There is a curious attribute of rotatio of a form we take in our stride, but nevertheless is very interesting. As an object rotates it defines an axis of rotation. The notion is hardly explainable without a demonstration, and is an example of a recursive definition, a convolution or a tautology!

It seems everything to do with rotation is circular,but this notion needs to be avoided: we are dealing with closed loops or near closed loops of any path, we could include spirals, but we have to recognise the dependence of their definition on radials and closed loops. The circle is only one form of closed loop, and a very special one.

So we have an axis of rotation whnever we rotate that is apprehend a closed loop, and with that notion of axis is an unbreakable link to a radius or a radial line if this line is not constant in length.

So a closed loop, an axis of rotation and a radial line are tautologically linked to any definition of rotation. We simultaneously observe and appreciate measure and compute all 3 when we perceive rotation.

This simultaneity develops an innate sense of synchronicity. This relate Chronos to rotation. Their are other forms of simultaneity but unless they loop back they go out of observable range, therefore closed loop simultaneity is best for Chronos measurement.

Chronos measurement requires a unit as does all measurement. In this case i can use a unit rotation around any closed loop. We rhen reference any closed loop rotations against this.

This of course produces scalar multiples, but leads to part rotation issues, and synchronicity issues. There are also issues with regard to rotational quickness and constancy of quickness.

We apprehend quickness innately through our fundamental sensory mesh processing cycle frequency. Thus our basic iteration cycle frequency is an inate measure of quickness of change, and therefore all motion, including rotation.

Rotation cannot be measured by smaller closed loops, it gives the same result, and rotation frequency requires a unit rotation to establish, so the only thing left to measure rotationally is part rotations.

Part rotations lead to the notion of arc, not necessarily circular, which leads to the notion of angle between radials.

The precision way to divide angles into smaller and smaller parts has taken centuries to perfect. and is based on the circular arc.

With the angle measure established we can talk about angular magnitude as defining to rotational magnitude

Originally posted by author:

Angular magnitude can e aggregated as i have discussed in the previous posts on aggregation rhythms, but because it relates to a closed loop measurement of a dynamic rotational motion the mod() arithmetic is particularly appopriate.

The Babylonians used a mod(60) clock arithmetic and a mod(60)60

^{n}aggregation structure which include n=-1,-2 for minutes and seconds of arc.Thus there whole aggregation rhythm was tied to this rotational scheme by setting unit rotation=360.

Using mod(360) as a base it was easy to switch between everyday aggregation needs for commerce and astronomical aggregation needs. These astronomical needs related to yearly cycles of the sun, stars and moon and provided a strong astrological link to everyday commerce. and the notion of lucky numbers and good days and bad days on which to venture.Today we tend to restrict our use of angle measure to a range of (-π,π] but there is no reason why we cannot establish an aggregation rhythm mod(π) and an aggregation structure mod(π)π

^{n}and use mod(2π) to factor out the rotations.If [tex]0 leq theta_n <pi[/tex] we can write [tex]theta_n pi^n[/tex] for the terms.

i have concentrated on the rotational and angular magnitude of the rotational motion, but have yet to deal with the synchronicity.

I can use a unit rotation as a frequency measure. One aspect of measuring rotation is to establish a radial as a marker an some reference external to the rotating radial and a counter to record the radial marker lining up with the external reference. This can be counted, enumerated, or experienced as a duration.

Duration is an experience which is proprioceptively induced. The normal mesh clock rhythm for processing data is attached to some simultaneously moving object by a process of reference. Certain proprioceptive responses require severs processing cycles to complete others are instantaneous. The sense of duration is therefore an artifact of the processing processes, which is associated simultaneously with some moving or changing object..

The final strange thing is the circumference of a circle. If as we do we have a radial arrangement of a disk of material and that rotates, it is hard to believe that it will stay together due to the increased circumference regions have to travel to remain in synchronicity.

Lazarus Plath's circle box controls use rotational frequency ratios to determine the shape of the trochoid. This relates precisely to roots of unity!

This is not very clear, or thought through properly, but i sense that it may be possible to define synchronicity by a lagrangian set of constraints for a rotating body made up of regions.

the constraints would take equilibrium as the fundamental notion and the regions would rotate relativistically around local centres of rotation, as well as around a centre of rotation for the whole collection or regions.

a rule of contiguity would define the maximum displacement difference between a region and its local rotational centre before which rotational constraints could be applied to determine the local relative position, and after which the rotation around the body centre determines position of the region, or rather what i mean is the position of a region is determined by the dynamic magnitude formed by the body central rotation and the local centre rotation, with a cut off where the local centre no longer affects the position of the region because the body centre rotation creates to big a displacement, At that time the region is no longer conected to the body, as this is achieved through local equilibrium conditions, and therefor the body no longer has a rotational influence on the region.

At this radius the body simply breaks up into separate regions which may clump together hrough local rotational centre effects, independent of the main body.

This is a kind of gravitational system transmitting equilibrium and synchronicity through local rotational centres, with Lagrangian constraints.

I can see it but i have not fully described it.

The question is begged: what happens when a region "breaks free" from a local rotational centre? This till leaves the bodies rotational centre as the centre of rotation for the region, but without the constraint of synchronicity. This means that a region in this state is free to rotate around the body centre, but is notsynchronized to the centres rotation, so it has an extra set of degrres of freedom, precisely the "broken" lagrangian constraints.

However why would it rotate around the centre of the body? I cannot answer that as it is a mystical, divination question which is equally answered by: "equilibrium constraints" or "god would have it so!"

Newton chose a combination of the two and called it "action at a distance" and described how it works.

The Equilibrium constraint relates again to Shunya, yoked roots of unity and rotational ratios, and therefore will be logarithmic in nature. Newton's estimate of the proportions was an inverse square "law"(presumably of god) of the radius yoke, but it is in fact a negative logarithm law of the roots of unity using radius rotation ratios of some form. Depending on the frame of reference the paths of motion of the "broken free" regions would describe trochoids .

OMG! Lazarus has organized the control boxes in the circle box so they use the same radii. The rotational ratios are varied and that defines the trochoids for each of the 2 control boxes, but only one of the trochoids is displayed on the main screen and only one of the rotational radii frequencies is shown or picked out; the common one between the circle boxes not within the circle box. However the relative motion between the two trochoids is displayed, so the main trochoid rotates against the background, but by a function of its relative position to the undisplayed one.

Then Lazarus does something amazing. When a third circle is brought into play he calculates the coordinate or rather vector position of the centres relative to each other and calculates their common trochoidal path in 3d or rather in generalised coordinate representation.

It is brilliant!

I aggregate to measure.

Thus all my aggregation, manipulation, summation and calculation is to measure.

My algorithms, mathesis , ruminations and procedural rituals are so i can measure.

All my tools, preparations and conventions, directed and agreed practices, points of view and perspectives are simply so i can measure.

What and how i measure then is not solely up to me. and carries with it a ritual a convention a history and a mathesis. It has adherents, proponents antagonists and pugilists, as well as heroes and sages, lords and serfs, in short it is a thoroughly human enterprise, all for te goal of measuring, and through that process manipulating what is going on around and among us.

It is to develop a proprioceptive basis to our understanding of our existential experience.It is to measure the mystical and mysterious and make it mundane.

But in our measuring we make one condition: our unit whatever it is must be fixed, not slippery, still not dynamic, stable not changing. And that condition, understandable as it is , fails us and dooms us to failure and approximation; or rather, more cheerfully, provies us a launching pad to iteration, and an ever changing reality of lights.

When we aggregat we aggregate visually, and that gives us spaciometric magnitude but only in part. The other spaciometric magnitudes are bi-aural aggregations, bi-olfactory aggregations , and if we would let them grow a bi-whisker aggregation.

The close circumspect circumflex-kinesthetic aggregation contributes to a grounding of all the other aggregates of intensity- magnitude. So magnitude that we measure has form and/or intensity or unending unbounded space, which we can explore and investigate forever!

Originally posted by author:

b a*(a-b)/a

2b a*((a-b)/a)^2

3b a*((a-b)/a)^3

4b a*((a-b)/a)^4

5b a*((a-b)/a)^5

………………..

nb a*((a-b)/a)^narithmoi logos

The arithmoi is a geometrical form. In this case it is a rectangular cuboid, very thin, but increasing in length. It serves to record a regular plethoration of a unit 1b

The logos is the proportion written as a ratio.

Napier simply observes that a regular plethoration is in step with an iterative proportional decrease. The logos therefore had an associated arithmoi, it was the logos's arithmoi which melded to logarithm.

He had to draw attention to the arithmoi, because he noted that adding the arithmoi mutiplied the logos.

The idea of function is a later notion, but in everyway a table is the archetypical function definition, and Napier was developing a table. becaus of this he was able to develop a table look up notation that would define a function relationship in later eras.

Look up a certain value in the table and trace its logarithm:

we write log(a*((a-b)/a)^n) gives n, and eventually over time it becomes =n[tex]log_{frac{a-b}a}((frac{a-b}a)^n) = n[/tex]

In this case a-b/a= sin(π/2-[tex]epsilon[/tex]) [tex]epsilon[/tex] being a very small radian value.

[tex]log_{sin(pi/2-epsilon)}(sin(pi/2-epsilon)^n) = n[/tex]

[tex]log_{1}(sin(pi/2-epsilon)^n) = n[/tex]

[tex]sin(pi/2-epsilon)^n[/tex] is used to work out a table of values for each n.

b a*(a+b)/a

2b a*((a+b)/a)^2

3b a*((a+b)/a)^3

4b a*((a+b)/a)^4

5b a*((a+b)/a)^5

………………..

nb a*((a+b)/a)^n

Allows us to derive logarithms for any base =(a+b)/a .Thus for base 10 a=1 b=9

[tex]log_{10}(10^n) = 9n[/tex][tex]log_{cos(0+epsilon)}(cos(0+epsilon)^n) = n[/tex]

[tex]log_{1}(cos(0+epsilon)^n) = n[/tex]

[tex]cos(0+epsilon)^n[/tex] is used to work out a table of values for each n.[tex]log_{cos(pi/2+epsilon)}(cos(pi/2+epsilon)^n) = n[/tex]

[tex]log_{0}(cos(pi/2+epsilon)^n) = n[/tex]

[tex]cos(pi/2+epsilon)^n[/tex] is used to work out a table of values for each n.

[tex]log_{sin(0-epsilon)}(sin(0-epsilon)^n) = n[/tex]

[tex]log_{0}(sin(0-epsilon)^n) = n[/tex]

[tex]sin(0-epsilon)^n[/tex] is used to work out a table of values for each n.[tex]log_{i}(i)^n) = n[/tex] would imply a-b/a=+&-1=i

and therefore b=a(1-i)For an arithmoi +&-1 means that the form is a unit square with one side +1 and an orthogonal side-1 this gives a unit area of -1 which is the yoked unit to +1.

Therefore using b= +&-1=i

gives (a-b)/a=(a-(+&-1))/a = a(1-(+&-1)/a)

If i set a=1

1-(+&-1) becomes the base

The basic algebraic rule separates different kinds of aggregated arithmoi

[tex]log_{1-i}(1-i)^n) = n(i)[/tex]

also for

(a+b)/a=(a+(+&-1))/a = a(1+(+&-1)/a)If i set a=1

1+(+&-1) becomes the base

The basic algebraic rule separates different kinds of aggregated arithmoi

[tex]log_{1+i}(1+i)^n) = n(i)[/tex]

Arithmoi allow me to understand units and scalars.

The greek arithmoi are spaciometric forms. The forms are magnitudes and to measure, a unit arithmoi is freely chosen and used to plethorate a form. The basic unit is a cuboid, and in fact the special cuboid, the cube has a foundational place in the proportioning of things, and the development of scalars.

From the cuboid are abstracted (or dimensioned): Area as surface area, and broken into directed face "vectors"; and length as perimeter and again broken into directed length "vectors".

A particular algorithm called in general rooting gives the unit length vectors of a cube form a common corner. In this case they happen to be orthogonal, but in general they are at the angles of a generalised coordinate system. The best way to visusalise roots higher than 3 are as roots of unity of a unit sphere.

Thus the solutions to the general polynomial theorem will lie on the surface of a unit sphere, and these 'imaginary" magnitudes are no longer numbers but vectors in the unit sphere vector space. Some of them may form the corners of 3 dimensional polyhedra/polytopes.

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!

Shunya.

Somehow beautiful shunya came to me as a woman, a rich indian princess,serious in purpose and bountiful. Her face is adorned with jewels and her hair braided into a tiara. Her gait is determined, and none who touch her remain, consumed into her like the finest of dusts, the most essential of oils and the most aromatic of perfumes. Behind her a wispy trail of stars in the blackness of the night sky slowly fades away into the morning light.

Look, but do not touch. Watch her go by in awe. Say her sweet name and taste it on you lips, for you will never possess her, and she shall consume you.

Shunya

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 shunya or "negative" numbers, but he certainly put them both on the map for all eternity.

Originally posted by author:

Shunya is in its "majority" not computable or attributable to a surfaced form or set of closed surfaced forms of magnitude ; that is arithmoi.

Shunya is however in its majority computable and attributable to intensity magnitudes, dynamic magnitudes for which i have not yet found a greek name, but prefer an indian name: maybe "shunyasutra" after the vedic verse that give advice on calculation.

Shunyasutra then are dynamic magnitudes of intensity with translation , rotation and reflection attributes. and yoked roots of unity. Their dynamic intensity reflect static and dynamic stabilities or Equilibria. And while it may appear that we have no trigonometry for them we in fact do through a transformation called homology. They are an interesting set of magnitudes to study and we in fact study them as fluid, electro magneto thermo gravito dynamics.

[img]http://nocache-nocookies.digitalgott.com/gallery/5/410_14_01_11_9_31_49_3.png

There should be a logos-shunaysutra giving magnitudes of intensity a logarithmic assignment to aid calculation.

There are a few new view points: for example rotation is a magnitude that is attributable to direct apprehension by counting and usually shows no plethoration beyond the whole form and the sensation o spinning. a rotating intensity would be very similar, and marked by the stability of the intensity.

Units of intensity would have to counteract one change present but not noticed in visual units, the tendency to dissipate through diffusion leading to a decrease in intensity. Plethoration would have to be distinguished from diffusion by measurement of intensity levels. Very likely plethoration is accompanied by diffusion in both intensity and visual magnitude units, which makes sense of a probabilistic description of unit position during motion.

As unlikely as it sounds we have calculated this already in our sensory mesh computations, but for each of us it is a unique measurement we come up with, and this helps define our uniqueness as individuals.

So where do sequences and series come from? Do they exist or are they tools of measurement devised in a fractal scaling tool by human animates?