The Larson Research Center

Broadly speaking, the history of applying mathematics to physics can be characterized as the long struggle to generalize the concept of number in order to incorporate the concept of magnitude in physical systems; that is, to find an algebraic way of consistently operating on the magnitudes of measurement in a multi-dimensional system that contains magnitudes of points, lines, planes and volumes. The most modern incarnation of this effort is found in Hestenes’ Geometric Algebra (GA), based on the fourth in a series of multi-dimensional algebras W.K. Clifford developed, known as the Cl₃ algebra.

However, to appreciate the significance of this development in the context of constructing physical theory, it’s important to note that, historically, the fundamental difference between physics and geometry has been that geometry primarily deals with numbers and magnitudes of space, whereas physics primarily deals with numbers and magnitudes of both space and time, i.e. motion. Yet, in Newton’s research program, the application of mathematics has not been directly applied to issues of space and time, but rather it has been applied indirectly via the concept of the motion of mass in the context of space and time. Thus, in this capacity, it necessarily regards space and time as a fixed background structure that defines the motion of particles and the evolution of fields, in some cases, or else that defines space and time as a spacetime manifold that itself interacts with matter to produce gravitational motion, as in general relativity.

In Larson’s Reciprocal System, on the other hand, the identity of scalar motion with the operationally interpreted number system makes it possible to apply mathematics directly to physics, in the form of a systematic investigation of the mathematical relations of discrete instances of space/time that constitute manifestations of physical entities with properties of radiation, matter, and energy. To say it another way, Larson’s definition of scalar motion, as the reciprocal aspects of a universal space/time progression ratio, brings the concept of multi-dimensional time into the realm of mathematics on an equal footing with the concept of multi-dimensional space, in the form of multi-dimensional motion, and, thus, it redefines our understanding in terms of both mathematics and physics, by circumscribing them into one great whole.

Numbers and Magnitudes

In the beginning, the ancient Greeks (especially Euclid) kept numbers separated from magnitudes. Numbers to them were quantities, while magnitudes were geometric measures of lengths, areas, and volumes. Numbers were useful for counting magnitudes, and just as they could be used for counting books, lumber, or people, they could be used to count lengths, areas, and volumes, but they couldn’t be lengths, areas, or volumes.

The history of mathematics in physics is largely the attempt to generalize the number concept enough to merge it with the magnitude concept. The capability to algebraically manipulate geometric magnitudes as easily as numbers should be very useful. However, while magnitudes and numbers are both quantities, magnitudes have two other properties that numbers don’t have. The properties of magnitudes are:

1) Quantity

2) Polarity

3) Dimension

Hence, adding the dimension and polarity properties of magnitudes to the property list of quantitative numbers is the long-sought goal. If we can expand the scope of numbers to the point that they have these two additional properties of magnitudes, then we should be able to explore the magnitudes of geometry algebraically. As explained in the previous article, Hestenes’ GA seems to do this, but it does so in terms of the n-dimensional numbers based on the concept of vectors. The numbers of Hestenes’ GA are given a new property, called “orientation,” which is derived from the direction of its n-dimensional vectors and reflects the non-commutative nature of the outer, or wedge, product.

Since GA’s orientation property stems from the direction property of its vectors, and the direction of these vectors can be interpreted as polarity, they can be used to construct the spaces of geometry in a manner that mimics the magnitude property of polarity. However, assigning the “polarity” of vectors to the “polarity” of spaces in this manner is not part of the geometry itself, because, as Newton asserted, the science of geometry only requires that its spaces be drawn, from “principles brought from without;” it has nothing to say as to how they are actually drawn. Therefore, while the Euclidean spaces seem to have a polarity property in the sense that there are two directions to each dimension of space, the space magnitudes of geometry are all positive scalar magnitudes of length, area, and volume. While it’s true that we can assign a direction to these magnitudes by selecting a reference point at the beginning of a line segment, or at the beginning of the constituent line segments of a plane or volume, such “orientation” is only an attribute of an assumption as to how the line, area, or volume is constructed, that is to say, on “principles brought from without.”

Hence, the “polarity” property of orientation is not an intrinsic property of Euclidean space magnitudes, but only a property that is inferred from how the spaces are constructed. The fact is, neither the magnitudes of Euclidean spaces, nor mathematical vectors used to construct them, have a true polarity property: There is no such thing as negative magnitudes of space, or negative magnitudes of time, or negative magnitudes of vectorial motion, or negative magnitudes of energy. This is true, even though, in special relativity, the metrics of Minkowski spacetime mimic the property of polarity of magnitude with the concepts of spacelike, timelike, and lightlike vectors.

The Symmetry of Equal Proportions

However, there is another property of geometric magnitudes that the Greeks were fond of and that modern man has rediscovered. This is the property of symmetry. The symmetry of nature can be seen everywhere and forms the core of what we consider beauty of form and harmony. Today, it is a guiding principle of mathematics and physics. Therefore, since symmetry is so powerful, let us start with numerical symmetry. This means, initially, finding the symmetrical relation of quantities, since that’s all we have to work with. The most obvious mathematical operation that will do this is an operation favored by the ancient Greeks, proportion; that is, equal proportions are the ultimate expression of symmetry: This can be expressed symbolically as n:n, which is different from the more familiar identity relation, where n=n. We can characterize this difference by noting that the identity relation equates the value of quantities, whereas the proportional relation evaluates the relative value of quantities. Thus, n = m is the same as m = n. However, n:m is the inverse of m:n, which is the simplest mathematical expression of the symmetry property obtainable.

Amazingly enough, though, there is one case, and only one case, where m:n = n:m. Of course, this occurs only when m = n. Obviously, this is the simplest and oldest mathematical relation known to man. The ancients used it in the form of a scale, or a pan balance, to measure the relative proportions of trade goods. When the weight of goods on one side equals the weight of goods on the other side, the scale is balanced. If the weight of one side is more than the weight of the other, the difference is the same regardless of which side of the balance the goods are placed. Modern chemists and pharmacists still use this ancient instrument in their professions. Thus, we can see how the beautiful principle of symmetry relates the relative values of two quantities and, in effect, measures them.

The Symmetry of Equilibrium

Now, if the quantities we want to evaluate are constantly changing, then the operation of proportion evaluates the rate of change of two quantities, rather than the tally, or weight, of quantities of substances. On this basis, equal rates of change are in balance. The numerical expression of this is: dn:dm = dm:dn = 1/1 = 1, where n and m are the rates of change of two, reciprocally, related quantities. In other words, in this case, the number 1/1 = 1, instead of representing the equality of two quantities of substance, is actually representing the equality of the change-rates of two dynamic quantities in equilibrium. It is a mathematical expression of the balanced, or symmetrical, condition of a dynamic system, analogous to a balanced static system.

Now, since we want numbers that can express the quantity, polarity, and dimension of geometric magnitudes, we should be very impressed with a number capable of expressing a dynamic symmetry, because geometric magnitudes can only be measured dynamically; that is, we have to move something to measure length, area, or volume. For instance, one way to measure length, is to move a measuring device of known length until it is parallel and coincident to the length we want to measure.

So, what can we “move” in our symmetrical, dynamic, number, dn:dm = dm:dn = 1/1 = 1, to measure length magnitude? Well, obviously, the answer is either m or n, since these are the only two rates in our number. Ok, then, let’s change the rate m. Let’s double it. We get:

dn:dm = 1/2.

If we change the rate n instead, we get:

dn:dm = 2/1,

but what does this have to do with length magnitude? Answer, everything. Think of dn:dm = dm:dn = 1/1 = 1, as a “point,” a balanced “point.” Now, the two unbalanced “points” dn:dm = 1/2 and dn:dm = 2/1 are unbalanced in two, opposite, “directions” from dn:dm = dm:dn = 1/1 = 1, the balanced “point.” If we plot these two “opposite” magnitudes on a line, we get

1/2, 1/1, 2/1,

where the imbalance between 1/2 and 1/1 is one unit on the line, and the imbalance between 1/1 and 2/1 is one unit on the line as well, but on the other side of 1/1, so their respective imbalances offset one another and therefore the combination itself is balanced. Hence, what we have here is a numerical expression of a length magnitude; that is, this number has three properties:

1) quantity

2) polarity

3) dimension

The value of the number’s quantity property is four. The value of its polarity property is two (the base in 2¹). The value of its dimension property is one (the exponent in 2¹); that is, the three terms of quantity (1/2, 1/1,and 2/1) constitute two opposed properties measured from 1/1, like the two opposite ends of a unit length (maybe a stick or a rod), measured from its center. We can express this as a combination of integers as follows:

(1/2 + 1/1 + 2/1) = (-1 + 0 + 1),

but where it is regarded as one composite number, not a total of three separate numbers. In other words, we can think of it, as we think of complex numbers, which were invented using the “imaginary” number i and have the form:

(a + ib),

which is one composite number consisting of two different types of numbers that don’t sum to a total quantity of one type, but express the result of combining two related types of numbers. Thus, we can think of our three reciprocal number (RN) terms, as a new complex number with the form:

(aL + bM + cR),

where L, M, and R, indicate left, middle, and right respectively. Recall that these complex reciprocal numbers are numbers representing a symmetrical condition. Hence, they are numbers with three properties, only one of which is quantity. There are not just three quantities here. There are two, opposing, quantities, the sum of which balance. In this type of number, the symmetrical condition can be either balanced, or unbalanced. If it is unbalanced, it can be unbalanced toward one end or the other, but not both. In the case of (1/2 + 1/1 + 2/1), the number is balanced. Therefore, the imbalance is zero, but not the number itself! We say that the zero sum of its two polarized quantities means that it is in numerical equilibrium, not that it doesn’t exist. Thus, while the integer value of

(-1 + 0 + 1),

is the sum 0, the sum of

(1/2 + 1/1 + 2/1),

is 4/4 = 1/1 = 1. In other words, this composite number is a one-dimensional, balanced, number, a triform consisting of two opposite polarities (dyad), with respect to a monopole (monad).

The Complex Reciprocal Number

Since ancient balances have been replaced by more modern methods of measuring proportions, we haven’t used these types of numbers much in modern times, but in constructing physical theory, we are looking for numerical symmetry, as a starting point, and this numerical symmetry has amazing powers. We will call this triad, formed by combining the monad and dyad, a complex reciprocal number (RN).

We can see how the symmetry property of the complex RN just keeps on giving in the binomial/trinomial expansion, where the dimension property of a reciprocal number determines the value of the other two properties, its quantity and polarity properties. Recall that the dimension of our complex reciprocal number above is 1 and the number of its properties is 3, where one property corresponds to the number’s quantity and another to its polarity; that is, it has three quantity terms, two of which are polarized with respect to a third, non-polarized, quantity. It is a complex number composed of two types of numbers. In other words, just as the familiar complex number is a composite of two types of numbers, a real type and an imaginary type, the complex RN is also composed of two types of RNs, a unipolar type (monad) and a bipolar type (dyad). Therefore, we can say, in general, that the value of a complex RN consists of the values of its two properties, quantity and polarity, which are determined by the value of its third property, its dimension property. In the one-dimensional case, then, the value of the polarity property is 2¹ = 2, and the value of the quantity property is 4¹ = 4.

The Duality of Symmetry

In the case of the ordinary, non-reciprocal, numbers denoting quantity, which we are all familiar with, increasing the dimensions from 0 to 3, by increasing the index of the ad hoc complex root, is interpreted as a general change in the type of the number; that is, the type of number goes from real to complex, from complex to quaternion, and from quaternion to octonion, etc. Each type of number has a different set of properties and algebraic rules, called its normed division algebra.

However, because the complex RN is a numerical expression of symmetry, the result is different. As the dimensions increase, the type of number doesn’t change per se, but two of its properties, quantity and polarity, change value as a function of the its third property, dimension. With the non-reciprocal numbers, on the other hand, the invention of the imaginary number compensates for the lack of natural symmetry found in the reciprocal number. In the reciprocal number, the dual quantities, arising out of its symmetry, are a result of natural reflection. We can place signs on the two opposite quantities and call the unbalanced term on the left, negative, and the unbalanced term on the right, positive, and the balanced term in the center neutral, or one bipolar term and one unipolar term, the dyad and monad, respectively. However, quantitatively interpreted scalar numbers, being quantity only, don’t have this duality naturally, so, historically, the polarity, or “direction,” property had to be invented for them.

Thus, the way we get to the opposite quantity with quantitatively interpreted scalars is to just change the sign and say we did it by multiplying it by the square of an imaginary number, ‘i’. In this way, we can make two types of numbers (positive and negative) out of one type of number (positive). It seems kind of hokey now, but it has worked superbly for two centuries, and today it is regarded as arguably the greatest leap of imagination in the history of mankind. Go figure!

Exploring Higher Dimensions

Anyway, once this was done, why stop there? If we think in terms of one-dimensional motion and regard ‘i²’ as a 180 degree rotation from the positive side of zero to the negative side of zero, then a rotation of ‘i’ is a 90 degree rotation. But then, we may ask, “how do we increase the dimension of these numbers?” The answer is, we just increase the number of imaginary numbers, of course! In other words, increasing the dimensions of ordinary, non-reciprocal, numbers increases the different types of numbers in the number system. It can’t be denied that we gain a lot by doing this, but at the expense of destroying the harmony that was cherished by the ancients.

For example, increasing the dimensions of the non-reciprocal number from 0 to 1 increases the quantity of imaginary numbers from 0 to 1, creating a new type of non-reciprocal number, the complex number. With both a real number and an imaginary number, the one-dimensional complex number can define four polarities in two dimensions. Furthermore, incrementing the number of dimensions from 1 to 2 adds two more imaginary numbers, or complex roots, as they are called, to the real and the imaginary number of the 1D complex numbers, which can then be used to define eight polarities, because with a real and three imaginary numbers, one can define both positive and negative magnitudes in the three orthogonal dimensions of a 3D spatial reference system (the upper and lower halves of four quadrants for instance).

These numbers are called quaternions. Within the quaternion number system, three types of numbers can be identified, the ordinary, or real, number type; the complex number type, with one imaginary number; and the quaternion type, with two additional imaginary numbers.

Finally, incrementing from 2 to 3 dimensions brings us to the octonions, but these are regarded as a combination of two sets of quaternions, since the quaternions have all the imaginary numbers required in a three-dimensional system. This might seem complicated to explain, but we can put it all together in the first four levels of the binomial expansion known as Pascal’s triangle:

0) 2⁰ = 1 = 1 type (1 2⁰ (real))

1) 2¹ = 11 = 2 types (1 2⁰ (real) and 1 2¹ (complex))

2) 2² = 121 = 3 types (1 2⁰ (real), 2 2¹ (complex), 1 2² (quaternion))

3) 2³ = 1331 = 4 types (1 2⁰ (real), 3 2¹ (complex), 3 2² (quaternion), 1 2³ (octonion))

Now, clearly there is geometric information in these numbers. If you start with the 2⁰ positive scalars (reals), you can regard them as geometric points that have no polarity, then comes the 2¹ complexes. Think of these as 1D lines (a line between two points - there are three of these). Next the 2² quaternions are 2D planes (four lines between four points - there are three of these), and then the 2³ octonions are cubes (eight lines between eight points - there is one set of these) formed from two intersecting planes (quaternions), forming the three, orthogonal, axes of a 3D volume. It’s all kind of messy and unsatisfying and mysterious, but perhaps we can see why: the principle of symmetry is missing from this interpretation of numbers. The ad hoc invention of imaginary numbers has enabled mathematicians to compensate for the lack of symmetry in their numbers, but, as a result, the union of number and geometric magnitude is incomplete and confused, while the sense of harmony is destroyed by these scabbed on imaginary numbers.

Higher Dimensions of Symmetry

Ok, so let’s see the same thing now, but this time in terms of the complex RN, the numerical expression of the equilibrium stemming from the symmetry of proportions. Remember, these numbers also have two properties, quantity and polarity, the values of which are a function of the number’s third property, its dimensional property, a characteristic that emerges from the intrinsic symmetry of the reciprocal number. As the dimensions increase from 0 to 3, the value of the quantity property increases exponentially with base 3, because the symmetry of the complex RN consists of its combination of a monad and a dyad, a triad. The value of its polarity property increases exponentially with base 2, because it consists of the duality of the dyad, an inseparable component of the triad, or symmetry of the complex RN.

0) 2⁰ = 1 = 1 polarity (balanced polarity-unity of monad), 3⁰ = 1 quantity (0 triad)

1) 2¹ = 1+1 = 2 polarities, 3¹ = 3 quantities (1 monad+ 2 dyad = 1 triad)

2) 2² = 1+2+1 = 4 polarities, 3² = 9 quantities (triad squared)

3) 2³ = 1+3+3+1 = 8 polarities, 3³ = 27 quantities (triad cubed)

Here, we have a binomial/trinomial expansion, as the two properties of the complex RN, quantity, and polarity, expand exponentially, as a function of dimension. Now, behold the magic of symmetry:

1) Line 0, above, is a 0D complex RN corresponding to a geometric point magnitude, a balanced number equivalent to the magnitude of one point, with no, or zero, dimensions, and 1, unified, monopole (the Greek monad):

RN⁰ = (1/1) ~ 2⁰ = 1 pole ~ 3⁰ = 1 quantity, where ~ stands for “equivalent.”

2) Line 1 is a 1D complex RN, corresponding to a geometric line magnitude, a balanced number equivalent to the magnitude of unit length, with one dimension and 1, unified, monopole and 1, dual, dipole (the Greek monad plus dyad that together equal a triad number, the first number for the Greeks):

RN¹ = (1/2 + 1/1 + 2/1) ~ 2¹ = 1+1 = 2 poles ~ 3¹ = 3 quantities

3) Line 2 is a 2D complex RN, corresponding to a geometric plane magnitude, a balanced number equivalent to the magnitude of unit area, with two dimensions and 1, unified, monopole, 2, dual, dipoles, and 1, four-quantity, quadrapole (the Greek triad number squared):

RN² = (1/2 + 1/1 + 2/1)² ~ 2² = 1+2+1 = 4 poles ~ 3² = 9 quantities

4) Line 3 is a 3D complex RN corresponding to a geometric volume magnitude, a balanced number equivalent to the magnitude of unit volume, with three dimensions and 1, unified, monopole; 3, dual, dipoles; 3, four-quantity, quadrapoles, and 1, eight-quantity, octopole (the Greek triad number cubed):

RN³ = (1/2 + 1/1 + 2/1)³ ~ 2³ = 1+3+3+1 = 8 poles ~ 3³ = 27 quantities

Thus, the long, elusive, goal of mathematical physics, to unify number and magnitude, is reached at last through the principle of symmetry. To fully appreciate this will take some time, but the harmony of the ancient Greek tetraktys, inherent in the n-dimensional complex RNs, should be strikingly apparent.

Moreover, recall that the three quantities and two polarities of the 1D complex RN completely define a unit line as two opposite RNs, 1/2 and 2/1, equidistant from the center RN, 1/1. Now, the 2D complex RN must do the same for the unit plane, and the 3D complex RN must do it for the unit volume. If they do this, the complex RNs and corresponding geometric magnitudes are equivalent. (The Greeks believed that “anything that has a middle is triform, which was applied to every perfect thing.”) We can easily show that this is the case, using matrices:

1) The second power complex RN, corresponding to the plane unit magnitude, has four polarities (2² = 4), and nine associated quantities (3² = 9), which can be represented as a 3x3 matrix, or combination of nine quantities, each with its corresponding polarity:

We have to use our imagination here a little, because we’ve separated out the three, orthogonal, dimensions of the 3x3x3 matrix for simplicity, but we should be able to see that the four, quadrapole, poles in both planes of the 2D number can be combined with an orthogonal dipole pole, or 1D number to form the eight 3D poles of the octopole in the 3D number :

1) +++

2) —-

3) ++-

4) —+

5) -+-

6) +-+

7) +—

8) -++

Alternatively, the four, dipole, poles in the two planes of the 2D number can be combined with an orthogonal dipole pole of the 1D number to form the twelve poles of three sets of quadrupoles in the 3D number:

1) ++

2) +-

3) —

4) -+

5) ++

6) +-

7) —

8) -+

9) ++

10) +-

11) —

12) -+

Finally, there are obviously two sets of dipoles in the 2D number, that, combined with the one set in the 1D number, form a total of six, uncombined, monopoles in the 3D number:

1) +

2) -

3) +

4) -

5) +

6) -

The interesting and unusual feature of all these complex RNs is that they each contain the monopole at the center, which, of course, is the source of their symmetry, and, as such, is indispensable, because it enables the complex RNs to conform perfectly to the ancient view of harmony. When we include the 0, or unpolarized, monopole at the center of these polarities, the total number of the polarized quantities of the 3D complex RN is:

RN³ = 8 + 12 + 6 + 1 = 27 = 3³

Again, we can see that the scalar interpretation of the 3x3x3 matrix is equivalent to the vectorial interpretation, by expanding the cubed complex RN:

[1/2 + 1/1 + 2/1) + (1/2 + 1/1 + 2/1) + (1/2 + 1/1 + 2/1) + (1/2 + 1/1 + 2/1)] = 16/16,

and the complex RN³ is equivalent to the sum of 16 complex RN¹s:

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

[1/2 + 1/1 + 2/1) + (1/2 + 1/1 + 2/1) + (1/2 + 1/1 + 2/1) +…(1/2 + 1/1 + 2/1)] = 64/64,

which is to say that the twelve poles in the four exterior sides of the 3x3 multiplication matrix, formed by joining four lines (RNs) end to end around the perimeter of the matrix are the same twelve poles formed, if the 3x3 matrix is obtained by crossing four lines at its center. However, four of the exterior lines form double poles at the four corners where they meet, so we count only 12-4 = 8 scalar poles in the 3x3 matrix.

In contrast, if we form the matrix by crossing the four poles at the center of the matrix (two orthogonal and two diagonal), four of the twelve poles meet at the center, or the crossing point, and are superimposed in the interior matrix, so we count 12-4 = 8 poles in the 3x3 cross matrix. However, 3x3 = 9, not 8, so what’s happening is that, in the cross product, we are counting the four middle poles as one (4/4 = 1/1 = 1). Thus, 8 poles at the ends of the four crossed lines and one in the middle gives us 9 poles, but three others are hidden in the middle so-to-speak, so there are really twelve poles in the cross product too, when we count the hidden poles.

Subsequently, when each of the four sides of the 3x3 matrix is extended, orthogonally into the 3x3x3 cube, the number of exterior lines is increased from four to sixteen (twelve parallel and four perpendicular). With three poles in each exterior line, this is a total of 3 x 16 = 48 poles, but eight poles are double poles at the corner, and four poles are double poles at the center plane of the 2x2x2 stack of cubes formed by the 3x3x3 matrix.
Therefore, we count 48-12 = 36 scalar poles in the 3x3x3 exterior matrix. This is straight forward enough, but what happens when we form the 3x3x3 = 27 cross product with the twelve pole (9+3, 3x3 matrix)?

To form the same 2x2x2 stack of cubes of the 3x3x3 matrix with the cross product, we have to rotate the 3x3 matrix into a third plane twice, once into the orthogonal horizontal plane, and once into the orthogonal vertical plane. This triples the four crossed lines in the center from 4 to 12, but now, not just the crossing points are redundant, but the 3 lines in each orthogonal plane are redundant as well. So, we have 6 redundant poles in the middle of the 6 faces of the cube. Counting up the the poles of the cross matrix, then, we have:

1) Eight poles at the eight corners of the cube, plus

2) Four poles at the centers of the four edges of the cube, plus

3) Six poles at the centers of the four faces of the cube, plus

4) One pole at the crossing point, plus

5) Eleven hidden poles at the crossing point, plus

6) Six hidden poles at the faces, for

7) A total of thirty-six poles in the cross matrix.

However, again 3x3x3 = 27, not 36, so how come? Well, of course, the reason is that 9 poles (3x3=9) are redundant, which we can see, if we remove the three redundant lines caused by the two rotations. Two poles times three lines is six poles, but these three lines also cross at the center, so removing them also removes three poles from the eleven at the crossing point, making a total of 6 + 3 = 9 poles removed from the total of 36, or 27 poles plus 9 hidden poles.

Hence, we see that the difference between the cross multiplication of vectors, and the scalar multiplication of scalars, is not in the result, but only in the operations: The vectorial operation consists in two sequential rotations of direction, while the scalar operation consists in two sequential expansions of “direction.”

We will see eventually how that this perfect symmetry of the complex RNs of the RSM give the RST an enormous advantage, something undreamed of in current theories. In the next part of this article, we will see how this power operates to create n-dimensional scalar magnitudes and to explain Bott periodicity, as a necessary consequence of the RSM’s 3D nature.

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