## Division Algebras and Motion

Discussion concerning the first major re-evaluation of Dewey B. Larson's Reciprocal System of theory, updated to include counterspace (Etheric spaces), projective geometry, and the non-local aspects of time/space.
blaine
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### Division Algebras and Motion

When Dewey Larson discussed ratios of space and time, he was probably thinking in terms of the field of reals, $\mathbb{R}$. This is a mathematical object which contains the multiplication operation, and its inverse, division.

Division is required for motion (s/t, t/s) so its reasonable to say that any mathematical system which is able to express division will be useful for describing aspects of reality in a universe of motion. Thus, we should focus efforts towards the mathematical expression of such a universe in terms of the division algebras. Specifically, we are interested in finite dimensional division algebras in order to describe finite aspects of the universe.

There is a well known mathematical theorem called the Frobenius theorem which states that there are 3 "representations" (isomorphisms) of finite-dimensional associative division algebras over the real numbers. These are the real numbers $\mathbb{R}$, the complex numbers $\mathbb{C}$, and the quaternions $\mathbb{H}$.

There are many examples where quaternions are important in the various physics fields. Bruce has discussed these in several posts. But while quaternions generalize the imaginaries and the reals (allowing for the representation of the noncommutative 3-d rotations), they still are not sufficient in describing more complex phenomena. In RS2 it is posited that Octonions are needed to describe life physics. While the quaternions are associative but not commutative, the octonions are non-associative and non-commutative. This implies that the order of operations matters, even if they are displayed in the same order. In the Reciprocal System, and as recently discussed in a particle physics paper, the variables in this equation are both the states and the operators - in other words, it is all motion transferring between different types of motion. Thus, if the states are non-commutative, then they must also be non associative. We know that 3-d rotations are non commutative so this combined with the previous statement would imply that we are not generically interested in associative algebras. This is why octonions and potentially other algebras could be important.

There is another mathematical theorem that could provide guidance as to what other division algebras may be of import. This is Hurwitz's theorem, which states that
In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic form. The theorem states that if the quadratic form defines a homomorphism into the positive real numbers on the non-zero part of the algebra, then the algebra must be isomorphic to the real numbers, the complex numbers, the quaternions and the octonions.
The key here is unital algebras, because physics often deals with conservation laws. Unitary transformations are useful for representing these. Also, a consequence of this theorem is that
The only Euclidean Hurwitz algebras are the real numbers, the complex numbers, the quaternions and the octonions.
. Hence, the octonions encompass the non-commutative and non-associative possibilities of motion. Since it can be shown via the Cayley–Dickson construction that octonions can be constructed by composing quaternions in a special way (just as imaginaries can be constructed from reals, and quaternions from imaginaries and so on). This would seem to imply that determining the mathematical foundations of describing a universe of motion would necessitate the utilization of octonions.

bperet
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### Re: Division Algebras and Motion

Good summary.

My original take on this subject is posted here: The Reality of the Imaginary.
Every dogma has its day...

dbundy
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### Re: Division Algebras and Motion

There is something important to consider here: Using imaginary numbers in an attempt to unify mathematics and geometry can be very useful, but misleading at the same time. To avoid confusion and unnecessary complication, it is useful to combine our understanding of the dimensions of Euclidean geometry with the numbers of the Greek tetraktys.

There are several aspects of this endeavor that we should consider carefully. First, the four numbers of the tetraktys (1, 2, 3 & 4) correspond to the four dimensional objects of the geometry (point (requires 1 point), line (requires 2 points), area (requires 3 points), volume (requires 4 points)). Second, each geometric dimension has two reciprocal "directions," with respect to the point, i.e. forward/backward, left/right, up/down (the point itself also has two reciprocal "directions," i.e. in/out.)

When we assign these dual geometric "directions," to the four numbers of the tetraktys, we get 20 = 1, 21 = 2, 22 = 4 and 23 = 8. This is known as the binomial expansion of the tetraktys, and it is highly significant and relevant, not only to the division algebras, but also to the fundamentals of the reciprocal system.

In the case of the four normed division algebras, the reals correspond to dimension 0 (20 = 1), the complexes to dimension 1 (21 = 2), the quaternions to dimension 2 (22 = 4) and the octonians to dimension 3 (23 = 8), following the geometric dimensions of the tetraktys. However, mathematicians, not recognizing that each of these dimensions has two "directions," mistook the "directions" of the algebras for their dimensions, introducing the ensuing confusion and hiding the wonderful connection between the tetraktys and geometry and, ultimately, what we might call multi-dimensional scalar algebra.

Most physicists would balk at such terminology, because they are taught that scalar magnitudes have no direction, as opposed to vectors. They are quantities of magnitude only, applied to vectors. However, it's easy for the students of the reciprocal system to understand that each dimension of Euclidean geometry, and the Greek tetraktys, has two scalar "directions," which are analogous to the two scalar "directions" of numbers, in each of the four dimensions of algebra.

The 0 dimensional Reals, of course, are easy to characterize, especially, using rational numbers. The positive numbers are simply the reciprocals of the negative numbers and the identity number of the group (1/1) can be thought of as unity or zero (i.e. zero displacement, to use Larson's terminology).

The 1 dimensional Complexes (with 21 = 2 "directions") introduced the incredible world of rotation to mathematics, but the 2d quaternions, with 22 = 4 "directions") have only become useful in modern times, with the use of computers, and the 3d octonions (with 23 = 8 "directions") have become objects of research relatively recently, in conjunction with string theory, mostly.

The progressive loss of algebraic properties with increasing dimension in the division algebras (order (the 1d complexes have no intrinsic order), commutativity (the 2d quaternions have no intrinsic order and are not commutative) and associativity (the 3d octonions have no order, are not commutative nor associative), has plagued the physicists for many decades now.

The essence of the problem, believe it or not, is due to the limitations of vector motion, which is the only motion the LST community recognizes. Vector motion can only exist in one "direction" of one dimension at a time, and when we algebraically combine magnitudes in each of these dimensions, the algebraic pathology just described emerges.

However, there is an algebra corresponding to the physics of expanding/contracting lines, areas and volumes and it is a scalar algebra, with no known pathology, due to multiple dimensions. We only need to recognize it. When we do, we can easily explain the concept of quantum spin, and, along with it, the 2d magnetic moment of charged particles, and soon, hopefully, the 3d mass of all particles.

Just saying.

dbundy
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### Re: Division Algebras and Motion

I'd like to make further comparisons of multi-dimensional division algebra and multi-dimensional scalar algebra, if I may.

The Euclidean "norm," as used in the term "normed division algebras," simply refers to a magnitude, or length, of a vector. It's most usefully thought of in terms of the Pythagorean theorem, where the hypotenuse of a right triangle is equal to the square root of the sum of the squares of the sides.

However, as I noted in the comment above, algebraic dimensions, for mathematicians, do not refer to geometric dimensions, creating a disconnect between Euclidean geometry and normed division algebras of the Hurwitz theorem and mathematics in general.

Moreover, Euclidean geometry is limited to three non-zero dimensions, while algebra is not. Indeed, mathematicians consider finite dimensional algebras as only a subset of infinite dimensional algebras. This has caused some consternation in Wikipedia authors trying to sort out the nuances of the consequences (e.g. See here).

Again, it's important to recognize the role of the LST's concept of motion in the development of multi-dimensional division algebras. It is strictly a concept of vector motion, the norm being the length of the vector, representing a magnitude, many times a "unit" magnitude, as in the unit circle, where rotations of the unit magnitude, in either direction around the circle, are used to algebraically manipulate complex numbers, such as z=(a+ib).

Because the number z consists of two terms, it is considered two-dimensional, when, geometrically speaking, it corresponds to a one-dimensional line. This disconnect famously caused Hamilton years of grief, when he sought to multiply what he thought were three-dimensional equations; that is equations with three terms.

It wasn't until he realized that he needed four terms for three dimensional equations that the spark, leading to the modern idea of normed division algebras, fired in his brain. However, what is still not recognized by modern mathematicians, as far as I can determine, is the fact that the multiple terms in the algebras do not correspond to geometric dimensions, but to the "directions" of the non-zero dimensions.

When a point expands in one dimension, it necessarily expands in that dimension's two "directions" <------0------>. This is outward scalar motion in one dimension. To algebraically manipulate it, we need only express it as an increasing (decreasing) magnitude. Specifying direction, or orientation, in a fixed reference system is not necessary. Similarly, we can express a 2d expansion (contraction) of magnitude only, over time, with a simple equation, and likewise for the 3d case.

These scalar equations use expanding, contracting, Euclidean norms in one, two and three dimensions, which correspond to the changing radius (diameter) of 1d, 2d and 3d magnitudes, which can be defined by the linear, quadratic and cubic magnitudes of numbers, by recognizing that these also define corresponding geometric equivalents.

Had Hamilton known of the connection between the Greek tetraktys, which he adored so much, and the geometry generated by the duality of geometric dimensions, together with the reality of scalar motion, it's conceivable our world of theoretical physics and mathematics would have grown into something much more magical and wonderful than what we have today.

dbundy
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### Re: Division Algebras and Motion

One of the major advantages of the new multi-dimensional division algebra (i.e. scalar) over the traditional multi-dimensional division algebra (i.e. vector) is that the dimensions can be seen as an integrated whole; that is to say, if we begin with three non-zero dimensions and reduce them to 1, rather than begin with 1 and build them up to 3, we can gain an important insight: the lower degrees exist within the highest degree.

Let's use the usual polarity symbols to illustrate building them up:
1) 0 degree (20) = + or -
2) 1 degree (21) = + & -
3) 2 degree (22) = ++ & +- & -- & -+
4) 3 degree (23) = +++ & +-+ & +-- & ++- & -++ & --+ & --- & -+-

It's easy to see that the second degree consists of combinations of the first degree and that the third degree consists of combinations of the first and second degrees. Another mathematical way to look at the same thing is:

20 = 1
21 = 2x1 = 2;
22 = 2x2 = 4;
23 = 2x4 = 8.

However, if we start with the geometry of 23 = 8, by assigning orthogonal dimensions to each degree, we get a 2x2x2 stack of 8 unit cubes, and within that stack, we can identify three radii, or norms, using the Pythagorean theorem:

1) √(12 + 12 + 12) = √3
2) √(12 + 12 + 02) = √2
3) √(12 + 02 + 02) = √1

That each of these three norms forms the basis of an n-dimensional division algebra, isomorphic, as they say, to that of the Reals, seems to me to be straightforward, when one recognizes the existence of scalar motion.

Remarkably, when expanding/contracting (oscillating) as a 3d whole, over time, each norm expands to unity, or contracts to zero, at the same point in time (space), even though their magnitudes are not the same. It's mathematically impossible to be otherwise.

Hence, we conclude that this new set of scalar division algebras, unlike the traditional set of vector division algebras, is a well integrated set, with no algebraic pathology, and thus should be well suited for researching the scalar motion combinations, and relations between them, in the new reciprocal system of physical theory.

bperet
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### Re: Division Algebras and Motion

blaine wrote:
Sun Sep 23, 2018 10:44 am
The key here is unital algebras, because physics often deals with conservation laws. Unitary transformations are useful for representing these. Also, a consequence of this theorem is that
The only Euclidean Hurwitz algebras are the real numbers, the complex numbers, the quaternions and the octonions.
. Hence, the octonions encompass the non-commutative and non-associative possibilities of motion. Since it can be shown via the Cayley–Dickson construction that octonions can be constructed by composing quaternions in a special way (just as imaginaries can be constructed from reals, and quaternions from imaginaries and so on). This would seem to imply that determining the mathematical foundations of describing a universe of motion would necessitate the utilization of octonions.
Here's what I have found on this topic:

There exist two sets of dimensions, yang (linear) and yin (angular). We are familiar with the yang dimensions of line (1D), plane (2D) and volume (3D). The yin dimensions correspond to the same concepts but using angles instead of lines, resulting in the complex (1D), quaternion (2D) and octonian (3D) structures.

Might be confused here as to why the yin dimensions are "off"... first, I am only looking at the "vector" component, not the scalar (real scalar + <imaginary vector>). So a complex number only needs a single magnitude to express one, angular velocity. A quaternion only needs TWO magnitudes to express a sphere (latitude, longitude, for example)--you only need the 3-vector when you treat an angle as a line. A octonian only needs THREE magnitudes to express a hypersphere, which decomposes into a 7-vector in linear algebra.

Unital algebras only apply to a compound/composite motion when it is in balance (net motion has zero displacement, unit speed). When a motion is not in balance, such as an atom or molecule that is ionized, you end up with extraneous dimensions (3, 5, 6, 7). This causes the system to seek balance through the pushes and pulls of scalar relationships, until it reaches dimensional stability (1, 2, 4 or 8). In RS2, the combining of angular velocities cause dimensional reduction or expansion, such as Nehru's concept of birotation. This is how the dimensionality of a motion becomes unital.

The dimensional datum is 4 for a sector (w, x, y, z). The material is s3/t and the cosmic is t3/s. In a yang, linear form, it is called a "homogeneous coordinate" and in the yin, angular form, a "quaternion."

The projection of the inverse sector has a dimensional datum of 2, a complex quantity that expresses the net motion across the unit speed boundary (real = linear push/pull, imaginary = spin). Larson only considered the real aspect in his work, the push-pull, ignoring the spin component.

This means that the sectors are composed of complex and quaternion structures. So what of the "1" and "8" dimensions? They are "boundary conditions."

1-dimensional motion is literally "scalar" in that it can only contain a single magnitude, nothing else. The linear form of 1D in space is "distance" and in time, "duration." This is what Larson calls "scalar speed," the ratio of distance/duration, since it has no structure, only magnitude. And when this value is unity, it is the progression of the natural reference system--the "local" datum of the Reciprocal System.

The other extreme, the octonian, was never considered by Larson or Nehru. But we know the RS is about symmetry... so if the 1D motion is the natural datum, the "center" of the system, the 8D octonian (a 3D angular velocity) is the inverse of the center--the "sphere at infinity," which we see as a volume.

So in summary, the dimensional relations for the inanimate level of existence are (D=linear, d=angular):

1D/0d: progression, the datum or "center" speed
2D/1d: the projection of motion of one sector on the other
4D/2d: the structure of motion that we perceive as "stuff"
8D/3d: gravitation, the inverse datum or "sphere/plane at infinity"
Every dogma has its day...