larson’s cube, but also because it hides the real implications of Bott periodicity.

Baez explains Bott periodicity as the reason the number 8 is one of his three favorite numbers, but he can’t explain why this number is so significant, no one can, unless they admit that it reflects the fact that the physical universe

is limitedto three dimensions, geometrically speaking, as stated in the RST fundamental postulates.If they would only admit this limitation, they would understand that lines 5 and up of Pascal’s triangle should be interpreted as compounds of line 4 of the triangle. In other words, lines 5 and up are multiples of the 3D line 4.

1) 1 = 1

2) 1+1 = 2

3) 1+2+1 = 4

4) 1+3+3+1 = (6+2) = 8

5) 2(1+3+3+1) = (12+4) = (1+4+6+4+1) = 16

6) 4(1+3+3+1) = (24+8) = (1+5+10+10+5+1) = 32

.

.

.

Thus, the true nature of Bott periodicity is seen in the scalar expansion of three dimensions in the form of Larson’s cube, which can best be understood as the linear expansion of a 1-unit dipole as the independent variable, with the 4 quadrupoles and 8 octopoles, as dependent variables.

Also, since two of the three dipoles and quadrupoles are degenerate in the nxnxn stack, we only need scale 1 each of them, as we go up each step of the triangle. Hence, starting with line 4 and renumbering, the Larson cube expands as:

1) 1^1(2^1) 1^2(2^2) 1^3(2^3) ~ 1 2x2x2 = 1dp, 1qp, 1op

2) 2^1(2^1) 2^2(2^2) 2^3(2^3) ~ 1 4x4x4 = 2dp, 4qp, 8op

3) 3^1(2^1) 3^2(2^2) 3^3(2^3) ~ 1 6x6x6 = 3dp, 9qp, 27op

4) 4^1(2^1) 4^2(2^2) 4^3(2^3) ~ 1 8x8x8 = 4dp, 16qp, 64op

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.

.

We can see from this that the doubling of the sums of the steps in Pascal’s triangle, is due to the counting of the redundancy in the expansion of the 2x2x2 stack: there are ALWAYS 3 dipoles and 3 quadrupoles in the expanded stacks, so if we count all these entities, we get (counting just the number of entities, the bolded coefficients):

1)

1(2*0) +3(2^1) +3(2^2) +1(2^3) = 82)

2(2*0) +6(2^1) +6(2^2) +2(2^3) = 163)

4(2*0) +12(2^1) +12(2^2) +4(2^3) = 32.

.

.

Hence, we see clearly what has happened. The numbers in the expansion of Pascal’s triangle seem to be a reflection of the 3D expansion of Larson’s cube, not an n-dimensional expansion of a hypercube, as Baez and company see it.

Nevertheless, while the sums of the four 3D coefficients in the expansion of the initial LC (1331) match the sums of the successive lines in Pascal’s triangle, when they are doubled, the sums of the

productsof the coefficients with the respective n-dimensional entities do not match, except in line 4 (our line 1.)1)

1(2^0) +3(2^1) +3(2^2) +1(2^3) = 8 (27=27)2)

2(2^0) +6(2^1) +6(2^2) +2(2^3) = 16 (54 vs 81)3)

4(2^0) +12(2^1) +12(2^2) +4(2^3) = 32 (108 vs 243).

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As explained in the previous post, to match the sums of the products, the coefficients of line 4 must be tripled in order to match the triangle, not doubled. So, to match the sum of the triangle’s coefficients (2n), we double the 3D coefficients, at each line (a power of 2), but to obtain the sum of the products of the coefficients and the n-dimensional entities, we have to triple the coefficients (a power of 3).

It’s weird, but it’s probably due to the fact that the coefficients contain the degeneracies of the dipoles (3) and quadrupoles (3) within the LC. To reconcile this conflict, we can use the triangle’s coefficients and the LC’s n-dimensional entities, even though we can’t identify them explicitly in a geometric figure, yet.

1x2^{0},3x2^{1},3x2^{2},1x2^{3 }= 1 + 6 + 12 + 8 = 271x2^{0},4x2^{1},6x2^{2},4x2^{3},1x(2^{1}x2^{3}) = 1 + 8 + 24 + 32 + 16 = 811x2^{0},5x2^{1},10x2^{2},10x2^{3},5x(2^{1}x2^{3}),1x(2^{2}x2^{3}) = 1 + 10 + 40 + 80 + 80 + 32 = 2431x2^{0},6x2^{1},15x2^{2},20x2^{3},15x(2^{1}x2^{3}),6x(2^{2}x2^{3}),1x(2^{3}x2^{3}) = 1+12+60+160+240+192+64 = 7291x2^{0},7x2^{1},21x2^{2},35x2^{3},35x(2^{1}x2^{3}),21x(2^{2}x2^{3}),7x(2^{3}x2^{3}),1x(2^{1}x2^{3}x2^{3}) = 1+14+84+280+560+672+448+128 = 2187Notice, that we have broken down the terms with dimensions greater than three into subterms of dimension three or less, to respect the limits of Bott periodicity and the RST.

Even though it’s difficult to see how all these n-dimensional entities could possibly be assembled into one compound geometric figure, one thing is clear: The sum of the products of the triangle’s coefficients and the n-dimensionals of the cube are powers of 3, starting with the initial LC, 1331.

*Since we can now obtain the same result with the LC, we can re-write the expansion as:

LC0 = 33

LC1 = 3(LC0) = 34

LC2 = 3(LC1) = 35

LC3 = 3(LC2) = 36

LC4 = 3(LC3) = 37

which is a lot better than writing:

1x2^{0},3x2^{1},3x2^{2},1x2^{3 }= 1 + 6 + 12 + 8 = 273x2^{0},9x2^{1},9x2^{2},3x2^{3 }= 3 + 18 + 36 + 24 = 819x2^{0},27x2^{1},27x2^{2},9x2^{3 }= 9 + 54 + 108 + 72 = 24327x2^{0},81x2^{1},81x2^{2},27x2^{3 }= 27 + 162 + 324 + 216 = 72981x2^{0},243x2^{1},243x2^{2},81x2^{3 }= 81 + 486 + 972 + 648 = 2187What the implications are of this clarification of scalar magnitudes, directions, and dimensions is hard to tell at this point, but my guess is that they are significant.

* Update:Turns out that this is not correct. I should have checked it. Please see my reply to Horace in the comments below.