# Re: Fractal Foundations of mathematics: Axioms notions and the set FS as a model

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## Newton's Third Law and Circular Motion

When the trajectory of an object travels on a closed path about a point -- either circular or elliptical -- it does so because there is a force pulling the object in the direction of that point. That force is defined as the CENTRIPETAL force. It has not been more simply, or directly stated than by one Isaac Newton in his famous "Principia" (definition 5): "A centripetal force is that by which bodies are drawn or impelled, or any way tend, towards a point as to a center."

This force can be demonstrated by twirling a ball on a string, and either actually or conceptually cutting the string. The ball's trajectory is then a straight line tangential to the closed trajectory at the instant the string is cut. This is also illustrated by what happens to the ball in the "hammer throw" of track and field. The athlete spins the heavy ball around several times then lets it fly. It takes off in a straight line (not quite, because the hammer is actually not spun parallel to the ground, but that is not relevant).

That is really all that is necessary. The term CENRTIFUGAL force appears to have come about because of a mistaken perception that there is a force that operates in the opposite direction as the CENTRIPETAL force. But that is a misconception. The "pull" that is felt by the ball on a string or by the hammer thrower is the force that has to be applied toward the center, to keep the ball from flying off tangentially, not radially.

Unfortunately, the terms are often used interchangably, or incorrectly. Newton's term, which I think should take the prize is CENTRIPETAL.

Vince Calder

centripetal force and centrifugal force

centripetal force and centrifugal force, action-reaction force pair associated with circular motion. According to Newton's first law of motion, a moving body travels along a straight path with constant speed (i.e., has constant velocity) unless it is acted on by an outside force. For circular motion to occur there must be a constant force acting on a body, pushing it toward the center of the circular path. This force is the centripetal (“center-seeking”) force. For a planet orbiting the sun, the force is gravitational; for an object twirled on a string, the force is mechanical; for an electron orbiting an atom, it is electrical. The magnitude F of the centripetal force is equal to the mass m of the body times its velocity squared v 2 divided by the radius r of its path: F=mv2/r. According to Newton's third law of motion, for every action there is an equal and opposite reaction. The centripetal force, the action, is balanced by a reaction force, the centrifugal (“center-fleeing”) force. The two forces are equal in magnitude and opposite in direction. The centrifugal force does not act on the body in motion; the only force acting on the body in motion is the centripetal force. The centrifugal force acts on the source of the centripetal force to displace it radially from the center of the path. Thus, in twirling a mass on a string, the centripetal force transmitted by the string pulls in on the mass to keep it in its circular path, while the centrifugal force transmitted by the string pulls outward on its point of attachment at the center of the path. The centrifugal force is often mistakenly thought to cause a body to fly out of its circular path when it is released; rather, it is the removal of the centripetal force that allows the body to travel in a straight line as required by Newton's first law. If there were in fact a force acting to force the body out of its circular path, its path when released would not be the straight tangential course that is always observed.

http://en.wikipedia.org/wiki/Centrifugal_force

Centrifugal force
Not to be confused with Centripetal force.

Centrifugal force (from Latin centrum, meaning "center", and fugere, meaning "to flee") is the apparent outward force that draws a rotating body away from the center of rotation and is caused by the inertia of the body. In Newtonian mechanics, the term centrifugal force is used to refer to one of two distinct concepts: an inertial force (also called a "fictitious" force) observed in a non-inertial reference frame, and a reaction force corresponding to a centripetal force.

The term is also sometimes used in Lagrangian mechanics to describe certain terms in the generalized force that depend on the choice of generalized coordinates.

The concept of centrifugal force is applied in rotating devices such as centrifuges, centrifugal pumps, centrifugal governors, centrifugal clutches, etc., as well as in centrifugal railways, planetary orbits, banked curves, etc. These devices and situations can be analyzed either in terms of the fictitious force in the rotating coordinate system of the motion relative to a center, or in terms of the centripetal and reactive centrifugal forces seen from a non-rotating frame of reference; these different forces are equal in magnitude, but centrifugal and reactive centrifugal forces are opposite in direction to the centripetal force.

The History

History of conceptions of centrifugal and centripetal forces
Main article: History of centrifugal and centripetal forces

The conception of centrifugal force has evolved since the time of Huygens, Newton, Leibniz, and Hooke who expressed early conceptions of it. The modern conception as a fictitious force or pseudo force due to a rotating reference frame as described above evolved in the eighteenth and nineteenth centuries.

Centrifugal force has also played a role in debates in classical mechanics about detection of absolute motion. Newton suggested two arguments to answer the question of whether absolute rotation can be detected: the rotating bucket argument, and the rotating spheres argument.[1] According to Newton, in each scenario the centrifugal force would be observed in the object's local frame (the frame where the object is stationary) only if the frame were rotating with respect to absolute space. Nearly two centuries later, Mach's principle was proposed where, instead of absolute rotation, the motion of the distant stars relative to the local inertial frame gives rise through some (hypothetical) physical law to the centrifugal force and other inertia effects. Today's view is based upon the idea of an inertial frame of reference, which privileges observers for which the laws of physics take on their simplest form, and in particular, frames that do not use centrifugal forces in their equations of motion in order to describe motions correctly.

The analogy between centrifugal force (sometimes used to create artificial gravity) and gravitational forces led to the equivalence principle of general relativity.[2][3]

Action and Reaction are general terms for the interaction of bodies in dynamic situations. Thus principally they refer to situations in which both force and velocity are involved, that is acceleration and velocity. Newton's conceepts of force and quantitiy of motion as Measures were ahead of his time, but not quite what we are taught today, even as Newtonian Principles.

We have to distinguish between the conception and the measure of the same name. The concept of the measure as a conjunction of measures is different to the force as a concept of liveliness. Thus the liveliness in a dynamic situation Newton called in translation an sction or a reaction, and observed that they were a pair, that there was generally no action without an equal reaction in the opposite direction.
But Newton;s disciples were eventually convinced to modify this principle subtly because of circular motion.

Why this occurred is because no one including Newton accepted "action at a distance". There was no model that explained it convincingly and every model which otherwise worked involved some tensile rope or resistive wall.

The centripetal(pressure) and centrigugal(attractive) forces acted the same way according to Cotes, when clearly and experimentally they did not. circular motion existed in dynamic situations where a centre seeking force( a curvature force) was balanced by a centre fleeing force(an evolute force). In Newton's day such forces were not thought to exist in space. Only collisions and that mysterious sticky behaviour of particles in crystals, ropes etc which were lumped under the notion of tensile forces.

Consequently a lot of obfuscation started to cover over Newton's insight. Clearly magnetism, and electrostatics were not considered mechanically until Faraday.

http://en.wikipedia.org/wiki/Electrostatics

Since classical antiquity, it has been known that some materials such as amber attract lightweight particles after rubbing. The Greek word for amber, ήλεκτρον electron, was the source of the word 'electricity'. Electrostatic phenomena arise from the forces that electric charges exert on each other. Such forces are described by Coulomb's law. Even though electrostatically induced forces seem to be rather weak, the electrostatic force between e.g. an electron and a proton, that together make up a hydrogen atom, is about 40 orders of magnitude stronger than the gravitational force acting between them.......
History

Coulomb's torsion balance

As reported by the ancient Greek philosopher Thales of Miletus around 600 BC, charge (or electricity) could be accumulated by rubbing fur on various substances, such as amber. The Greeks noted that the charged amber buttons could attract light objects such as hair. They also noted that if they rubbed the amber for long enough, they could even get an electric spark to jump. This property derives from the triboelectric effect.

In 1600, the English scientist William Gilbert returned to the subject in De Magnete, and coined the New Latin word electricus from ηλεκτρον (elektron), the Greek word for "amber", which soon gave rise to the English words "electric" and "electricity." He was followed in 1660 by Otto von Guericke, who invented what was probably the first electrostatic generator. Other European pioneers were Robert Boyle, who in 1675 stated that electric attraction and repulsion can act across a vacuum; Stephen Gray, who in 1729 classified materials as conductors and insulators; and C. F. du Fay, who proposed in 1733[1] that electricity comes in two varieties that cancel each other, and expressed this in terms of a two-fluid theory. When glass was rubbed with silk, du Fay said that the glass was charged with vitreous electricity, and, when amber was rubbed with fur, the amber was said to be charged with resinous electricity. In 1839, Michael Faraday showed that the apparent division between static electricity, current electricity, and bioelectricity was incorrect, and all were a consequence of the behavior of a single kind of electricity appearing in opposite polarities. It is arbitrary which polarity is called positive and which is called negative. Positive charge can be defined as the charge left on a glass rod after being rubbed with silk.[2]

One of the foremost experts on electricity in the 18th century was Benjamin Franklin, who argued in favour of a one-fluid theory of electricity. Franklin imagined electricity as being a type of invisible fluid present in all matter; for example, he believed that it was the glass in a Leyden jar that held the accumulated charge. He posited that rubbing insulating surfaces together caused this fluid to change location, and that a flow of this fluid constitutes an electric current. He also posited that when matter contained too little of the fluid it was "negatively" charged, and when it had an excess it was "positively" charged. For a reason that was not recorded, he identified the term "positive" with vitreous electricity and "negative" with resinous electricity. William Watson arrived at the same explanation at about the same time.

http://en.wikipedia.org/wiki/Electric_charge

http://en.wikipedia.org/wiki/Robert_Boyle

Scientific investigator

Boyle's air pump

Boyle's great merit as a scientific investigator is that he carried out the principles which Francis Bacon espoused in the Novum Organum. Yet he would not avow himself a follower of Bacon, or indeed of any other teacher. On several occasions he mentions that in order to keep his judgment as unprepossessed as might be with any of the modern theories of philosophy, until he was "provided of experiments" to help him judge of them, he refrained from any study of the Atomical and the Cartesian systems, and even of the Novum Organum itself, though he admits to "transiently consulting" them about a few particulars. Nothing was more alien to his mental temperament than the spinning of hypotheses. He regarded the acquisition of knowledge as an end in itself, and in consequence he gained a wider outlook on the aims of scientific inquiry than had been enjoyed by his predecessors for many centuries. This, however, did not mean that he paid no attention to the practical application of science nor that he despised knowledge which tended to use.

Boyle was an alchemist;[11] and believing the transmutation of metals to be a possibility, he carried out experiments in the hope of achieving it; and he was instrumental in obtaining the repeal, in 1689, of the statute of Henry IV against multiplying gold and silver.[12] With all the important work he accomplished in physics – the enunciation of Boyle's law, the discovery of the part taken by air in the propagation of sound, and investigations on the expansive force of freezing water, on specific gravities and refractive powers, on crystals, on electricity, on colour, on hydrostatics, etc. – chemistry was his peculiar and favourite study. His first book on the subject was The Sceptical Chymist, published in 1661, in which he criticised the "experiments whereby vulgar Spagyrists are wont to endeavour to evince their Salt, Sulphur and Mercury to be the true Principles of Things." For him chemistry was the science of the composition of substances, not merely an adjunct to the arts of the alchemist or the physician. He endorsed the view of elements as the undecomposable constituents of material bodies; and made the distinction between mixtures and compounds. He made considerable progress in the technique of detecting their ingredients, a process which he designated by the term "analysis". He further supposed that the elements were ultimately composed of particles of various sorts and sizes, into which, however, they were not to be resolved in any known way. He studied the chemistry of combustion and of respiration, and conducted experiments in physiology, where, however, he was hampered by the "tenderness of his nature" which kept him from anatomical dissections, especially vivisections, though he knew them to be "most instructing".......
1674 – two volumes of tracts on the Saltiness of the Sea, Suspicions about the Hidden Realities of the Air, Cold, Celestial Magnets, Animadversions on Hobbes's Problemata de Vacuo
1676 – Experiments and Notes about the Mechanical Origin or Production of Particular Qualities, including some notes on electricity and magnetism

Thus we find that Boyles speculations were not thought relevant in Newton's time to the problem of orbits, and not until Coulomb who immediately recognised the Newtonian gravitational form. No one however was willing to accept, nor even remebered Boyles observation about action in Vacuo. It was considered as some kind of fluid for a while until Faraday reolaced he idea of a fluid model with the conception of a field.

The difference is profound, for the field acts like a fluid, like space, and yet there is no fluid observable. There is no gas abservable or containable etc etc, and it worked in vacuo as Boyle said, In vacuo meant all the gas was extracted.

Now we are not so naive as in Boyle's time, neither was Boyle naive to think all gas could be extracted from a space. But there was and is a quandary as to what is transmitting the force. The posit of a small particle called an electron by Lorentz, and its subsequent discovery after exhaustive experimentation and arguments provided a possible gas particle for the so called vacuum. In the meantime, the "Field" terminology of Faraday gained popularity

1.. Introduction
In modern textbooks there is a great polisemy as regards the meaning of electric
fi'eld and magnetic/ield, 1'1], [2], [3] and [4]. Field appears defined as a region
of space, as a vectorial function, as something which propagates in space, as
something which stores or contains energy and momentum, as a substance that
mediates interactions between gross bodies etc.
Here we anaJyze how the field concept was presented by Faraday and Max-
well, as these two authors are normally considered the modern initiators of this
concept. Although we restrict our analysis to these famous scientists, we agree
with lIeilbron when he mentioned that "the electricians of 1780 lacked the
word but not the concept, which they called 'sphere of influence', sphaera activitatis,or Wirkungkreis", [5].

http://qedinsight.wordpress.com/2011/01/24/the-field-concept-in-physics/

https://www.aip.org/history/electron/
http://en.wikipedia.org/wiki/Lorentz_ether_theory
http://en.wikipedia.org/wiki/Rudolf_Clausius
http://en.wikipedia.org/wiki/J._J._Thomson
http://www.aip.org/history/electron/jjelectr.htm
http://science.howstuffworks.com/dictionary/famous-scientists/physicists/hendrik-antoon-lorentz-info.htm
In the rapid pace of Theoretical, technological and experimental development much of the old theoretical or hypothetical or metaphysical models were considered obsolete. In particular the universal acceptance of a medium called an aether was simply by passed to put an end to egregious controversy. Action at a distance was subtly changed to field effect, and by so doing the conceptions of both antagonistic schools of thought had a common ground of agreement on which to proceed.

This led to the fractured description of "Sphaerae Activitas " we have today.

along the way the third law of Newton was severely modified in the case of circular motion to make centrifugal force fictitious, without reviewing the new discoveries in Electromagnetism as rrelevant to the issue..

A force that pushes away is a centrifugal force in circular motion. A force that pulls toward a centre is a centripetal force. Both such forces exist in the electrostatic model. The fluid/field description obscures this obvious experimental fact. Thus for a dipole we have the kind of force field needed to explain circular motion. Newton's third law is applicable in this way to the dynamic situation, and the weak torque action can be modeleled by centrifugal and centripetal models of tensile ropes and circular boundary "pressure" surfaces.

## Motionsequent Processing: The notions of Meet and Join

Grassmann next moves into the notion of processing motion sequents. The idea "Stoss" is the root of it for him, while for me it was Hamilton's "pure time" analogy. In any case the idea of morion sequents processing is functionally different. When you aggregate motion sequents you get a motion, as in the case of a film, or a video frame aggregation. The film is a subjective experience when the frames are aggregated in a special way. The rules of the way frames are aggregated are called editing, and in editing the meet and the join of film frames is crucially important.

Now all of this is am analogy for a much wider application of motion sequents: motion sequents as "frames" of reality, of real experiential continuum as opposed to, but hardly distinguishable, dream experiences.

This processing analogy holds good for all motion sequent processing and is the general basis for explaining Grassmann's more philosophical explanation of Ausdehnungs groesse, their combinations and terminology.

The first thing to realise is that combinatorics, editing and constructing a reality of form in motion are all related, and distinguishable. The terminology to distinguish is important, and subject related. Sometimes different terms are used for the same process, and i am going to examine meet and join as examples of this. suffice it to say that contiguity and continuity play n important role in distinguishing, and out of the frame of mathematical musings, have an immediate and natural meaning that carries through into any subject. The distinction suffices to support thr distinction between multiplication and addition, and at this level entirely explains the subjective difference , it also explains why in general
an≠ba and
ab =-ba i a functional terminology of process that has sense and in particular applies to Strecken.

Suppose i take an object a, that object actually is an output from a subjective processing sequence, and is called a compass multivector network, which as a related set of vectors is the equivalent of an Ausdehnungsgroesse. The spatial position relative to every other object b, c, d, e, ... is recorded by a common vector or Strecken. The objects may actually be bonded by a path or some other relation, physical or otherwise. Using transformatiional geometry i can describe relationships between symbols as objects.

Transformational geometry deals with affine transformations: rotations,translations and reflections. In addition to this i would need to add notions of meet and join.
http://en.wikipedia.org/wiki/Linear_transformation
http://medialab.di.unipi.it/web/IUM/Waterloo/node39.html

Order is important!

Suppose i move a to meet b a->b = a+b
then the relative motion is to b

If i move b to meet a a<-b =a+b
Then the relative motion is to a.
Thus meeting obscures two different cases of relative motion.
Suppose a rotates around b with b as a centre of rotation a<?>b or a<¿>b, with ? and ¿ indicating contra rotations, then i have an action on a relative to b wherever a and b are relative to each other.

Thus a<?>b(a....b)= b....a means the action has rotated a about b by a half rotation
a<¿>b(a....b)= b....a means the action has rotated a about b by a half rotation but in the contra direction.
Thus half rotation obscures 2 different cases of relative action.

To be rigorous i would include other cases, particularly the rotation of a relative to itself, so that i do not have to distinguish just yet "a" from "upside down a" or any other variant.

If a meets b and then i perform a rotation on the meet relative to one a<¿>b(a+b)= b+a

a<?>b(a+b)= b+a then the meet is not the same
a+b≠b+a

Because the relativities are now contra each other by rotation. a moves to b but the resulting meet is different by rotation.

Thus if i wish to write a+b = -(b+a), the = must be defined as "gives the same meet as" , and the minus must be defined as "rotate second relative to the first by a half circle", or "rotate relative to the centre of relativity".

There is another consideration: if the centre of rotation is not specified, and a process of half circle rotations takes place around the 2 possible centres arbitrarily, then the meet(a+b) may have a translation relative to some third object c. Thus the motion sequents have to consider inter relationships with all objects to have universal application.

The join is now a stronger form of the meet, in which the objects that meet are "glued together" to form a new object.

Thus a^b = joined(a+b)

This affects everything. Although i can still rotatate relative to a or b it is now a^b that has to be considered as rotating, and "upside down a" cannot be avoided and a<?>b(a^b) is null.

However there is a weaker case which we generally allow and that is a+b rotating relative to a or b. In this case upside down a can be reversed by a half rotation relative to itself. The outcome of allowing this type of system of rotations is the same as just rotating one object relative to the other, so why do it? The reason is illogical , but there, to make writing notation easier! Consequently we start with a join, loosen it to a meet, rotate it as a join , then rotate the individual objects relative to themselves , and then rejoin.
Alternatively w e may start with a join, rotate the join, loosen the result to a meet, rotate the individual objects relative to themselves and rejoin.
There are other alternative systems, but the end result is the same. Thus our notation loses coherence with real objects and we have to inspect the results of manipulation of notation against actions on objects in space.

If we regard this symbolically we are setting up the superstructure of relationships. A building analogy helps to clarify: we first establish the blue prints before we take action. ab =-ba is such a blue print instruction. When these two instructions are established, there combination is defined to lead to a null result. Thus moving the object a to the place of object b is the same as moving object b to the place of object a, unless b rotates relative to a by a half circle, in which case the spatial motion results in the opposite or contra motion with a different outcome. Using a third object as a common relative we can then define this resulting situation as being contra to each other, and relative to this third object, performing both actions sequentially results in a null action

In every way, by hitting upon the relationship in 3 points and 3 Strecken Grassmann had a natural and sturdy platform to develop his general construction blueprints for the Ausdehnungslehre, in which every important relationship was laid out clearly on the table before him. Hamilton, in his couples laid out the deductive relationships between 2 moments of time, and more, but would have missed this invariance because his scheme was rooted in objective relationships, not subjective invariants, ie analogous forms that could hold more than one set of "meanings" at the same time.

The importance of triangles, the so called triangle numbers and the tetractys were known to the Pythagoreans. To impeach their intelligence as mystical meanderings in this matter shows a misunderstanding of the keen empirical sensibility hey possessed. These relations, carefully worked on by Grassmann, establish the importance of having a regard to the subjective-objective interface between man and space, and the magic within the gematria of many mystical philosophies, from the Yi Ching, Through the Brahmaputsidhanta,and the Pythagorean Theurgy to modern Quantum and String theory.

There are 2 other aspects of an "object"/symbol to consider, now this rough plan has been laid out. The first is the internal relativity of an object, in compass multivector terms the subjective experience of the network for an object, thus its own relative right and left, up and down etc; The second is the subjective memory of an object, in motion sequent terms its frames as sequenced in the subjective memory, Both these aspects have direct bearing on processing relevant information, ie extracting and abstracting what is deemed relevant or important.

The consequence of considering these things is the increase in complexity, but also the realisation that the accepted terminology is barely accurate, but it is sufficient for the "wise" one to achieve a limited goal: the relative position of any assigned point in space after an action assuming continuity of motion: the relative intensity of any volume in space assuming a continuous variation.

No object should be considered stationary either rotationally or translationally. Thus a static form is only relatively static to a certain scale, determined by the output processing accuracy. It often happens that as we increase accuracy, ie magnify the output, that local variations become apprehensible. Thus what one apprehends as one object may resolve itself into a multiple form of objects that meet or join, and are in relative motion. The stability of the system, dynamic or static, is in major part what contributed to it being apprehensible as single form.

The ausdehnungsgroesse and the compass multivector networks are applicable to this fractal state of affairs.

## The Motion Trace

In going from the gematria of the semeion to the gematria of the gramme we add one fundamentally important subjective processing tool: the motion trace.

This tool traces the path(Ω) of any selection of points in the sequential motion of the selection synchronously(usually) or otherwise. Such a trace is identifiable with the gramme, and thus a dynamic representation of every form in Euclid's Stoikeioon is revealed. Euclid is therefore never static!

We may identify this motion trace with Grassmann's idea of Strecken, and thus bring to the notion of Strecken a more general path than just a straight line or a cyclic dependency.

With motion trace we may see the independence of Length and Direction and indeed Orientation, and thus attain to the "true" or full freedom of subjective processing.

The notion of subjective vector herein advanced is defined on orientation with the transfer of information clearly being defined on direction: the path traveled in transference. There is no real implication that this path is a "straight line" although for convenience it may e thus represented, until that of course becomes or leads to substantive inaccuracy.

I may by differential means accommodate any path to a limit sum of such "straight lines", but this is done in order to apply calculation schemes based on straight lines to curved ones. If i may arrive at a calculation scheme based on curved lines i may apply it accordingly in a manner appropriate to accommodate it to the curve being investigated. Whichever is the simple in notation and process i may employ to th arriving at the relationships of form, whether they be in equilibrium, static or dynamic, or in ever changing flux of vorticity.

To this end, the attachment of the notion of monas to every aspect of form and to every aspect of motion trace will afford the user with a way of accounting or many relationships of form by a sequence of counting, and a construction of form with a sequence of ordering and arranging and translating, rotating relative to some semeion of reference or the subjective processing centre. To these actions magnification and deformation as well as the very special reflection through a centre of rotation, and a useful set of tools may e had to describe most dynamic forms and systems. The notion of length is but one monas that may be employed. Whereas the notion of motion race may be employed at all levels to give or remove extra"dimensionality" to a form to descrie some motion or position or situation or change.

Reflection occurs only in a centre of rotation. Thus a reflected image represents a summation over the centres of rotation of the reflected "ray". The processing of these "rays" to form the reflected image introduces its own processing order constraints.

It is to be added to the fundamental attributes of the subjective process in interacting with space, that the subjective vector, being equivalent to the semeia or analogous in strict terms, returns information by the contra path of the subjective vector and on the basis of the subjective vector orientation and focal distance etc. information is input into the subjective process. Thus the motion of the semeia is input via changes in the compass multivector network and on this asis path and direction which is not coequal or collinear with a subjective vector path is distinguished in the subjective process, and coincidence is distinguished.

Thus a set of attributes are inherent in the subjective process based upon orientation path and direction, information transference, and processing of sequential comparisons of the information. Special paths have been idenihfied and these go with special motions in the ever dynamic situation: translation, rotation and reflection throufg a centre of rotation. but what i draw attention to here, before continuing is that for any trace of motion collinearity must not obscure the contra direction of motion nor for any "point" as a semeion must coincidence obscure the differences in rotational status of a point. Thus 2 or more lines may be collinear, but stiil not be the same line, due to designations of direction, length,axis of rotation,etc, and more particulaly 2 or more points may be coincident but not equal due to rotation direction or status, or weight inherent within a point.

Thus we may proceed to deformation of form in space due to magnification and shear and bend and twist as attributes of motion in space. Motion trace is of singular assistance in these situations to add dimensionality to subjective processing that enhances cogent manipulation of form. Today motion trace is significantly enhanced by video capture of motion.

## Gravitation 3

, , , ...

There are 3 structures positive , negative and dipole. Dipole is often called neutral or neutron.
These are static structures

There are a scatter set of dynamic dipole structures. The set can be sequenced by dipole size, but essentially they form a fractal structure with inherent regional boundaries fractally arranged at all scales.

The dynamic dipole structure is fractally entrained to the 3 static structures in motion. Thus a fractal structure has many levels which may be combinatorially distinguished by static and dynamic descriptors,

Pole
Contrapole
Dipole
dynamic Dipole
dynamic contrapole
dynamic pole

The dynamic pole and dynamic contrapole are very likely to be extreme versions of the dynamic dipole

From these distinctions the theory of electro magneto gravito dynamics can be modelled and the strong nuclear and weak nuclear forces will be found within the models built.

The models must be fractal and iterative.

The notion of density is a version of the notion of intensity and the subjective nature of this experience can be modeled by intensity per unit space functions.

The model must be a compass multivector construction from the properly identified monads for each form, and the exposited relationships between the monads of the form both in spatial sequence relationships and motion sequence relationships.

Iterative computation of surface outputs is expected entrained to each motion sequent, and the other intensities, haptic, auditory in particular should also be computed and output.

Other sensory intensity outputs must eventually be included to complete the dynamics.

The goal? The prize of becoming powerfully in tune with our universe,and being able to dance on the "waves", and drink the sweet wine of Orion.

A rotating and translating dipole field formulation should be able to account for all known observations. The phenomenon should be a function or depedent on the speed of rotatio and the radial distance of rotation and the speed of translation of the dipole. The multi dipolar effect should show strong relativistic effects at very small scales. By relatavistic i mean reference frame relativity not space-time relativity.

The forms should mimic classical notions at larger scales and below light speed translations and rotations. Scaling up will have to be able to accommodate deformation, that is different scales to different regions at different times, as opposed to uniform magnification only..

Find the form
Strip it down
and put it back again.
What are the relationships?
What is the relativity?
What is the sequence?
Oh my god of Fractals,
what are the invariants?
And now...
What if........?

## Sequence Theory

Sir William Rowan Hamilton, in his paper on "couples" develops what i call a sequence theory. He calls it a science of pure time, but i recognise it as a specific example of a more general Theory of Sequences .

The sequence is a subjective experience of motion order. That experience may be dynamic or static, explicit or implicit inferred or deduced, interpolated or extrapolated, periodic or aperiodic,finite or infinite,closed or open,subjectively or objectively manipulated or processed, self referential and relativistic. Inductionable and editable a sequence reflects one of the key fundamental attributes of subjective consciousness.

There is no sequence devoid of relative motion and no motion devoid of relative sequence. Thus any definitions are bound to be tautological. However, that being accepted an initial point of view in the tautology is to say motion is the subjective experience of relative spatial change, with particular relational change characteristics, these relations reflecting the sensory networks that perceive them. In addition a relevant symbol of this conceptual construct is the semiotic change in a network/field of compass-bivector symbols of regional space.

The compass-bivector more probably compass multivector nature of the subjective apprehension defines all relative motions as trochoidal.

A good example of a compass-bivector subjectively apprehended motion is demonstrated by:-

And That of a compass-multivector network is demonstrated by:-

There are 3 dimensional analogues of these demonstrations, but the second one does involve 3d disposition. Lazarus Plath provides several apps on his blog site to play around with and a browser version.

These important apps demonstrate the aspects of a multipolar shunya field for closed loop motions. I hope to ask Lazarus to make an open looped version.

There are 2 poles to the subjective sequencing experience: conscious and unconscious. This represents a kind of continuum or step level of attention. At the unconscious pole sequential processes beget the basic Logos Kairos sumbola Sunthemata,Summetria Response which acts like a basic operating system that sequences the input output processes in the sensory neural network meshes. At the conscious pole a sequencing system that governs the external sensors and the intermediate internal sensors in terms of input and output, provide the basic notion of sequence from which i may infer all sequences.

Fundamentally Sequences are fractal structures, and inherent within all things i do or apprehend as an animate. The 3 possibilities are: real sequences that impose sequence on everything; No sequences in reality except in my subjective processing; Subjective sequencing that imposes sequence on all my experiential continuum, that is my experience of space.

Why 3? 3 is the minimum distinction i feel any analysis should make so that some resolution may be arrived at if desired, and so that truth is not a defined or applicable concept, but rather probability is.

For those who love Truth, be sure that you have not been fed a lie!, otherwise all things are probable and anything is possible.

Hamilton successfully builds the theory of couples using Eudoxian Trichotomy, and so that will be necessary and sufficient for my purposes, or inappropriate.

Of the 3 i feel that the first case is the most probable, and so i will "concentrate" on that.

The act of focusing and concentrating is a sequence creation act.

Thus a general sensory network will receive a "flash signal", that will be immediately processed at every processing centre along its sequenced path to the main processing centre. Thus the sensory network immediately sequences the signal at different levels and provides a sequence trail for the main processing centre to select.. Following internal subjective processes, i may experience three sub processing centres as selecting, by a neural response triggering a synaptic process that sends, in sequences, 2 sorts of network signals to exterior receptors : one a weak reference signal of the original signal the other an appropriate response to that reference signal.

So for example i see a face, it is sequenced, the face is sent back to the eye as a weakened signal image with a response sequenc to focus. This clearly generates a new stronger signal response which goes through the same process until homeostasis is reached or a signal interrupt is given.

The signal interrupt may come from an intermediate or higher level process that for example needs to output on the signal status achieved or to output a response say from a recognition algorithm or process, ie perception at whatever level. Full perception may require more processing and more extensive sensory network processing and involvement.

So internally and subjectively a model of consciousness can be rationally described, using the cybernetic and fractal paradigms that rely on a tautological notion of sequencing and its synonyms, processing, structuring sampling etc etc.

Now the objective signal that started the internal processes is of course "subjectively conceived", and so why would it not appear sequenced?

That fully taken on board, i nevertheless recognise a subjective objectivity that more naturally fits within my experiential continuum. Thus if i establish an object as a metron, the compass multivector network that symbolises it records variation that is consistent in any relative position i adopt at will, by a "free" choice. This invariance in behaviour i define as objective behaviour.

Accepting objective behaviour means that i have to accept the deductions and inductions that follow, and one of the deductions which is empirically established is that most signals in the electromagnetic and auditory spectrum change sequentially with regard to a metron that measure amplitude and frequency and an allied metron that moves to allow sequence to be periodicalised, that is made periodic.

Thus it all fits with the first case of a sequencing reality that imposes sequence on every thing.

To attribute to space, the shunya field, the sequencing function and imposition of the same has fundamental and far reaching consequences.

The immediate ones are motion, memory,computational consciousness, causality, and instantaneity. Everything that inheres sequence is by this attribution a product of space or the shunya field or tautologically equivalent to it.

Thus by the definition of motion the shunya field is in motion and imparts motion ceaselessly . The spatial change is subjectively experienced sequentially, thus this is by attribution an imposition from the shunya field,and that being accepted the shunya field itself is in spatial change.

Again i reiterate, this is a tautology and it serves to describe points of view or grounds for certain language useage. The probability of this description has already been assigned by me without reference to these additional forms,because it is a subjective choice.

Similarly, i perceive auditory and intensity change sequentially, but this is only understandable if i have a subjective standard against which to compare these changes, and this subjective standard i simply call memory for the purpose of this discussion. The sequential nature of this memory function is experienced subjectively, but it is now possible to objectify the experience by comparison with film or photographic or recording technologies in general. When this is done the truly remarkable functional ability of animate and inanimate memory, is explorable objectively and iteratively giving awesome insights.

Thus memory is fundamental to processing sequential signals and is fundamental to a computational response to the signal processing that takes place within the sensory mesh. This computational response is the basis of computational consciousness, which again is both subjectively and objectively experienced within the shunya field as sequential.

The attribution of an imparting nature to the shunya field means that causality rests in the shunya field. However this causality is sequenced and thus not instantaneous. There can be no distinguishable sequence when instantaneous action occurs. Instantaneous action is therefore not excluded from possibility, it is just incompatible with sequenced causality. Causality is therefore everywhere in the shunya field but not instantaneous in the shunya field. A sequence thus "flows through the shunya field triggering regional causalities, as "flows" of dependent sequences. In such a scenario it is possible that in a separate region an identical sequence flow is originated, which by a third sequence flow is determined to be synchronous, that means at the same instant/sequent in this third flow. No possibility can be excluded,but our subjective determination of these things relies on a comparison structure i will relate in another post.

Now the notion of quickness can be defined in term of a comparison between 2 or 3 sequences. if 2 sequences are compared, then the 'flow" in both sequences are being compared. if the sequences are finite or periodic, then a trichotomy can be established which will be the basis of the notion of quickness: which on finishes or repeats before the other. if 3 sequences are used then one is used to compare the others by relating the others to sequents in the third. Thus again one may achieve a given sequent by the finish while another achieves a different sequent, given that both are measured from the same sequent in the third sequence.

In this case very fast sequences may only be distinguished from "instantaneous " ones by comparing against a faster sequence.

Infinite sequences may not be comparable except by a third reference sequence, and then the comparison is only valid for the "length" of the reference sequence, or the quickness of the reference sequence.

Am i deliberately trying to avoid "time"? No i am trying to free the mind from the notion that time is more than a subjective comparison of motion. Sequence Duration and sequence causality may be taken as concepts of time but they are what they are, and they do not flow in any necessary linear way. However it is clear that some sequences have periodicity, and again that my be used as concept of time, but there is again no reason for that periodicity to be constant. So the notion of a constant uniform linear time is not a very necessary one, no matter how useful.

As case in point, most analysis (simple) using complex numbers will use the cross product and or the dot product, the outer and inner products so called. Even the quaternionic form will use these algorithms The "shape" formed by such a action exhibits rotation and expansion or compression, very useful for describing many phenomena in space. However, if the "time" coordinate is included in these product algorithms, then "time" is no longer linear in the product. Time like the space coordinates is subjected to expansion or compression.

The point is this does not in reality matter, but for an observational result an observer may have to wait variable amounts of "time" to experience the calculated result.Alternatively variable "rates of time" may need to be employed.

Having accepted the attribution to the shunya field of the sequencing activity and the imparting of sequencing activity and actions, i am left with the "role of apprehending the nature of sequencing, choosing particular sequences and sequents, and utilising seqoencing, sequents and sequential systems of action or recording.

## The compass-Vector Network

The gematria of the semeia being a subjective experience of objective locations in space, both relatively static and dynamic, highlight the subjectivity of the notions of point, encapsulating it within certain indicator activities which derive from subjective motivations,and experiences.

Nevertheless we still can and do build a representational, recording and on some way countable or mensurable and definitely paradigmatic set of "standard experiences. These are subjective objective links,by which we may apprehend space ,somewhat objectively or through objective means which nevertheless are entirely dependent on subjective processing.

One structure that arises is the compass vector model of a semeion or a semeiotic experience. In this model both changes in orientation and orientation are encompassed as a whole experience. Changes in orientation are otherwise known as spaciometric rotation.. There is also a subjective a priori notion of distance, and consequently the notion of vector attaches to this change in focus and parallax.

The full definition of a compass vector involves a network of vectors relative to one or more compasses, and in itself forms a basis for all fundamental subjective spatial notions including perspective.

It is therefore possible to asses the notion of vector in terms of compass vectors that associate to relative perspectives ,and any dynamic within space corresponds to a dynamic change in compass vector network perspectives.

This more general notion of a compass vector network as encompassing associated perspectives and parallaxes , meand that a compass vector network is sufficiently complex to describe any motion in space subjectively without recourse to special spaces such as planes and angles. Thus every motion is describable by a compass vector network with perspectives and parallax and the associated changes of the same.

From these basic compass vector networks, i may Then develop the notions of planes reference frames and fixed centres of rotation and relative motions and relativity itself.

The compass vector network with perspective and parallax enables any motion to be described by an iterative sequencr of compass vector networks with relative parallax and perspective changes, and thus the constiute for me a definite notion of the motion sequent, introduced in an earlier blog.

The compass-vector model however does not naturally include the twisting and bending proprioceptive senses that underpin rotation and change of orientation and the fixing of orientation. Thus a compass vector is a resultant of a subjective process of relative strain statuses within my main body. These relative strains not only inform orientation but pitch and roll and yaw. This revels that any vector carries a lot more subjective information than just rotating to orientation. In fact at least 2 centres of rotation are necessary to describe any orientation. In practice if i pick any semeion it is located by at least 2 subjective processes: one involving the eyes the other involving the body and ears.

## Shunya and Phusis/Harmonia and Rhea

Shunya is infinite potential, pregnant possibility, that void of buzzing expectancy. It means full.

Why was it translated as empty?

My guess is that these attributes of Phusis and Rhea were associated in the western greek, Pythagorean philosophy with the monad and the dyad. But more importantly the Arabic collators believed that one was associated with all these properties, as an epithet of Al illah. There seems therefore to be no sentiment to translate shunya as anything like its true meaning.

At the most fundamental level, the Monad is the Primordial One and the Indefinite Dyad is Primordial Matter, because Prima Materia is the indeterminate, formless, quality-less foundation of all being; She is Sub-stance -- She who stands underneath. Like the One, Primordial Matter is ineffable, obscure, dark; therefore They are both called Abyss. Thus, the Goddess of Matter is also called Silence (Sigê), because Silence must precede the Word, the in-forming Logos, embodying the Ideas of the Craftsman (see below, on the World Soul). Her role as Mediator between the Father of the Gods and the Demiurge is confirmed by the Chaldean Oracles (fr. 50):
between the Fathers is Hekate's Center borne.

(Here the Female Principle is called "Hekate," which is pronounced "heh-KAH-tay" in ancient Greek.) Primordial Matter is much deeper, more profound, than the matter studied by contemporary physics. Hers is Potential Corporeality, not a "stuff," but the unlimited Power to Be.

One of the most common names of the Female Principle is Dynamis, which means Power and Potential. This is the aspect of the Indefinite Dyad in which She is Unlimited, Unbounded, and Infinite, for Hers is the Infinite Potential to Be. She is Potentiality at all levels of Being, for She dares the Monad to Proceed and Become. She is more powerful than the One, which is something, for She is the limitless power to be anything; She is all possibilities.

Therefore She is also the prolific, generative source of all creation. She is Multiplying, for without Her the Monad would be just One; She leads the Monad to proceed into fruitful plurality and substantial manifestation. Thus, on the lower levels of Being She is called Life-Giving (Zôogonos), which brings us to our next topic.
The Mother of the Gods

As remarked above, the Dyad, by bringing multiplicity to the One, creates the plurality of Unities (Henads), who are the Gods. Thus Rhea becomes the Mother of the Gods by substantiating multiple images of the Father, Kronos.

The ancient Pythagoreans called Rhea "The Ever-Flowing" (To Aenaon) and connected Her name with Rheô (to Flow) and Rhoê (Flux, Flow, Stream), a derivation confirmed by modern linguistics, which traces them all to the Indo-European root sreu- (to flow). This is because Primary Matter is fluid, for it has no determinate boundaries, within or without; Matter is ever changing, always in flux.

Another word correctly derived by the ancients from this root is rhythmos, which means Rhythm, but also recurring motion, measured motion, and time. This is because the Indefinite Dyad creates Otherness, and therefore all the oppositions governed by Kronos and Rhea: Unity/Multiplicity, Light/Dark, Male/Female, and many others. Whenever there is a tension between opposites there will arise an oscillation between them, a cyclic approach to One then the Other. Therefore, Rhea transforms measureless Eternity (Aiôn) into determinate Time (Khronos), symbolized by the cyclic alternation of Light and Dark. (By creating Time, She also creates Space.) Further, Rhea governs all cyclic processes, on Earth and in Heaven; She creates the Universe as a Harmonia of opposites. (See Opsopaus, "Lib. Oct. Mut." for universal structures in the tension of opposites.)

However, Rhea Herself exists outside of Time, and thus She governs Motionless Motion. This is because She is concerned only with cyclic change, and therefore with the numerical ratios among the rhythms of these changes; She governs their Harmonic Relations. (In modern scientific terminology, we could say that She oversees the "frequency domain" rather than the "time domain," which is the province of Hera, Her daughter.)

The ancients also connected rhythmos to arithmos (Number), but modern linguists trace arithmos to a different Indo-European root, rê(i)- (to reason, count), from which we also get such words as reason, rational, ratio, rate, and rhyme. Nevertheless, the ancient connection informs us about how Pythagoreans understand Rhea's responsibility for Number. This is natural, for the Indefinite Dyad is the principle of Plurality itself, which separates one thing from another, but also of the Matter that allows one thing to be different from another, by substantiating multiple instances of a Form.

Rhea governs the levels of Being above the Intellect (see Proclus' Seven Levels of Reality, below), which explains why a total grasp of Number is beyond our intellectual abilities. (Modern mathematics addresses only an impoverished shadow of Number.) The properties of Number fall into two classes, corresponding to the two phases of Emanation: Procession and Reversion. For Number comprises both the Power to Generate everything but also the Power to Unify everything. In particular, Number discriminates the holistic thought of Kronos into the distinct Ideas, the articulate Logos, of Zeus. But also, by Reversion, Number redirects and reunifies the Ideas toward the One.

This leads us to the important concept Noêsis, which is usually translated Intellection, but is better understood as a process of holistic intuition, especially at this level, which is prior to Time, and therefore prior to sequential thought. Rhea, as Mediator, is the intellective (or noetic) process connecting the Divine Intellect (Zeus) at the next lower level, with the object of His intellection, Pure Being (the Monad), at the next higher.

But Rhea's domain is also the level of Life, and therefore Proclus says, "Life is Intellection" (Zôê Noêsis). On the one hand, this means that Life is fundamentally identical to holistic Intuition. On the other, it means that the Ideas are themselves living Archetypes (not static concepts). The Chaldean Oracles (fr. 56) tell us,
Of Blessed Noerics Rhea is the Source and Stream;
for, first in Power, in Wombs Ineffable all things
receiving, on The All She pours this whirling brood.

(The "Noerics" are Archetypes living in the intuiting Divine Mind, and may be identified with the Gods.)

The etymological connection between Mother and Matter is well-known, but it is worthwhile to look at it more closely. The Indo-European root mâter (mother) is the origin of our word Mother, as well as the cognate Greek and Latin words (Mêtêr, Mater). The latter is the source of Matrix, which originally meant a mother of any species, and by extension the Womb or anything else in which something originates, develops, is nourished, or is contained. Matrix, in turn, is the root of Matter and Material, which referred originally to an originating, nourishing, or sustaining substance. I will have more to say about Mother, Matrix, and Matter below.

We have seen that the Mother is the Life-Giving Goddess, the source of Ever-Flowing Matter. Therefore She provides the Quantities (cf. Number) of Matter needed for the sustenance of everything in creation. She is the Goddess of the Primordial, Life-Giving, All-Sustaining Earth. Demeter's name means Earth Mother (Dê-Mêtêr, from Dê, an alternative form of Gê or Gaia). Pherekydes calls Her Khthoniê, which means "She of the Earth"; it is an epithet of Underworld as well as Earth Goddesses (rightly seen, They are hardly different, for creation is sustained from within the Body of the Earth Goddess). Indeed, Demeter and Persephone are called "The Khthoniai," and Khthôn is a name of the Earth Goddess. Her special realm is dark Tartarus, which stands opposite to shining Olympus, ruled by Kronos. Tartarus is the hidden region of dark Primordial Matter, the Foundation of Existence. The Black, Dark, Obscure Earth (Khthôn or Gaia Melaina) was proverbial in ancient Greek.

The Father and the Mother, having become Two, now must dance to Rhea's Rhythm. And this dance will bring Them together again, for the Father has Lust (Orexis) for the Mother's Body, and She desires to reproduce His Form. Through Their Conjunction, the Monad is divided by the Dyad, and Matter is unified by the One. From Them come the Creator and Creatrix of the Material World. Thus the First Generation, Kronos and Rhea, yields to the Second, Zeus and Hera. Indeed, after the Wedding, according to Pherekydes, Khthoniê is replaced by Gaia, the Goddess of the Earth as we know it, but that is our next topic.

Brahmagupta was familiar with greek philosophy and Pythagoras's influence, therefore he was aware of pythagorean notions of Phusis and Rhea. These correspond to Shunya philosophically. Brahmagupta therefore comments on this in his advices on shunya and misforunate numbers.

The Brahmasphutasiddhanta specifically sets out to correct these "wrong" or non traditional "greek" notions, as they influenced Indian traditional Astrology.

Because the notion of shunya was associated with cipher, arabic sifre, wind. the notion of emptiness and futility took the place of fullness and potential. Thus Brahmagupta is not wrong as one unwitting commentator states when Brahmagupta divides by shunya .

I have dealt with this topic elsewhere, but it naturally follows on from these discussions of pythagorean philosophy.

It is an observation, however, that the Indian scientists took to the gematria with great skill and passion and developed through this the "tractys" number system! For later followers of Pythagorean Philosophy this proved an irresistible presentation of the power of the tractys and a recommendation for the worldwide adoption of this system. Its facility for commerce was also a sign of its harmony with the workings of the universe.

It is a wonder that the pythagoreans of ancient greece did not suggest it, and indeed we know only a small amount of their inner teachings. However, history has placed the preemeinece of this system at the foot of Al Kwharzim, who being so impressed with it wrote a book on it which became the basis of Algebra. Thus we have the link between the gematria and Geometry and the gematria and Algebra, and a clear link through gematria to the realms of, shall we say higher powers. and astrology. The Astronomy dealt with the void of space where all the gods resided in popular belief, but not as simplistically as that" there were several layers to the void, some we could see, most we could not. Thus the void was the source of all that is, and in Indian philosophy that void was called shunya. In Pythagorean philosophy that void was associated with Phusis and Rhea and controlled by Hekate, and also symbolised by a circle or a wheel.

Brahmagupta, i have observed, was a fuddy duddy, but he did not believe in perpetual motion machines, he, like Pythagoras belived in the perpetual motion in the void. Where they differ is down to cultural influences, but essentially they agree on all their major points.

## Manipume and the Theory of Moment Sequents

I awoke this morning with a clear idea of how the structure of moment sequence pans out
We start with a definition of a motion sequent being a spatial distribution relative to a reference motion.
The reference motion that is chosen is a periodic rotation extension of a flow or region around a centre of rotation. Each spoliation of the region along the rotation locus is a sequent position. The rotation and any action is linked to sequentsand space not solely to space as spatial distribution is. Therefore for each position along the curve of rotation I may describe a sparial distribution. Depending on what unit I choose for the rotation extension is how fine a structure of mogul sequent I construct. I may visualise this as picking lengths along a helical spiral and slotting a spatial distribution at that length.

Although the analogy to a film frame is exact and helpful it will not be used in defining the spaciometry of the mogul sequence. One can already see that a helical curve description is preferred.

Within spaciometry curved motion will be common. The definition of a curve will be a set of centres of rotations which may or may not be contiguous or collinear but which are not coincident.. If a set of centres of rotation are coincident then the radial extension from that centre of coincidence becomes dominant and defines a fine structure of nested circular, spherical or dynamically modulated forms. Because I do not utilise points the non coincidental curves are constructed from partial curved or linear regions or from contiguous regions in either a continuous or discrete definition.

In the coincidental situation it really depends on whether the regions are linked regions of object or forms in space as to whether the fine structure created exists at all. If it does exist it has non collinear extensions which may be defined iscretely as rays of varying magnitude or continuously as discs or sectors of discs or shunyasutra forms rotating around a centre coincident with many others and contributing mass to that centre..

A relation is a spatial comparison modulo sequents- we discount the actual sequents involved in the actioon of making the comparison. Of course that makes the comparison suspect, or approximate, but we can live with that.

An action is a transdormation of space, spatial attributes that defines sequents. In this case we include all sequents involved in the action and refine the sequents down to the essential few. This may involve removing sections of sequents to make clear the action,but again this makes the action dodgy or approximate. Usually we can get away with it, but sometimes an ingrained cultural bias leads to the leaving out of a crucial set of sequents.

The analysis of the experiential continuum simply expressed is the description of spatial transformation, such as spatial attribute,or intensty or extensity or rotation based with respect to the sequents defined by the actions.

With actions defining sequents we can ssee that reciprocally sa series of equents of changes define actions, but this is a weaker relation, as subjetively a series of sequents may be judged to produce a random effect.

An example. The derivation of the square root of a general quadratic require a method of square rooting
The method remains open as there are several methods. But when the action takes place a functional deficiency arose which took 600 years to resolve. The resolution is rotating the relevant part of the geometric figure relative to shunya.The result is a figure from which a reading action can account or reference the resultant .
ths we have a convoluted action which defines sequents which result in a form that can e accessed to give the required result. to replace this with a symbol i is to mislead and confuse. To call it a number is to obfuscate. The clearest notion is a vector and the clearest exposition of that is Grassmann's.

So the action is to move in a certain direction. The motion that achieves the transformation is not specified, but it could be a pure anticlockwise π/2 rotation. Only Grassmann could isolate this extension in a vector algebra.

The rotation action defines a series of sequents, a vector action defines a set of sequents. The equovalence of these series is what needs to be established to determine the motion sequent algebra.

e = cosø + isinø

The rotating vector arises naturally from this equation, which makes perfect sense in a vector algebra, and no sense at all in a number system. Two things combine: a rotation extension and a direction vector i. This equation arises naturally from the logarithm construction of the sine proportions, modified to include the differentiable properties of the exponential function.

The rotating vector is not the same as a rotation extension, but it is a system that includes the rotation extension, and its link with the vector i is announced. In Grassmann a vector bound to a point defines a relative point. Here a rotation vector bound to a bound vector gives a bound vector with that orientation, or a point position on a curve. The rotating vector can be used to define the sequents of any action.

When using vectors or ausdehnungen one has to acknowlrde an underlying set of conventions which principally establish an agreed orientation, an agreed standard measure of space and an agreed standard measure of sequence. But one also has to agree a standard reference frame.

Because a reference frame is subjective an external reference frame has to be agreed, and this is where we can confuse one another because of the understanding of what a reference frame is and what it can be reduced down to.

A subjective or natural reference frame consists in LRUDBF, usually fixed relative to the torso and a ø,Ω radian measure relative to the head and the torso reference frames. to be clear i have a Left Right Up Down Back Front relative to my arms and chest and back, and a LRUDBF relative to my eyes ears and back of head.

The relative orientation BETWEEn the 2 reference frames i measure using the great circles in a sphere with a centre of rotation that is the region i call my head.

within that sphere i conceive of a ray of orientation that points/extends in the "direction" i am looking or have moved my head, and in this case i perceive the great circle in the horizontal plane and the great circle in the vertical plane as being in a sphere fixed relative to my torso reference frame. Effectively i have 2 spheres resting a short distance apart but in a fixed relative position, free to rotate relative to each other, one effecrively inside the orher.

When i move my head the one fixed relative to my head moves as if my head was the centre of rotation, while the larger one fixed relative to my torso remains unmoved. I then unconsciously compute the radian measures in the great circle of the larger sphere which give me the relative movement of the smaller inner sohere as Ω and ø.

Now i can compute the change in terms of the relative change in the smaller inner sphere, and which i unconsciously use is determined by he action i am involved in.

Within the smaller inner sphere my eyes play the role of a pointing ray furthter identifying the region of interest, and i compute solely within the smaller sphere radian measures based on its great circles å,ß giving the relative position of my eye orientation in that sphere.

This is the minimum system that i unconsciously use to describe my experiential continuum, and i can construct more complex systems. However, since the point is to agree a reference system the simplest should be used. We have no de facto agreement of which system we are using that reflects the natural system of reference, and so helps avoid confusion. For example, we are mostly taught to do plane feometry with two "cartesian axes fixed at right angles when in fact we should be using a minimum of 3 with any 2 fixed relative and the 3rd free to roam. This is because we exist in 3d, not 2d and any conceit that we can fully comprehend 2d is misleading if we ignore the dimensions out of the plane. We also do not need to fix the axes at right angles, and when we do not we releas the fleibility within the systems and move into the realms of vector algebra, natural dynamic geometry of vectors and generalise coordinate systems. The right angle is fundamental,but so is the generalised gnomon. What is not fundamental is the Carewsian reference frame. However the spherical trigonometry i have evoked is necessary to describe my experiential continuum and thus fundamental even if it is not sufficient. One can consider it a fusion of Cartesian and Polar coordinates.

So always sk which frames of reference are we agreeing on in any discussion about relativity, and recognise that if spherical geometry is not included then they are on shaky grounds.

http://en.wikipedia.org/wiki/Relativistic_Doppler_effect

## The vector gnomon

Vector addition is based around the trigonometric attributes around the triangle, thus the gnomon in the unit circle is very instructive.

The gnomon in the unit circle ties together the three sides of a triangle and the associated arcs. In vector terms any two vectors when added rotate and extend or diminish each other . Thus addition is effectively a rotation transformation.

The summation of vectors is defined geometrically around a form and it is an action. The action is to get from A to B. Why aggregate around a form? The behaviour of space and the motion in space respects form or deforms form. Respecting form means we have to aggregate distances and going around a perimeter does that. Deforming space means the lengths or edges have to be elastic and defining a unit vector with a scalar represents that. So noting the form produced y ana ction records all these conditions. So why highlight one dimension as the resultant when the geometric form contains the information?

When i perform the action of measuring or traveling my interest is the direct measuremnt, but i do not want to lose the other information. so the resultant vector is not always the goal.The systematic application of operations produces a record, which can reveal combinatoric information and derive formulae. Vectors begin to highlight effective organisation of the permutation and combination of actions in space.

Suppose we concentrate on the resultant, then addition traces an extension and then a combined rotation and extension. This describes well the effect of non parallel forces acting on the same object. Thus the form we trace out by drawing motion holds true for the motion of larger objects. The trace records motion under the same type of force distribution scaled down. Thus the vector trace also records model forces, and suggests applicability. The geometry in this case is secondary to the force model used to trace the geometrical figure.

Using this analysis we can define a rotation vector that represents a drawing force that rotates a centre. Again the motion can be characterised as an extension followed by a extension around the circumference of a circle. In this translation there is no further radial extension. Not shown usually in drawing the circle is the vector triangle that is in a pair of compasses, or the tension vector in a taught string anchored at the origin. or the action and rection vectors as one draws around a circular form, Every one of these vectors needs to be recorded to describe the actual situation. When that is done then the proper vector description can be noted and the proper geometrical form results which gives the desired results. We can also se how a rotational unit vector is interrelated to other vectors in the situation.

In one case vector addition is the same as just an extension, when the vectors are parallel or contiguous, but as i have said before position of a vector has also to be recorded otherwise the missing positional information can lead o a distorted result.

When i look at the analogical frame of references that encapsulate the analogies vectors are linked to i see that vectos as extensions of forms are used to model motion But we have no direct way of using motion as an analogical basis in our current systems. The tautology always starts with extensions of space or form, and we represent motion by a trace which is an extension of form called a trace line or a locus.

The only pure way to capture motion is in sequents and these seqoents have to be ordere and displayed exactly as in a film sequence or a series of frames. Motion sequents are the only analogical base that uses motion not extension, abd i will continue to explore its impact on describing reality.

The research question is using the definition of a motion sequent as an image frame in a fim reel how do i devise a notation system to describe what is happening in sequence of frames. How do i generalise this to 3d, and how do i directly relate this to electrnic storage of film sequences? Do the different forms of recording alter the nascent algebra ? Does the media editing industry already have a model i can use and generalise? What is the significance of the computing paradigm and the influence of computer science? Has computer science and programming languages already developed an algebra in flow charting? Does iteration and convolution have a role in such an algebra? Does fractal geometry< and the work of Benoit Mandelbrot?

The possible answer is out there. I can feel it in my flowing water!

## The Fundamental axiom of relative existence in a motion field.

, , , ...

The fundamental axiom of existence is that space must be spinning to be perceived.

The spinning space requires a perceiver to be denoted or languaged as "existing".There is no limit on the manner or distinctions in the way space spins as relateable or perceivable by the perceiver

Existence therefore is a shorthand way of describing the relative relationships of the spinning spaces and the perceiver of them. The perceiver of spinning space may language the perception as it or they see fit to define.

The distinction between all other motions is derived from this fundamental perception of spinning spaces as "existence".

The non existence of a spinning space or set of spinning spaces is perceived as no "spin". This is not the same as no relative spatial motion. Relative spatial motion is thus defined in terms of relationships to sets,collections, aggregations of spinning spaces. The language used to denote the plurality and distinctiveness of spinning spaces is also as the perceiver or perceivers see fit.

Properties of periodicity, regionality, permanence, intensity, centre of rotation,axial rotation, orthogonal axial rotations, orientation,direction, direction of rotation, radius of regionality, rapidity,dynamism, stability, equilibrium -dynamic or static or explosive, are all fundamental attributes of spinning space, and thus of spinning spaces, attributed by the perceiver as perceivable distinctions.

Thus general relative motion in a motion field is fundamentaly spin rotation around some centre of rotation whether local or at in-finite distances radially, and spinning spaces, having local centres are perrceivable against the "background spinning space having non local centres. It is this perception against the background which is languaged as exstence, and spinning "regions" that distinguish themselves, ie are distinguishable by the perceiver as emerging from the background are languaged as "coming into existence" and those which are diverging or immersing back ino the background spin as "dieing" or "ceasing to exist".

Thus the motion field background is in constant motion of spin and various local motions of spin emerge or condense out of it or divergr , dissipate or evaporate into it.

With this set of distinctions as axioms i may assume the position of the perceiver and begin to Rock!

Let's Rock and Roll!

This is related to Shunya , with shunya being the background rotation, and local rotations emerging in pairs with opposing spin directions, and dissipating by contact with oppoaing spin directions,but with varying rates of spin - slow dissipation, or meeting an exact opposite twin - explosive dissipation.

Also transfer of spin rotation to a slower spin rotation is a momentum effect as is the "elastic" rebound effeect. Permeability and viscosity also are describable in terms of spin momentum transfers.

December 2013
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