Reciprocal System Research Society

Recently, we've been taking a look into what the rotational operators ("imaginary" numbers) actually are, since 2/3^rds of electronics won't work without them; obviously they are representing some kind of physical quantity. Initially we thought that the rotational operators, to be consistent with the concepts of yin-yang, represented temporal displacement as an angular velocity. This is the approach Nehru took with his papers on quantum mechanics and how atomic rotation could be represented by the quaternion. Our initial conclusions brought us to the complex quantity as a representation of motion, with the real aspect being the local, and the imaginary being the non-local.

Daniel's paper on extradimensional structure build upon this idea to explain the 7-dimension structure of the life unit, which matches quite well with esoteric research. It also led to this conclusion at the inanimate level, the complex quantity is representing two concepts:

(space + i time)

where 'time' is the temporal displacement of a space:time speed, and the rotational operator (i) is indicating that the result of the temporal aspect is expressed in equivalent space, so the complex quantity is actually:

(space + equivalent space)

The rotational operator is nothing more than a symbol to identify a quantity is an equivalent space quantity, rather than a normal space quantity.

Larson considers the time region as a 2-dimensional region, where space (s) is replaced by time (1/t), so speed (s/t) becomes ((1/t)/t) = 1/t². I've always had a problem with that explanation, because motion is a relation of space TO time, not of unity to time. So "space" is still in there, just fixed at a unit value. So it is still an s/t speed, where s=1 and t varies. BUT, it inside the time region, a unit-sized bubble that contains ALL of time, from zero to infinity. This is what the Chinese Taoists call the yin-side (inside).

In the linear, outside region, one can run a straight line from zero to infinity. Take that same line, and stick it on the yin-side, and you end up with a circumference, where zero and infinity coincide over a unit distance. This gives rise to Nehru's concept of a "bounded region" that repeats, either as a simple, harmonic motion or a rotation. This projection of a straight line onto a circumference is the equivalent space, requiring TWO spatial dimensions to represent a single, angular velocity inside the time region.

The concept of "dimension" is also misunderstood. s³ is just (s x s x s), or in a geometric notation, a location of (s,s,s). The "cubed" exponent is just giving the number of coordinates required to express an object, geometrically. t² is just the coordinate, (t,t). We get away with being able to use an exponent because the numerical value for the variable (s or t) is the same for each coordinate in the set. If the variable did not have the same magnitude, then you would have to represent them as a coordinate set, for example: (1s, 4s, 3s) is not the same location as s³, and (2t,1t) is not the same location as t².

This is what gave rise to the necessity of imaginary numbers and complex quantities, because the complex quantity is a pair of coordinates that represent a 2^nd-power function, which cannot be expressed as a simple, single-variable exponent.

Magnetism demonstrates this, as a consequence of this "space + equivalent space" structure. Magnetism is not 2-dimensional, but rather a 1-dimensional temporal displacement being observed and measured in the 2-dimensional region of equivalent space, and therefore easily represented by the rotational operator (the imaginary quantity that appears throughout electronics). Technically, magnetism is a 1D angular velocity because it is the expression of a 1D motion in time--the yin aspect--that is projected as 2D because we are measuring magnetism in the only place we can measure it: equivalent space, which translates that 1D yin speed into a 2D yang rotation.

That is why the imaginary operator is rotational--the underlying temporal speed is NOT rotational nor linear, it is just magnitude. That speed is observed as a complex quantity in space, with the imaginary part being the yang shadow of the yin temporal motion, not the motion itself.

There is a consequence of this line of thought: time and the clock are two, different concepts, that are used by the consciousness of the observer to convert speed into distance and duration.

Consciousness is responsible for things like unit speed, the unit space boundary, and the unit time boundary. This explains the clock function--it is the normalization of Euclidean projection to an absolute scale of unity. The time region, for example, has space fixed at unity and time varies... but that is not a true picture. It is just motion, ns/mt. Our consciousness is normalizing the numerator from "n" to "1", and the scale factor to do that is what we call clock time. (To normalize speed, s/t to distance, time must be factored out. Hence, clock time becomes the scale factor, t/1, so when multiplied by speeds s/t... s/t x t/1 = s. Speed becomes distance. The same holds true for energy, t/s, which is normalized by clock space, s/1... t/s x s/1 = t. Energy becomes duration. The unit speed boundary is where clock time meets clock space. Also notice that the "clock" units are the reciprocal of the "regions" (time region, 1/t, and space region, 1/s).

The simplest expression of this "equivalent" relation is the photon: not 1D speed, but non-unit speed in one of three, scalar dimensions. (As discussed elsewhere, the dimensional datum is 3, not 0.) The photon has two projections into coordinate space and coordinate time: "±space ± equivalent space" and "± time ± equivalent time." I use ± because there is no preferred direction in any case, so it would take all values--linearly, a bivector, rotationally, a birotation. There is no "natural" preferred sector, but the normalization process of consciousness will create a preferred sector to obtain the clock component, and therefore, a frequency.

The ONLY property the photon has is speed. Normalization of speed into extension space or time produces the clock. All the other properties of the photon are properties of the interaction of the photon with other motions--not a property of the photon, itself.

The photon has no inherent shape; it is not a wave. It is simply a speed. However, the process of normalizing space using clock time, and normalizing time using clock space results in two, "equivalent" functions inside the unit space of the time region The first is the 2D projection of equivalent space; a 2D rotation in Larson's calls "magnetic." The second is the 1D projection of equivalent time, a 1D rotation that Larson calls "electric." Together, the 1D electric and 2D magnetic projections form the photon: electro-magnetic radiation.

Now if you understood this, please explain it to me.

I've been working on a computer model of the RS for some years and have never had much success until last night, when I recoded to use the rotational operator (imaginary numbers) as an equivalent space function, rather than a temporal function, the result being (space + i equivalent_space).

Talk about working like a dream... I just deleted about half the code I was using to extrapolate equivalent space from temporal displacement. The complex number, used as a rotational spatial function, just fits perfectly. The real component becomes the "absolute location" in space, and the imaginary component, being orthogonal to space and 2D, becomes the "field function" that exhibits all the properties of force fields.

Geometrically, real space becomes a locust of points to create physical structure, and imaginary equivalent space becomes the planar glue that attracts or repels as an invisible force, since the effect is planar and manfests between absolute locations, not at them (where the nuclei of atoms sit).

I still have a couple more things to accomplish with this simulation: first is to update the equivalent time functions, and second is to remove all the hard-coded "unit boundaries" (unit speed, unit space and unit time) and change them to a normalized clock function. That's going to be tricky, as the functions are recursive.

While thinking of how to code that normalization in a computer, which takes place during the metric and Euclidean strata of projection, some interesting concepts have presented themselves... namely, that the unit space boundary that provides the repository for atomic rotation is a function of projection--not an absolute. Anything that can normalize speeds to a unit value for clock space would literally precipitate particles and atoms out of nowhere. (Well, they are still there, but existing as "potential" versus "kinetic.") The simplest structure would be the photon--raining "light orbs", spherical, since there is no preferred direction. Some interesting esoteric consequences there. When I get further along with the programming, it will be interesting to see how it is precisely done.

I'm quite excited about this... I have not gotten this close to having a viable RS-based virtual model before. When I get the coding finished, I'll make the class libraries available for experimentation. Currently, they are in PHP 5, since that is the language I deal with the most. I will be converting to Java classes, keeping compatible with OpenGL so interactive graphics can be used with them.

Could not have done it without the imaginary number... poo-poo it all you want, but not only does it WORK, it works EXTREMELY well.

This is all fascinating. I am not sure if I understand the notion of "equivalent space" (is this just representation of time like the fictional "time axis" applied in the graphics and diagrams of conventional physics?). Modern people are used to consider time as linear, but its understanding as rotational was inherent in many ancient cultures. Even now we say that something "turns into something" as synonym of "change into something" and the change as such is more or less an effect of time. Thus from the Sanskrit root vṛt ("to turn/revolve", "to move/advance" and "to live/exist") comes the word vartana which means both "turning, roling" and "setting in motion, causing to be". Would it not be possible to consider forces and fields as effects of time (or its primal "life-force")?

As I see it, the imaginary numbers introduce dynamics in the picture since the imaginary exponential growth (e^ix) is equivalent to rotaion while the integer powers of the imaginary unit mark the four phases of the cycle - i needs to be multiplied by itself 4 times in order to return to its initial value, that's why this corresponds to the quaternary divisions (like the four seasons, the four directions, the four elements etc.) that consist of combining two polarities - one static and one dynamic. It's like the division of the year in two halves - light one (with longer day) and dark one (with longer night) which are like positive and negative, - and then adding to that the dynamic aspect of increasing and decreasing of light and darkness (day and night), which gives us the four seasons (phases of the cycle). Or if we consider rotation in 3D space this gives us polarization of space in two static linear directions of south and north along the axis (from where the rotation appears clockwise and counter-clockwise respectively) and two dynamic circular directions of east and west transversely to the axis (toward and against the rotation respectively). The combination of real and imaginary quantities seems to be the most abstract way to describe such concepts mathematically.

I also think that the idea of "unity" is only an useful mental construct, not some real basis of existence. In my opinion the assumption that all quantities could have a common denominator (unit measure) would imply a deterministic universe. In order for it to be stable, i.e. dynamic, self-correcting and versatile system, the proportions should be in fact irrational. It is our consciousness that analizes it in discrete units assuming that they are commensurable while ignoring the infinite irrational reminder. The latter can be infinitesimal and not worth considering for all practical purposes, but it is what exercises influence in the long term (that's the "butterfly effect"). Maybe this is the way that the time-flow manifests - an influx of infinitesimal irrational quantity that causes disparity between space and time units, preventing them from "clicking" exactly and so separating moments as unique from one another. Thus we may take a full cycle to be the unit of time and the radius to be the unit of space (e^2πi = 1) to have a unit speed (1/1), but the real proportion of the quantities will be irrational (1/2π or its reciprocal). This is what Henry Bergson called "cinematographic nature" of cognition and science - our mind takes shots of reality and them combines them in moving picture, but what happens between the frames is what eventually determines the course of phenomena.

I am not sure if I understand the notion of "equivalent space" (is this just representation of time like the fictional "time axis" applied in the graphics and diagrams of conventional physics?). Modern people are used to consider time as linear, but its understanding as rotational was inherent in many ancient cultures.

Daniel's paper on EDsETs has a detailed description of the concept of equivalent space, in the section with the same name as this topic, Time and Relative Dimensions in Equivalent Space. You may find that of use.

In a more mathematical sense, one cannot represent the concept of infinite angle in a linear geometry (yang: unbounded line, yin: unbounded angle). In order to get around the rotational issues of the time region, Larson introduced the concept of equivalent space, as the spatial equivalent of the temporal rotation--which is an angular velocity. By using 2 dimensions of space to represent a single dimension of rotation, you can express the unbounded angle as a repeating rotation, n(0-360), forming a helical structure that has a representation in a linear, spatial system.

that's why this corresponds to the quaternary divisions (like the four seasons, the four directions, the four elements etc.) ...

but the real proportion of the quantities will be irrational (1/2π or its reciprocal).

Quaternary division also shows up with π. Curiously, in Nature, π=4 (see Topic: Quantum Pi). As such, 1/2π is a rational number, being 1/8 = 0.125. Using π=4 provides for a simple explanation of Larson's division of the elements in the Periodic Table.

I have been learning some interesting things in writing a computer simulation, that Jameela might find useful. Since a computer has no concept of a Universe of Motion, the first thing I had to do was to define the concept, such that the Universe would have some rules to follow:

1. Had to create the concept of a magnitude, which is a counting number with a minimum value of 1. Did that by using an unsigned integer [1..n].
2. Had to create a method to express the number of times one magnitude occurs within another. This is the concept of multiplication and its inverse, division.
3. With a magnitude and reciprocal relationship, a ratio can be created to mathematically express the system through symbols.
4. However, a ratio of magnitudes, by itself, does not do anything--not even provide a value, since in order to measure you have to have the measurement and the datum of measure, the origin. Since all I have is the multiplicative functions where concepts of zero and infinity do not yet exist, the choice for the natural datum is unity, 1/1, since it is the reciprocal of itself.
5. To define a measurable system, using only the concept of a ratio, the choices are limited to the double-ratio, a ratio of ratios that is conventionally known as the cross-ratio.
6. That makes the building block of a Universe of Motion the construct of the cross-ratio (which is the only invariant in all the strata of projective geometry).
7. Just as Larson does with displacement, we can just forget about the unit-datum measure and deal with the "change" from unity--that is the ratio of motion that the Reciprocal System is built upon.
8. With a basic set of rules available, one can now create an empty, virtual universe... or as we say it in programming, $uom = new Universe().
9. It is interesting to note that this universe is NOT created out of "nothing," because the concept of "nothing" does not exist. It is a pure act of creation, from the essence of the external universe--in this case, the ability of the binary computer.
10. And also notice--it does not create itself. It required an intelligent act of a creator.

Just something to think about.

Something else interesting has turned up in the use of imaginary quantities for equivalent space... the entire universe becomes analogous to an electric circuit, since the inductive and capacitive functions are a natural consequence of the structure of equivalent space. Resistance is just the spatial component of the "circuit", the real axis.

In that model, the EM radiation of the photon is just a tuned, L-C circuit, where the resonant frequency is just the frequency of the photon. It has no other characteristics, until the circuit goes out of phase and has a presence on the real, spatial axis--which we call heat.

Many people have worked with the idea of an "electric universe." I'm going to have to check out some of that research again, but as I recall, they only dealt with the spatial projection. Larson was the only one to postulate 3D time, outside of the Hermes Trismegistus (and that's going back a ways...).

Actually I made a mistake here - it should be 1/2πi, i.e. the proportion is both irrational and imaginary. I mean this partly as symbolic taking the notions of real/imaginary, positive/negative and rational/irrational to represent qualities notwithstanding that they are quantitative nonetheless. Thus the imaginary quantity represents the fundamental non-substantial flow that can be interpreted as "Time" or rotation. It's second power (in Greek - dynamis - lit. "force") is negative (= need, lack, deficiency etc.) and this represents the level of "energy" or "force" (i.e. the urge behind the phenomena) which power (dynamis) in turn is positive and that's the empirical level from which we abstract "things" creating our illusional world of "objects" which is symbolized by the emergence of the integers from the continuum as abstracts for allegedly existing countable "things". Discretization is a function of the mind because of its finite capacity for processing information that cannot handle the infinite amount of digits of the irrational fractions. I guess the discreteness emerges in the process of digitalization of the picture in the consciousness. (Maybe even "space" as such doesn't exist except on the "screen" of our consciousness where the picture of the empirical world is simulated). The irrationality of the proportion between the complementary aspects of distance and duration (hypostasized as "space" and "time") guarantees the indeterminism of the process showing that they couldn't be reduced to some common unit which both of them contain whole number times (thus any combination of the units is unique and never repeats even after an infinite number of cycles - that's the infinity of time in both directions - no beginning and no end). Whenever one of them is fixed at integer value, the other becomes blurred due to the irrational remainder as happens with the complementary quantities in quantum mechanics. I think it is possible that our bewilderment by the quantum uncertainty is due to our idea about the physical quantities as having "exact" value in themselves and independently of our measuring - that is we consider them like the numbers as point-like positions upon the number line, while maybe the real nature of these quantities is dual and it manifests either as "point" or as "interval" in relation to its complementary quantity. Now if all physical quantities can be expressed as integer (positive or negative) powers of length and time the units of which are defined by an irrational proportion, then we can interpret their uncertainty as manifestation of the flowing wave-like nature of the space-time itself (the wave-length and the period of which would be the natural units for distance and duration - on fundamental level those are the Planck units) - it is not a "thing" or substance, but a process of mutual transition between distance and duration. It's a part of how the nothingness maintains its ballance so to speak - the universe is not created from nothing, it is nothing as a whole, while the lesser energies (negatives) have only relative existence. The so-called "conservation laws" are not actually conserving anything (because there is nothing to conserve) but act as ballancing mechanism that maintains the net value of zero for the whole of the universe and that's why it requires existence of an opposite "anti-world" to compensate all phenomena. The illusion about "creator" appears from the process of creation, not the other way around. (This process is constant and continuous, not something that has happened once in the past). It's all in the way we express things linguistically by sentences comprised of subject and verb, while in actuality only verbs have some resemblance to referents. Nobody ever does anything anywhere, because nobody actually exists - there is only self-organizing process of fleeting relatively existing phenomena governed by the feedback loops we perceive as "karma".

(I am not sure if I am expressing intelligibly what I mean or even if I myself understand it... Maybe we'll need some language like that from Borges' fictional "Tlön" to be able to express such concepts.)

"In that model, the EM radiation of the photon is just a tuned, L-C circuit, where the resonant frequency is just the frequency of the photon. It has no other characteristics, until the circuit goes out of phase and has a presence on the real, spatial axis--which we call heat.

Many people have worked with the idea of an "electric universe." I'm going to have to check out some of that research again, but as I recall, they only dealt with the spatial projection."

It is very interesting that this system comes to the same conclusion. I guess you are familiar with the works of Steven J. Smith: http://www.oocities.org/electrogravitics/. He models the physical vacuum (= "electrodynamic space-time") as net of equivalent LC-circuits with infinite resistance: http://www.oocities.org/electrogravitics/eds1.html.

It is surely not a coincidence that the speed of light (electromagnetic wave) in vacuum happens to be the same as the fundamental ratio between space and time. So far I haven't found another explanation why this should be so.

Thus in General Relativity Theory they assume that a geodesic line, i.e. the shortest distance between two points in space-time coincides with the trajectory (world-line) of the light (the photon). Still, the space-time is considered as if existing there by itself before the light travels it. But what if it's a part of the same process - light doesn't "propagates" in space-time, but actually "creates" distances and durations while unfolding the flux of space and time which are complementary aspects of the wave too.

(The concept about "flowing" space produced by time was proposed by Melchior Palágyi in 1910 who build a 4D model of space+time even before Einstein and Minkowski, but unfortunately I couldn't find much information about his ideas).

"Larson was the only one to postulate 3D time, outside of the Hermes Trismegistus (and that's going back a ways...)."

As far as I know the earliest such model was proposed by P. D. Ouspensky in his book "A New Model of the Universe" (1931). He considers a 6D universe with 3 spatial and 3 temporal dimensions. The latter include our usual linear time (4th dimension), the circular time or Nietzsche's "eternal return" (5th dimension), and the alternative time-lines (6th dimension) that embrace the all possible developments of an object. His disciple Rodney Collin developed these ideas further and in his books "The Theory of Celestial Influence" and "The Theory of Eternal Life" he also finds many curious descriptions in the ancient books and traditions that fit well in this model.

Then during the 1960s R. Oros di Bartini and P. Kuznetsov proposed their "kinematic system of the physical quantities" that expresses all quantities as product of integer powers - positive or negative - of length and time (L^RT^S). This model requires 3D space and 3D time both of which are vectors:

"The elementary (3+3)-dimensional image can be considered as a wave and as a rotating oscillator being alternatively source and sink generated by the singularity of the transformations. In the oscillator is taking place a polarization of the components of the background - a transformation L→T or T→L depending on the orientation of the oscillator that creates branching of L- and T-extensions. The elementary oscillator is a charge that creates a field around itself and in itself" (R. Bartini).

Thus a charge is created by the mutual transition of time to space and vice versa being presented as "volume with angular acceleration" (L³T^-2). It could be electric, magnetic or gravitational charge (mass) depending on which two of the three temporal axes are taken to define the "time area" (T²) in the denominator (it could be product of "longitudinal" and "transverse" times with different scale factors or only of "transverse" times).

Recently Steven Weinstein from the University of Waterloo also investigates the implications of the wave equation in models with multiple time dimensions: http://arxiv.org/pdf/0812.3869v1.pdf.

It seems that this idea becomes more and more popular lately. Either we all are onto something here or maybe just hooked up onto the same mental matrix (or thought-form or archetype or whatever we may call it).

I am not sure if I am expressing intelligibly what I mean or even if I myself understand it...

Well, what I got out of that was that space and time are integers, because they are referenced to zero, but equivalent space and time are real numbers expressed as series expansions because they are referenced to infinity, so an infinitesimal must be used to access the value.

I looked at a number of those 3D "time" system you mentioned a while back, and most of them translate to 3 dimensions of clock time, which is the metric stratum of geometry. Rather than having [ x/w y/w z/w w ] as a coordiante system, they are using independent variables [ x/u y/v z/w 1 ]. Larson'sconcept of 3D time is a separate coordinate system--not a spatial denominator. (You can see that in the expressions of L³T^-3, and no analog to clock space).