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Of the continuing strangeness of imaginary magnitudes,but unnecessarily so.

So when Brahmagupta invented misfortunate numbers after meditating on the I ching he set in motion a wave or revulsion and confusion that exists till this day.

The greeks had drilled down to the core of magnitude, looking for unity, the indivisible unit of reality they called an atom.
For them, and for all cultures there was always something, never "nothing", and whatever this something was it was dynamic.

So Brahmagupta was traduced in translation! and then he was rejected through his unfortunate choice of examplars: misfortune and fortune!

Shunaya is the finest of tilths, the lightest of mists, the essence of perfume! but it was reduced to The cold blackness of misunderstanding.

Brahmagupa's operator, Bombelli's operator speak of yoked magnitudes, and the yoke that distinguishes them is the yoke of roots of unity, the yoke of rotation, the yoke of dynamic equilibrium.

The greeks , From Thales to Plato with their plethorate arithmoi, and their proportional relationships, developed the notion of scalar. The indians with their love of rhythm and vedic  sutras and verse developed the notion of compact aggregational structures.Both had a dynamic understanding of a motile reality, and responded to it dynamically.

So the concept of numbers has arisen as an amalgam of many notions but non more injurious than the conception first introduced by spherical geometrists and gradually developed in the west into the number line by Wallis and eventually Dedekind. The dynamic magnitude was being turned into a static symbol, a shadow of itself, and keeping all the bad associations without a hope of redeeming all the good dynamic qualities.

The dynamic qualities were spread abroad the landscape as one argument after another rigorously defined a number, leaving out this or that attribute. As the attributes of dynamic magnitudes  were discarded others came along collecting them into new entities called vectors, matrices, tensors and eventually a strange collection of measures were put in place of the dynamic magnitude of the indians and the Greeks.

So none treated dynamic magnitudes properly, and thus  weird notions proliferated and the weirdest was the notion of the "negative number" along with the notion of "nothing".

Brahmagupta understood from the I ching that unities emanated from shunaya in yoked pairs: the yoked pairs  swirled around like spokes in a wheel, in equilibrium, dynamic and or static. They were balanced and yoked and from Shunaya, the womb of all things, thus they were always the same as shunaya when aggregated, and they had their own rhythm.

These yoked pairs were balanced and therefore equal, but moving in a rotation and therefore constantly changing their orientation relative to "the world", but not to themselves. Brahmagupta mistakenly chose to distinguish them by fortune and misfortune, but we do not have to make the same mistake. I will distinguish them by their angular difference in radians 0 and π. or i could use π and 2π.

So now i can use 0 to distinguish all magnitudes which are yoked to π magnitudes.

If i use a boundary i can write these yoked magnitudes on the boundary with either a 0 or a π  and place them diametrically opposite. I could even use _0,π to further distinguish the yoke and let the magnitude and direction of the yoke be represented by a line. All lines cross at an intersection which i will call shunaya, the centre the origin, the balance point.

If as Brahmagupta advises all the magnitudes are unity we have 1₀---1_π

And these aggregated are shunaya, the centre from which they came and around which they spin.


We have to recognise the rhetoric of former times. Thus we did not use a confusion of symbols to discuss these things as they were not in use then. Instead i attempted to describe, distinguish and demonstrate the notions to which i draw your attention, just as ancient scientists did, and consequently their rhetoric was misunderstood and lost in translation.

The Sutras allow us to apply Brahmagupta's advice  in this way and we see that a dynamic magnitude which emanates from shunaya is balance exactly by a dynamic magnitude which emanates precisely π radians rotated from it. Thus shunaya is Equilibrium of all things, also.

Every dynamic magnitude has a radial direction and a relative rotational magnitude to its balance magnitude. In a yoked pair this relative rotational magnitude is π radians, but as we now know there are more relationships than yoked pairs emanating from shunaya as roots of unity!

The important thing is that a dynamic magnitude has a radial direction and a relative rotation associated with it, and the vector notion is thus shown to be inadequate.

This was the conception allowed by Brahmagupta's attribution of yoked pairs to shunaya. Thi is a conception that Brahmagupta placed before us, but though original it is derived from preceding notions abour space, or shunaya from other traditions and cultures.

When Brahmagupta applied it to astronomy, he added additional rotational attributes to the yoked magnitudes, namely radial axis spin in seemingly opposite clockface  rotations. He also constructed a table of sines to help with his astronomical calculations and measurements, which i think like the reek was related to right angled triangles and chords, but additionally to rotating circle radials by Brahmagupta.

Proprioception works like this: suppose I take a picture of a scene. If I take my picture, on my retinas as my basis I then proprioceptively step into the seen and attribute behaviours and properties to every object in the scene. I do this continuously through an iterative process of adjustment requiring me to compute scene changes and relations on a masive scale every moment. If I change my focus or attention, then proprioception takes that into account and provides valid scenarios for the new focus.

Sometimes the proprioceptively map does not quite fit the sensory data, and that is when I have paradoxes, illusions or misrepresentation of what is in the scene.

I may provide a perception of a scene proprioceptively which is lacking detail, but sufficient for purpose of navigation, situational awareness, echo location etc. But for detailed examination it is insufficient, and that is when I will focus on a small range of not FS a small region of space and enhance the data signal and develop a more detailed perception by iteration of the perceptual output . This new set of perceptions are organised into levels of percption so that I/ we can choose appropriately how much processing time we need to devote to recognising something.

Recognition then is perceiving at various levels of processing. Thus a perceptual process iterates through various levels adding more detail to the recognition and testinfpg the validity. Should the validity test fail recognition becomes confusion and should prompt reexamination at a slower and more detailed rate.

The rate of recognition is dependent on the numerosity of the perceptions and these are determined by the experience of the perceived, the fluidity of the processing state, and the decision/ control power the perceived has at the time. Certainly one can be more decisive after a rest, and with a clear head. So flooding the process with perceptions derived from sensory and proprioceptively meshes will not bring about quick recognition, but it may lead to the correct recognition with sifting of the data/ perceptual outputs or it may lead to new relations andn creative links.

There is a kind of homeostasis in our processing system, best expressed by occurs razor: the simplest recognition is the one we tend to use first. It may not be very good, but if it saves my life it is good enough!

Into this mesh of activity and perceptual behaviour comes the notion of method. Thus we develop individually a method of processing or managing all this computation that is unique to us. This method or collection of methods is then re iterates into the mesh and used in a self referential way to define our self identity.

Thus a particular perception behaviour distinguishes an individual, and the common perceptual behaviours distinguish groups of like minded individuals.

Thus scientists perceive in a particular iterative pattern, mathematicians in a distinct but similar pattern and so on for all the so called areas of study.

At the end of the day each one of us is a unique combination of perceptual traits that focus our thinking and behaviour in ways that are adaptive or maladaptive useful or dysfunctional, depending on the prevailing situation.

As my son is fond of remarking: if ever we get attacked by zombies he knows exactly how to save us all!


So relativity is firmly based on the proprioceptively perceptual mesh I use to apprehend a circumstance, and when I meditate on lazurus Plath's trochoids in motion I see this very clearly and intriguingly.

Limits and continuity are related through ratio, similarity and magnification, in the following way: Proprioception and other mesh sensory systems compute a spaciometry. The system attempts to compute this efficiently and quickly and uses fast transforms of any changing parameters in a scene.

This is very similar to video encoding algorithms, particularly the VP* encoders by Google/ON2 , a brilliant codec developer swallowed up inside google somewhere!

Thus frame, frame rates, keyframes etc are all appropriate and proportionally applied to the discussion of computing a proprioceptive spaciometry of notFS.

So the limits that we inately expect are iteration or computation limits. This means that the mesh expects to compute a frame within so many iterations. When this is done the surface computed is sufficent for the purpose it was computed, containing detail or artifact def pending on the intentional teleos of the whole system.

 therefore like Albert we may at first see a lion, but then on a second look see a different form all together. (In fact this is the poem i remember a snippet from from my youth! I never knew until today!)

The limit of the perceptual computation is linked to magnification by unit size. thus the unit size allows the same computational limit to compute detail for a smaller area/volume.

As the unit size is gradually shrunk as in magnification the copputation smoothly adjusts, giving the ene of continuity and contiguity. However speed of change soon dispels this illusion, For very fast change we will experience a blur or even no resolved image at all. This shows that the actual event from which the mesh is calculating a response to is changing faster than it can compute , or there is no contiguity and/or continuity at that region of sensory data, or some intense convolution exists there.

Thus at all times our/my notion of limit and continuousness is a computational artifact of my proprioceptive and other sensory meshes.

I can reasonably expect my sensory meshes through the CNS and PNS to compute fractal scenes at any magnification, or unit size as long as the signal input variation is there for them to process.  

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.

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)60ⁿ 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)πⁿ and use mod(2π) to factor out the rotations.

If we can write for the terms.

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

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!

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

arithmoi          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




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





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





is used to work out a table of values for each n.





is used to work out a table of values for each n.




is used to work out a table of values for each n.


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



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