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Arithmoi.

For me the concept of form as magnitude has freed my mind. The concept of unit arithmoi has made sense of measurement and aggregation. the concept of dynamic magnitudes, dynamic arithmoi has enabled me to respond to a changing reality as a moton field , a fields of attributed and attributable arithmoi in relative motion.

Form is dynamic, and it is magnitude and it is in relative motion in a motion field of forms. And the motion fields of forms is boundless, in that i cannot see the boundary of it , and i am part of it and influenced by it.

And i can at last acknowledge a boundless structure as a fractal structure with self similarity at all scales, and therefore boundedness is not a problem, because there are always greater and lesser boundaries which i can attribute without tangible influence on what is going on around me in my appreciable and apprehensible environment.

So my Logos Response provides me with computed spaciometries in response to a surrounding signal source called notFS, from which i compute the set FS and my basic response is a "logos" field of dynamic arithmoi in relative motion or equilibrium.

Arithmoi are attributed forms: forms that my sensory meshes have computed boundaries and surfaces and dimensions and directions and rotations and reflections for. And sadly i had been taught to ignore the computational attribution of my meshes, and encourage to think of these arithmoi as properties of reality, not attributes of an incredible computational network.

So bit by bit i have been taught to make measurement harder and harder to understand, follow or achieve. I have been blinded to the intuition that i have computed all measurements in my sensory mesh already, and displayed the computation in all sensory systems already, and experienced he answer to any any measurement solution already.


Arithmoi are the beginning of greek reductionism of magnitude, Logos the beginning of greek appreciation of proportionality, atom the beginning of greek ideas on indivisible dynamic unity, and the foundations of their science, their mathematikos, their manthema and manipulation of the forms amongst which they moved.

 The greeks had epistemology and Sophia, but from Thales they learned empiricism, observation of how things relate, and they learned it from some of the oldest and pragmatic cultures in the world at their time. And they had more to learn: from India, from Egypt, from Persia from China, but they went the way of all ascendant cultures, but their legacy remained to be mangled by the victors.

Without greek influence the west would never have recovered and gone on to surpass in technological pragmatism much older and wiser cultures, whose practical sciences far outstripped western sciences. But by pooling these cultures wisdom the west eventually regained its ethos and went on through its forms of government and patronage to establish centres of intellectual competency directly related to the scholasticism of the near and far east.

Jews and Arabs principally played the major role of transfusing these competencies into the west.

So our western culture owes much to ancient cultural influences, but the difference that made the difference was greek influence transmitted also through jewish and arabic influences.

Arithmoi are dynamic spaciometric forms of magnitude  with attributions of  surface boundaries,and dimensions of volume, surface area and edge lengths. And dynamic attributes of form orientation and translation and rotation and reflection, translational and rotational shear, which dynamic attributions effect surface orientation, edge orientation, surface and edge relative rotations and translations and shearing, relative density and compaction and relative equilbria.

Arithmoi thus are dynamic and relative and reveal on study how aggregation is structured in measurement, and how units plethorate by ratio and scale ,and magnification reveals more and more about the aggregate structure of what is around us.

It has to be said that most of what we know today was known in the earlier cultures, but its application to technology required the western industrial revolution.

One man stands out  In my mind in all the World. Napier.

What he did has lead directly to so many modern mathematical , astronomical, and scientific tools and conventions and at least two major inspirations in thought which are fundamental to the way we view the reality of our day. Both Roger Cotes and Sir William Rowan Hamilton claim inspiration from Napier's Logarithms.

Logos and Arithmoi are absolutely fundamental to Vectors, versors, and Quaternions, which ling Greek and Indian trigonometry to the very quantum chromo dynamic description of Quantum Dynamics , to Einsteinian geodesic descriptions of space time via spherical trigonometry.

And these things do not have to be hard won, we can arrive at them playfully, because we have already calculated them in our sensory meshes!

Shunaya

The finest of tilths, the essence of quintessential, the perfume of aroma and the merest hint of flavour, the thrill of the slightest touch, the mildest tingle of sensation , the whisker of a movement, he raising of the hairs on your skin.

Their are situations all around me which my sensory mesh cannot compute a boundary surface to, and yet my other sensors are computing their results to. The visual mesh computation produces no visible result to bound the magnitude in a form, so the other senses give their results as dissociated sensations, and yet not dissociated.

I have to remark that magnitude is predominantly a visual concept, and though i have referred to the other sesnors as also measring it i have pointed out that the measure intensity of magnitude or rather magnitude of intensity.

For a visual based person it is perhaps hard to accept the fundamental diference this makes to their apprehension of their reality. We all pay lip service to the blind person relying on their other senses. This is not an understanding, it is a concession , a way of saying you "see" through your other senses.

Think for a minute if you are visually based: can you smell through our eyes? Can you hear through your Eyes?

We watch a lot of videos, but we generally do not have a deep synaesthesia which enables us to smell a scene or hear a soundscape. We attempt to translate across through words and associated music, but this is an indirect fabricated experience not a direct sensory experience from or own unique sensory mesh.

Well here is a news flash. Some of us are not visually based! Some of us experience the world through the other sensory mesh taking the lead: some of us see the dead and smell the living and touch the face of God!

Generally then when a "substance" is too fine to compute a surface my sensory mesh can still produce computational attributes which are dynamic magnitudes of intensity with direction orientation and rotation and unit intensites just as before.

When i see a cloud, it is  a computed surface i attribute to it, but if i enter a cloud i may never know exactly where that surface is, but i can know the level of intensity associated with the other senses as i engage with the cloud. Who has not walked in a fog and seen it thin out around you but still look as thick in the distance, except a blind person? Who has not been in pitch blackness where you cannot see your hand in front of your face, but the intensity of the other sensory magnitudes lets you know how close it is?
Brahmagupta may not have originated shunaya or "negative" numbers, but he certainly put them both on the map for all eternity.

For the Greeks the basic arithmos i think may have been a plane figure. For me , however the basis arithmos is a solid.

Euler decomposed a solid into faces edges and vertices.

My first observation is that this is a basic polynomial aggregation.

So if i add the nomial  solid to the list i can write S for solid, F for faces, E for edges and V for vertices  .

Thus a cube aggregation would be


If i adopt the convention that V stands for Origins, E for vectors from origins and F for vector sums from those origins in 2d and s for vector sums in 3d i have a vector polynomial, or a polynomial which could be similar to a polynomial in 3 variables.

This gives say



where 0 is of course shunaya and so contains yoked roots of unity .

When we set a polynomial = to (0) shunaya we are in fact posing the question, what are the roots of unity of this solid? In a plainer explanation: we are asking which edges are connected to each other through a vertex/ corner?

The answer for a cube will depend on us identifying the edges, but what we do instead is identify the unit measure, so the answer is strictly 1and1and1 or 1&1&1.
This is  a yoked triple root from shunaya and therefore has 8 variants if we introduce Brahamaguptas yoked unities ±1 . 8 comes from the 2 choices(+ or -) and the 3 yoke giving 2³.

Why do i have a choice  of say red or black, fortune or misfortune,+ or - ? As indicated this arises from the application of these vectors to everyday life. We chose to give this initial yoke this choice significance. Thus if i choose to signify it as a rotation through π radians, i can as long as there is a mod (2) aggregation arithmetic underlying.

So i could use sin(π/2),sin(3π/2) or cos(π),cos(2π) because these are mod(4) relationships that can be made mod(2) by using P radians, but not sin(π/2),cos(3π/2) because the mod(2) relationship is out of phase. However sin(π/2),cos(π) can be used.

The relevance of these significations is that whole branches of scientific significances rely on them! They even include what some cal pseudo or irrational sciences, but which clearly come from the same root ideas!

For me, it is of significance that "imaginary numbers" have defeated their derisive jibe and shown themselves to involve and engage the imagination of human animates in ways that have liberated them from the tyranny of "correctness" , conformity and convention, and the encrustments of coercive power elites who seek to enforce what is unenforceable: their unified view of perfection. Unfortunately Pythagoras falls into this category.

If we are to learn anything from the "imaginary numbers" it is to let human animates imagination roam free.

So clock arithmetics, that is modulo arithmetics describe the ratios of rotational magnitudes, amongst other things, and reveal phase differences which we have come to denote by roots of unity. These phase differences are ratios linked through the trig functions to logarithms, as are all ratios/ scalars.

Modulo is therefore linked to logarithms by this means
And the meaning of roots of unity are found in the logarithms of trig functions. But the method of Napier links series: arithmetic to geometric; and so he encompasses the foundations of Calculus and one can easily derive calculus formulae by his methods, including Taylor, Mclaurin and Cauchy series expansions.



Cotes easily derives Euler's formula from studying Napier's Logarithms, but Napier did not use base e as is assumed by many.

Cotes may very well have derived e as the base of his logarithms decades before Euler, but what is of interest to me is that Cotes derived his formula in the context of astronomy and Newton's work on gravity. He died before he had time to explore its significance for astronomy and Newton's Laws of gravity.



I suspect he would have put it to Newton that his Laws were a first order approximation of a much more general Law of gravity.

Hypervertices are quaternions, hyper edges quints. hyperfaces hexants, hyper solids septants, Octonians are therefore super hypervertices.

Hamilton struggled to do the triples because you need yoked triples as roots of unity and you need 8 of them. Quaternions however are vertices so need no new roots and can select from the triple yokes or the yoked pairs or some combination. Hamilton knew how to do yoked pairs so he solved initially in yoked pairs with i triple. There is a solution which uses 8 triples, but i do not know what it is yet.

Quints have yoked quadruples as roots(2⁴), hexants have yoked quintuples(2⁵), and sexants have yoked hextuples(2⁶) and octonion would have a choice from all these.

Now the yoked tuples are yoked roots but they are not roots of unity which encompass roots of negative magnitudes. Thus the true root of unity is given in powers of 2.

Logarithms also have a much wider variation in format than what is usually portrayed, and are subject to modulo arithmetic constraints on the trig ratios.

OMG. so this is where Euler got his unit circle from!

Napier makes total sene now as does Cotes. Spherical trig and circular trig came first! what an advantage the Greeks and the Indians had over our method of instruction and introduction!

The more I think about it the more I realise the fundamental nature of the process I have called flip.

Flip is the sudden change of direction, orientation or evaluation as motion crosses a boundary marker.

The boundary marker has to be a measure Mark, and thus ideally a unit measure Mark. The units measured are magnitudes in the visual sensory mesh, or the bi aural bi olfactory sensory meshes for intensity magnitudes.

The flip is related to motion sense and direction of measured motion. So the process involves measuring magnitude from a measure Mark and sensing when that Mark has been reached, or a measure Mark has been reached. At that point the sensory mesh give a visible, tactile  kinaesthetic response associated with the flip. This is how I know a plop is occurring.

At that stage I can redraft the flip response count it or finish the measurement..

The flip response keys me to continue using the same orientation  or direction evaluation or indeed to "flip" to another orientation. Using this signal response I can measure motion and consider it the same in direction, or status until the flip.

Statices and directions I use are defined by me usually in binary distinctions:

Left right
Up down
Away to
Before after
Start finish
In out
On off
Here not here/there
Early late
Round up round down
Pass fail
Etc

 In terms of rotation I can set a measure to flip to  so for example flip to \pi/4  and flip back or flip another \pi/4

Flip begs the question? What do we do the flip for?

We do the flip for a stays dependent purpose.

So in computing we have choice points or if then statements and these provide a status dependent flip, or a conditional flip.

Depending on the condition or status flip controls subsequent activity or actions.

So now I have generalised the notion of flip and the flip marker. The flip marker may now in fact not be a Mark at all. It may just be a condition or status, used as a a test when to operate flip.

Once flip has been operated certain actions are associated.

Drilling back down to measuring I may operate a flip in direction of measurement or a flip to the next numeral count when a certain condition or Mark is met.

How does this apply to negative numbers?

Brahmagupta set out a condition for a magnitude to be yoked to it's paired unit. The condition is that they both come out of shunaya and when they meet they return to shunaya.

So the flip is shunaya is cut and has a magnitude removed leaving a gaping hole of the magnitude that has been removed. So the yoked pair is the magnitude and the hole. The magnitude and the hole can exist but not independently. The flip is proximity of motion. If they collide the flip is activated and the action is to annihilate them, that is return them both to shunaya.

So let's do this in Chinese. The Chinese had a red rod and a black rod and an account pot. If you paid money into your account  a black rod went in. If you took money put of your pot a red rod went in . There'd rod was the flip, the action that followed was to take out equal red and black rod magnitudes. What remained was your account balance. This was a double entry bookkeeping system, if you like.

So now what if you were left with red rods in your account pot? Then you owed the bank money!

I think the Chinese used black as money paid out and red as money paid in.

So the flip condition was red and black together meant action needed to follow.

So finally to our system of negative scalars. The Italians used men and piu as part of the numeral attribute when accounting. The only thing they needed for this system was the actions to follow on the flip.

Brahmagupta had delineated these actions and Bombelli had reiterated them. The flip was aggregation of the monies credited and debited.the action was to annihilate equal magnitudes of men numerals with piu numerals.

-5*-5, -5*5 are presented as an oddity, which they are when written in notation, but in dealings they are clear if I give away  five debts to someone else I am better off by that magnitude.it is the same as someone giving me that magnitude, that is piu that magnitude. Similarly giving away five credits leaves me worse off and is the same as accepting five debts, that is men that magnitude.

Let us not lose sight of the flip that occurs due to the condition debts and credits accruing and the action of accounting that follows the flip.in this case it is a delayed flip as the money going out is the flip to account,as is the money going in. We can action the flip immediately nowadays by computers, but in Bombelli's day the nearest thing to immediate reaction to the flip was double entry book keeping.

So the flip is dependent on the condition and action is dependent on the flip. This is a conditional algorithm, requiring a test to initiate. We can do this in cybernetic systems nowadays, but the significance here is that sign is a part of a conditional algorithm. Our fault has been to isolate it from it's conditional nature and to make out that it is a thing entire, extant in the world we call reality.

Sign is part of a conditional algorithm and it is the first action of the flip: designate the sign.what then is the preceding condition or status?

I have identified several statuses above, each of which may be a preceding condition.

Concentrating on measurement, it being a vector, a dynamic magnitude I may want to measur motion in one direction as opposed to another. The condition is then the direction, and the flip occurs when I have to measure in a different direction. The flip enjoins me in an action: Mark these measurements differently, and aggregate them differently according to the rules of combination.

Several sloppy conventions have to be pointed out. Applying - to numerals permanently isolated the conditional nature of the sign. Secondly what worked for accounting did not work for geometry without additional conditions. These conditions were understood by artisans, who really had no need for them, but not by mathematicians that well until Wallis, Newton's tutor who adumbrated the number line concept. Geometrically sign had to apply only to measurements rotated  by radians to one another.

This made the condition a rotation of π radians measured from a rotational axis. This caused a flip, a change in notation and a change in final accounting.

In geometry it is natural to ask what do we do with measurements that are rotated by less than π? There was no "sign" given for that flip.

Although Bombelli did not know how to explain it as such he knew enough to realise he could extend the actions that were dependent on a flip to cover the case when finding radicles or roots of a geometric figure. Roots of a geometric figure are presented to us a mysterious things that drop out of equations.but in fact they are orthogonal sides of a geometric figure, not mysterious at all , and the equation merely gives the magnitude, not the root!

Thus the root for most geometric figures were known to exist by inspection, it was simply how to calculate the magnitude. The signed magnitude carried information about the measurement direction, and where the flip occured. What Cardano could not stomach was rooting negative numbers, he had no measure to make sense of it, bur Bombelli did. He had a carpenters square and this made perfect sense of rooting negative magnitudes by, in effect flipping it about using neusis. Bombelli even created a notation for it : piu di mene and men di mene. This notation was in fact a "sign" for dependent action the condition was rooting a negative magnitude the flip was to put this "sign" on it and the action was to use the bombelli operator to do the accounting or aggregation.

Because of the loathing of negative numbers it took a while for people to tackle the issue, and by then some confusion between the sign and the magnitude had been established. Thus i popularised by Euler was a mixture of sign and magnitude. The sign here represents measurements flipped from the initial measurement direction. But the sign also represents magnitude, so it is not a pure sign like -. Once this was realised they made - into a mixed impure sign: a -1.

Several things can now be done: redefine sign in terms of a magnitude and a distinguisher. Restore the conditional flip. Define the dependent conditions and actions for each instance of sign. Generalise the notion of sign to cover every condition and specify the general flip and the general dependent actions.

This then makes i a sign and not an imaginary number and opens up the door for roots of unity.
i is the 4th root of unity and a dynamic magnitude having magnitude orientation, and rotation.

Yes. i changed the titling to make it reflect more what it is, and to invite more ordinary fractalers to comment, contribute an collaborate.

Yes i do recognise this thread looks like a blog!  So i can only apologise profusely and let you know it is not a blog! When i first started it in 2008 the phrase working document was current and it meant everyone could contribute and tear out pages by consensus if everyone thought it was needed.

It is and always has been a thread for those looking at the deeper foundation to fractals in maths, but i do admit that my entries have been a kind of lonely man's journal!

Sorry once again and see you soon!