# The Indian Mathematician…

Having no fear of astrological misfortune, and neither impeaching or besmirching any god, i gaze steadily at the void and the decompositon of it that BG has brought to my attention.

Firstly the attribution of plus and minus to a magnitude, referring to the fact that independent of our involvement, should such attributed magnitudes come into contact they would automatically subtract and leave a balance.By this BG advises us of a ceaseless activity in space that is computational.

However at present there seems to be no automatic emission or decomposition of the void into these components in what BG has formulated, at least by report.

However i do not worry, as this is a new apprehending of his insight and over "time" more will flower from it.

Now we may also look for other attributes like debt and fortune,+ and – which are of this bipolar structure and which give a "zero sum" in some sense. Dynamic equilibria as well as static equilibria are fruitful candidates.

Concentrating briefly on +1 and -1 as unities, the BG rules imply that all transformations in one apply equally to the other scalar arithmoi, the spaciometry of forms and their manipulation and transformation- in a simplified set of operation equivalent to an arithmetic.

Where we do have to nake alteration and be mindful is whre the two unities mix, and the rule arelaid out clearly there: -+=-; +-=-.

So bearing this in mind we have no trouble determining√-1.

Firstly √ has been defined as the ± of the root of the magnitude. A tautology is used in this definition which i think is simpler expressed as: the magnitude is without sign and is a scalar of unity. To this scalar we attribute sign as necessary. The rules then are rules of how we attribute sign before and after calculation.

That being said we have been defective in our definition of √. We have defined ± as plus or minus but have not defines plus AND minus +/-. As you can see i have had to switch them about when ±would more naturally mean plus and minus.

The difference this makes is simple : √-1 can now be defined consistently as plus and minus 1 (±1)

Where √+1 is now defined as +/-1(plus or minus 1).

These 2 definitions represent a dynamic equilibrium and 2 static equilibria.

Originally posted by author:

Shunya simply put means replete/swollen with every conceivable and non conceivable attribute.

So when Bhudda says the world is empty of self, he advises not to pursue self ,or that which does not exist independently. As all things are dependent, he advises to seek all things, a wider apprehension. By these advices he directs any listener to a life of active learning and appreciation of all things as an interdependent connected web.

Brahmagupta therefore invites a meditation on the source of all unities by introducing shunya into our basic conceptions of unity and their scalar arithmetic, or manipulation,or calculation.

There is one thing Brahmagupta advises that is overlooked, and it ought not to be overlooked, and that is the yoke he so deftly and lightly lays on the shoulders of these attributable unities. As light as it is yet it binds stronger than death quantiies and magnitudes in proportions of exactness, in relations of cunjugacy, adjugate companionships and bilateral even multilateral activity and actions and behaviours. Thus what we see intimated is merely the tip of a far more extensive range of attributable properties and decomposition of a very swollen void.

Without shunya we would not have the Cartesian Algebra as it is extended to today, neither affine transforms or tensor analysis,nor vectors in any recognisable form.

Originally posted by author:

Plato looked at science (Mathematikos) which was concerned with spaciometry, the measurement of dynamic space, and which hoped to apprehend the construction of the universe by the gods, if they existed.and whether they cared or not if they existed. Men(thinkers and manipulators and measurers) existed      anthropoi they believed to the Theoi (dharma, Dyaus sanskrit) that lived above mount Olympus, in the visible theos which is the ouranos, the heavens, and greek men dared to challenge the Theoi the children of Dyaus/Zeus if they could be bothered to fight! One day greek men would sit among the Theoi, or pull them down trying!

Plato looked upon all this in mathematikos (science) and contrasted it with all other forms of knowledge (epistemologia) and declared to Lovers of Sophia, Philosophers that the greatest wisdom, the gift of Sophia was in Science, and that all epistomologia (knowledges) also gifts of Sophia should get off its backside and organise itself like science. He defended that opinion for the rest of his life .

Symmetry is involved in the yoke of BrahamaGupta but along with symmetry comes that which is not symmetry, and these stand side by side as a decomposition of spaciometry. It is this decomposition that describes or is described by the fundamental group structure of the void.

As a representation of this i posit the relationship between the sphere and the spherical vortices that constitute the vortex torus form, or as found in oceanic forms the shell vortices. These shell vortices have always been valued as precious objects and beads, but of course there is a time of human exploitation using these precious items,to further imperial aim!

The relationships between spherical space and spherical vorticular space is the relationship between equilibrium, and dynamic equilibrium and runaway disequilibrium leading to boom or bust, bang or crunch!

Brahmagupta advises on symmetrical yokes, but also on non symmetrical ones: n/0 and 0/n.

What 0/0 advises beyond all voids diffuse/evaporate into 1 void( or rather a supervoid which generates voids- in other words a nested fractal)  i have yet to discover.

If I was to say that shunya meant dynamic space, what would you think? Nothingness? Hardly! You may think "wind", and indeed the Arab sifr does connote that. From this through various ethnological shenanigans we end up with zero. Confusion or what?

Now I mentioned dharma in conection with Zeus and I know this is a misconception, but it resonates so i am leaving it. I could not correctly recall the Sanskrit devah at the time.

Dharma is after all "the rules of perceiving" and that is very relevant to any science.

I have just seen a new banner logo : infinity is visible. Of course n/0 says that in a more cryptic way. So I guess 0/n means that infinity can be portioned and proportionate, where n/0 means visibility is due to disproportionate proportions,ie symmetry breaking. Finally 0/0 must mean "the possibilities are… Infinite"?

We sense, therefore we exist.
We perceive therefore our existence is distinguishable.
We distinguish sensors, therefore there is something to sense.
Because we sense and perceive and distinguish through and with sensors then an external internal sensory decomposition exists.
The decomposition of sensory data into internal and external components will be distinguished as space, therefore space exists, and has attributable properties, the most fundamental being internal and external to us.

Originally posted by author:

"  Origin and development of the concept of emptiness( wikipedia)

The theme of emptiness (śūnyatā) emerged from the Buddhist doctrines of the nonexistence of the self (Pāli: anatta, Sanskrit: anātman) and dependent arising (Pāli: paticcasamuppada, Sanskrit: pratītyasamutpāda). The Suñña Sutta,[6] part of the Pāli canon, relates that the monk Ānanda, Buddha's attendant asked,

"It is said that the world is empty, the world is empty, lord. In what respect is it said that the world is empty?" The Buddha replied, "Insofar as it is empty of a self or of anything pertaining to a self: Thus it is said, Ānanda, that the world is empty."

He goes on to explain that what is meant by "the world" is the six sense media and their objects, and elsewhere says that to theorize about something beyond this realm of experience would put one to grief."

Now suppose that the word empty is translated "swollen" or "replete".

Bhudda therefore advises one to go beyond the senses to comprehend the world, In other words be wholly empirical, and therefore dynamical, livng and experiencing each fleeting moment, from moment to moment, And any conclusions you draw from that experience must fail and turn to dust with you,for they are as dynamic and fleeting as you are. Shunya therefore has the curious attribute of being self reflexive, making sense in all its interpretations.

Brahama gupta is said to have drawn a perpetual motion machine, a wheel based design, therefore a yoke of symmetry, BG thereby advises not perpetual motion of the wheel but perpetual moion of the void, a dynamic void, for it is not a wheel but a symbol of the void he drew. `in this way BG extends his operator beyond scalar representation in arithmetic to geometrical representation in spaciometry. No doubt he applies it to the sphere.

The perpetual rotation of the yoked decompositions we are more familiar with in the sin and cosine version and the Euler Cotes formula

[tex]e^{itheta} =costheta+i sintheta [/tex]  where [tex]theta[/tex] represent the constant rotational dynamism.

This of course would not effect the perfect circle or sphere as the addition of its elements are zero!

As in the decomposition into + and minus there is a decomposition into clockwise and anticlockwise, and these are the effective constituent elements of a rotational machine. These i think would necessarily be spiral, and components would be trochoids of spirals. Both cannot exist in he same machine if it is to work, and therefore work has to be done to establish this state of affairs. Therefore we cannot get more out of a machine than we put in, unless we break the symmetry yoke in the system. In oing so the system will explode.

Newton observed, but few have fully understood that all forces we investigate  are equilibrium forces, and arise only in respect to matters of equilibrium. However there are disequilibrium states, but whether they are nested in a larger equilibrium system etc is only a guess at this momemt.

Originally posted by author:

The decomposition of the void is the natural subject of the philosophy of the I ching. For this subject chinese philosophers have devised maps or flow chart like diagrams called Taijitu.

Lai Zhida it is said introduced the taijitu into the yiology of the I ching in 15th century. In doing so he was introducing a "modern" scientific revision of the Yi Ching. 5 centuries earlier Zhou Dunyi introduced a version called  the Taijitu Shuo which applied it more generally,and particularly to human activity in all its forms.

Zhu XI particularly established this philosophical discipline.

According to the chinese Fu Xi is the originator of the Yi Ching, and we see here the spoked circle, the twisted vortex and antivortex and the cone out of which Nuwa and Fu Xi emerge, holding symbols of the heavenly powers.[img]http://upload.wikimedia.org/wikipedia/en/d/d0/Nuwa%26fuxi.jpg

The development of the Yi Ching continues through philosophers and astrologers to apply a divination method to more and more contexts.

Although dense and rhetorically complex the Yi Ching is basically simple. 64 Hexagrams have over time been established as the reference frame. This reference frame is a human measuring instrument of possiblities from the void, and as such represent a decomposition of the void into these 64 distinct groupings.

Thus the void is full of all potential but only 64 are studied by the Yi Ching.

The fundamental decomposition of the void in China  has always been accepted as yoked opposites, in this case Yin and Yang. It is my consideration that this decomposition influence BrahmaGupta, and in fact the Chinese Yi Ching was a curiosity of his in the 6th century AD around the time of chinese indian cultural exchange.

Using this fundamental yoked pair as a guide to each line of the hexagram, some activity is used to "draw" decompositions from the void, to Realise a supepositional state! However, it was thought that one realisation was not sufficient only giving one aspect of the superpositional state, so over time 6 lines were developed, and thus called the hexagram. As stated 64 hexagrams were developed as sufficient and necessary.

For each of the 6 lines in a hexagram the yin yang decomposition would have either a static or dynamic resolution! so the yoked pairs were recognised as being equilibria that are static and/or dynamic, and decomposed into anticlockwise and clockwise dynamic equilibria and original and reflected static equilibria: thus 4 states.

In terms of the Yi Ching this gives 64*6*4 =1536 different state readings and 16*6*64 exegetical interpretations for a pair of readings and 4^3*6 *64 for any triple of readings etc..

This was felt sufficiently complex for human affairs!

Zhou xi and Zhou Dunyi developed aspects of this traditon to give a sufficiently robust description of probable outcomes to Government decisons, so in that sense it was used as a kind of game theory.

The Yi ching has clear application to quantum chromo dynamics and other quantum descriptions.

In regard to Brahmagupta, i see that considering these things lead to his advice on decomposing the void, and brought from chinese influence the notion of shunya into indian philosophy in 6th century, through a long cultural contact starting in 4th century?.

Of magic squares, magic circles, the Yi Ching and the Yin Yang as decompositions of the void into yoked unities, the concept of which has given birth to Taijitu of the fundamental process of the universe.

Of how these elements have been algorithmically combined by the Yi ching and BrahmaGupta into the basis of Science from Quantun Chromodynamics to the vast reaches of the known universe expounded in Relativity theory.

TBC.

Originally posted by author:

Of cradles of civilisation and the mutual development of asian science and near eastern science and the western offshoot through greece. A cultural twinship against the aryan invasian theory and for the out of india theory

Civilisation and human presence not the same.

The pragmatism of Confucianism created a stable void in personal thought which the Buddha and Buddhism filled appropriately. The taijitu expressed this fusion of inner striving and outer order and harmony. Through the bridge of Buddhism the chinese had access to Hinduism and it s influence on Indian thought and practice. in the fusion with buddhism.

"Samkhya
Main article: Samkhya

Samkhya is the oldest of the orthodox philosophical systems in Hinduism. Samkhya is a strongly dualistic philosophy that postulates everything in reality stems from purusha (Sanskrit: पुरुष, self, atma or soul) and prakriti (matter, creative agency or energy). There are many living souls (Jeevatmas) and they possess consciousness. Prakriti consists of three dispositions known as qualities (gunas): activity (rajas), inactivity (tamas) and steadiness (sattva) which arises when the two other gunas are held in equilibrium. Because of the intertwined relationship between the soul and these dispositions, an imbalance in disposition causes the world to evolve. Liberation of the soul happens when it realizes that it is above and beyond these three dispositions. Samkhya denies the existence of God.[2] Western dualism deals with the distinction between the mind and the body,[3] whereas in Samkhya it is between the soul and matter.[4] The concept of the atma (soul) is different from the concept of the mind. Soul is absolute reality that is all-pervasive, eternal, indivisible, attributeless, pure consciousness. It is non-matter and is beyond intellect. Originally, Samkhya was not theistic, but in confluence with Yoga it developed a theistic variant.  "

The gunas correspond to motions and distinguish 3 sorts of motion: acceleration(rajas), static equilibrium(tamas), dynamic equilibrium(sattva). Though dualist the chinese notion of yin and yang (yoked pairs) is not present in indian's oldest philosophy as cited here, therefore i venture that yoked pairs is a unique to chinese philosophical idea that influenced later Indian thinking in the person of Brahmagupta.

Of chinese science/astrology and quantum chromo dynamics.

Hinduism is like saying Britishism or Americanism: it refers to a geopolitical regional entity not a faith. Buddhism again refers to a belief system based on "buddha's" rules of perception, and as such is a container of any cultural ideas of those who apply the rules of perception, the Dharma. Indian philosophical thoughts in all there varieties were thus brought into contact with chinese philosophical pragmatism or Confucianism.

Confucianism again is a carefully constructed container like a "dharma" for behaviour not perception, thus Buddhism and confucianism actually complement and attract each other.

Through this connective tube of buddhism and confucianism indian philosophies flowed into china and chinese pragmatic philosophies flowed into india. Thus Brahmagupta came to examine chinese science and astronomy and the Yi Ching. AT about the same time Lai Zhide examined indian science astronomy and astrology. Lai Zhide revised the Yi ching to put it on a more astronomical, astrological basis, that is a more scientific basis and redesigned the Taijitu accordingly. Zhou Dunyi revered the traditional taijitu in his later combination of the buddhist, confucianist taoist streams into neo confucianism.

Brama gupta in studying idea from the Yi Ching was lead to ad to the 3 motions of Shunya the notion of the yin and the yang, a balance that was yoked and could perform the 3 motions in a yoked form.

Fro this Brahmagupta derived the rules of yoked motion. These rules are and always have been a meditation and a sudhanta,

From india through the arab empire it spread throughout the world and impacted on everything. Through it the world has derived the so called "complex numbers", but which Bombelli called "adjugate" to numbers, that is yoked, and he recognised the pairs that he called "conjugate": the yoked pairs.

Through the Bombelli operator the rest is history because we end up at Quantum Chromo Dynamics through a wonderful and curious route.

However as i have pointed out there never really was a problem with the square root of -1, rather it was a problem with misconceptions of Brahmagupta's sudhanta.

For me the √-1 is simply +AND- 1.

Shunya means the void which i instinctively know is magnitude. However magnitude is also instinctively a measure but what measure?

I have to explore the concept of measuring, the activity i engage in voluntarily and involuntarily and identifiably as measuring. To do this i have to start with an axiomatic model of measuring.

Originally posted by author:

Indian concept of oneness.

The Indian concept of shunya has a deep meditative role in mathematics for indians , but the concept of 1 Eka  has a transitional role, that is it transitions from the individual to the whole as a universe,

Thus the concept of unity as a meditative  search into the meaning of the universe is entirely greek!  So the indians studied shunya, the greeks unity monos.

The earlier greek philosophers meditated on the world, the universe and came up with "unit" as the basis , The indians came up with shunya

"NUMBER VS. MAGNITUDE
In the mathematics of early Greece, there was a strong distinction between discrete and continuous measurement.
Number refers to a discrete collection of atom-like units.
Magnitude refers to something that is continuous and that can be infinitely subdivided.
Rational numbers can be expressed as decimals that repeat to infinity.

Early Greek mathematicians divided mathematics into the study of number, or multitude, and the study of geometry, or magnitude. The multitude concept presented numbers as collections of discrete units, rather like indivisible atoms. Magnitudes, on the other hand, are continuous and infinitely divisible. Because length is a magnitude, a line segment can be divided as many times as one likes. The Pythagoreans believed that magnitudes could always be measured using whole numbers, which would imply that lengths are not infinitely divisible. Other schools, such as the followers of Parmenides, known as the Eleatics, believed in the infinite divisibility of magnitudes.

Parmenides taught that true "being" is unity, static, and unchangeable. This is similar to the idea that "all is one," which implies that concepts such as multiplicity and motion are illusions. If everything is part of the same thing, then there are no "multiple" things and, consequently, no motion, which is the change in position of one thing relative to another. Pythagoreans believed in multitude and motion perhaps because these concepts are intuitive, part of collective common experience. A consequence of the Pythagorean notion of multiplicity is that magnitudes should be commensurable. To the Pythagoreans, the idea that between any two quantities in nature there exists a common unit of measure, a common denominator, may have been comforting. It perhaps suggested that the rational mind can always find a solid basis for comparison, and does not have to rely on guesswork to say definite things about reality.

It would be easy to dismiss the Eleatic view, if it were not for the arguments of one of Parmenides' most famous pupils, Zeno. As we shall see, Zeno argued against the Pythagorean notions of multiplicity and motion, using infinity to show contradictions in this view. Prior to Zeno, however, problems with the Pythagorean viewpoint arose from within their own ranks in the form of an independent thinker by the name of Hipassus of Metapontum. Hipassus showed that magnitudes are not always commensurable, an idea that upset his peers to such a degree that, as the legend goes, he was drowned for his heresy. In the next section, we shall examine the idea and consequences of incommensurable magnitudes.:"

a lexicon

mona- and mono- unit/single. page 331 and page 613. Entos is more unity by inclusion.But i have at last found the common counting namers in greek and en is "one" even if it is written α (alpha). En of course also means in, within, so the idea of wholeness being inclusive is apparent.

Before i leave this i point out that greek thought was based on units, that could be established at any scale, so the pythagorean notion of being able to define a unit that would measure any magnitude is intuitively correct. So what Hipassus and Theodorus showed is that irrespective of what unity you choose there will always be magnitudes that cannot be measured. Both i would say suffered the same alleged fate. Eudoxus covered over the crack by positing a theory of proportion, which satisfied all that proportioning and reasoning could always be done and so it did not matter as these magnitudes themselves could be used as unities. Therefore from the time of eudoxus it has always been known that arithmoi are scalars and that all proportions and ratios are essentially holding information about scale, All manipulation is manipulation of scales and rules found true for scalarswould universally apply. Thus there was no need to actually see an atom as the scaled up versions exemplify their behaviour.

Atoms accordingly would have different sizes and these sizes would form fundanental unities like the proto arithmoi. What these atomic magnitudes were was the focus of their continued research, to no avail. until the unity scalars could be arrange conveniently and themselves ino a scale of scalars, the continuous fractions in use t the time was tedious to the greek mind, however it was beloved by the indians like a rhythmical song.

Of Cantor and shunya.

Cantor in his exploration of sets came across the infinite sets, and on exploring them came face to face with shunya. It so unsettled him that he thought he had sinned against god and was going mad! after recovering he left us with the notion that there are different kinds of infinity. This is what Brahmagupta meant when he wrote n/0 the countable fractions of infinty!

Cantor through modern number concepts added the notion of uncountable infinity, but NOT different magnitudes of infinity. There is only one "magnitude" to "infinity" or shunya and that is shunya.

BG used an integer n when he expressed countable fractions of infinity. If we use a real number r then r/0 represents uncountable fractions of infinity. These scalars are intuitively (mod 0) and form a (mod 0) clock arithmetic.

OF gravity and electromagnetohydrodynamics and the nuclear forces and yoked pairs.

I found an interesting child's play centre from Holland which is just right for the budding nuclear physiscist.

Equilibrium is the source of all force vectors, and yoked pairs describe the first level of euilibrium action. I suspect there are prime ntuples of yokes!

For a while i will think about my response to shunya.

Instinctively i sense its magnitude (the Logos Response) its omnipresence (internal to external continuum) its transparency (optical attributes like opacity) its relativity(to me) and its penetrability, its enfolding enclosing lightness that does not smother but wisps of sensations tremble over me through its unrelenting touch.

I have a sensation: i have a thought.

I have an urge to measure, to experience the magnitude and to apprehend it, I have an instinct to organise my sensations: to make sense.

I have an urge to respond, to react and to act and to enact and to re-enact, to rehearse and review what i can recall of my sensations and to remember them to re-engage with them and to manipulate them as tools to remind me , to relate me and to distinguish me from the void, from shunya.

My fundamental measuring tool is me.

My Logos Response provides me with regions and boundaries and ratios and proportions. My response is to react and call out, to name each sensation by a response and to remember each response as a name and by this utterly instinctive iterative process to develop an internal language that models my experiential continuum.

And with my language tool i iteratively respond in convoutions to the void developing a mesh of networked ontological connections
Modeling as in a mirror the biological mesh of which i am slowly becoming aware, and ultimately the QCD mesh of motions of which i am gradually becoming aware.

With my language tool i measure and compare and distinguish and record through my biological mesh network and communicate within myself and without. I becomes we and me becomes us.

We come to know.

Of the myriad of proportions that we sense we pick as we may any as our unit for any particular urge that we have.
And so to measure shunya:

To measure shunya is a mystical activity, a daily meditation, an intense need that we have to experience and apprehend shunya. What tools we use become the basis of our science, our iterative convoluted exploration into knowing shunya.

What tools we use what tools we construct provide no greater knowledge of shunya, but rather provide us with a different aspect of the workings of our modeling of shunya. Shunya is only apprehendable by us as ourselves.

Shunya is the great mirror in which we see dimly reflections of our own selves.

There are many things in our world, and we learn hat if we pick the right unit we may measure them all. By this experience we learn the value of scale and the operations of scalars, and the peration of scalars is to aggregate and dis-aggregate: we must measure!

The art of measuring is to enjoy the iteration, and so our art is an expression of our iterative urge to measure.

We aggregate to measure and we number: pronunce the names of each aggregate stage in an eternal hymn; we make a count.
We aggregate in bundles to add rhythms to our chant, and our bundles multiply, and so we measure to the sum.
We aggregate in bundles of bundles and so we manipulate adding pattern and complex rhythms to our iteration,factorising our operations and so we sum, and dance and sing and make marks in the sand.

We drag a marker on some surface and this becomes another unit which we manipulate and aggregate and notice how they spin and move and clump together, and cover a surface and so we geometrise, all the while singing and dancing and drawing and building, putting unit blocks together, and learning through our science, through our art, through our play, our singing and dancing the wonders of shunya, the applicability of our model of measuring shunya, our place in shunya, and what we are relative to shunya.

The songs and rhythms of aggregation and their counterparts dis-aggregation are the songs of scalars, and they apply universally, but the patterns of aggregation, the rhythms of aggregation are endless and iterative and convoluted and beauiful to us. That one particular rhythm, the indian p-adic rhythm should be so beloved is only a stage in our exploration of the measuring of Shunya.

We have transcendental rhythms now, rhythms of π and e and yoked pair rhythms of i, and radial rhythms and the rhythms of trochoids, and the rhythms of permutations and combinations, and all of these are the rhythms of our response to our interation with shunya through unity.

We know. And now we know that we know…..

Originally posted by author:

It is hard to think of a spiral reference frame when you are schooled in cartesian and polar. Yet few of us have seen the full panoply of the polar reference frame!

However this to me is a great model of such a spiral reference frame.

I have moved deeper since those early days of searching and now look for a trochoidal reference frame. Look closer and you will see…

The yoked pairs and the trochoids that form them.

When facing shunya one faces on the inside a mirror that suddenly shatters one into the myriad of monads that one is.

En. Wahed. Eka. Eureka!

Halleluyah!

Shunya.

"I swear," said the sphere
"That my space is round!
But though i look
In every nook
This truth cannot be found!"

"I see your problem", said the cube
"And it, i think, is due
Perhaps an artefact, i say
Of a certain point of view!
Should you propose
To decompose
I think you will find its true."

You silly sphere,
Just like a dunce's hat.
To think that You could find what's round, like squareness
Just like that?

Originally posted by author:

The roots of unity in the plane sum ro zero, thwerfore the roots of unity are yoked as one would expect in interesting ways that create equilibria both static and dynamic.

One learns from meditating on the odd couple shunya and monad the structure of my model of the set notFS which I have called FS
And we propose it not as fact but as some concept more fundamental than that

That shunya emits the roots of unit measure but not as some have taught. The rectilinear and the round cannot be found in shunya except as members of a dynamic trochoidal class of rotating and rotational symmetries.
The spaciometric version of the roots of unity I will research by and by, but this we know
The yoke of roots of unity are their dynamic rotational symmetry, so that we might say the fluctuating void has rhythm, and swirling in it are rotations of a root of unity. But the roots appear and disappear through the resonance of their yoke, or rather stand as entities of a standing rotational wave.

Therefore the unity of our choice comes with it's own resonance and scalar aggregational rules, it's own combinatorial structure and it's permitted permutability. And it's ceaseless rotational frequency.

The rotation is thus quantised but as we know there is no basis to unity, thus we quantize it by our own sensors that measure, by our own tools that we use to measure.

In this way we feedback loop to ourselves and our unit limitations. The fluctuations of the void appear from our iterative convolutions in processing our signal response to shunya. The roots of unity inform us that this fluctuation is rotational and trochoidal, combining radial and rotational, indeed defining radial and rotational.

Therefore we imagine the motions of bodies in space, and space itself to arise from these rotational symmetries, like traffic jams and synchronous swimmers, to appear from the relative synchronicity of yoked roots of unity, and to move in coordinated fashion as if a wave, switching on what comes before and switching off what falls behind: a kind of animation through pixel manipulation or rather spacel or voxel manipulation.

And so our prime elemental substance lies in the relative rotational motion of regions of space even below the Planck length, these relativities being perceptible to us through our iterative and convoluted perception processing as attributes to space.

All motion and all stasis relies on these fundamental rotational symmetries which are the roots of unity dynamically manifesting to our sensors in a vast computational flux.

"This", as the saying goes.." this is how we roll!"

Originally posted by author:

[tex]e^{ i*theta} = (cos theta , sin theta )[/tex]

These are the roots of unity in the plane, but no longer the Cartesian plane, and yet still the Cartesian measure.

There is no plane, but a space and a tool to measure it with, the Cartesian measure, the Bombelli vector, the Pythagorean unit square diagonally split.

With such a tool we can set up our unit with which to measure a space that is flat and surface like.

And our units are orthogonal lengths 1,1; √2 a new surd unit at pi/4; a triangle area unit =  half a square unit. These are the units on our measure, and in addition we can do neusis with our measure, so everything is relative. Relative to our measure and contrariwise relative to the form we are measuring, and in addition relative to we who measure. This is the state of affairs as described relative to an observer outside of we who measure, with a measuring tool, a form parametric to us that we have dimensioned from shunya, a tool of units to be employed iteratively in some grand fractalian scheme of scalars.

And so to notation to record and to display, to calculate and to verify. These four necessary things succinctly put in a form that reveals and does not mystify…how is it to be done?  What praxis shall we use? What mathesis?

In the end we draw upon Descartes and bend him to our needs, in hopes that all familiar with him may follow our line of playful thought.

So now we are nearly ready. On my measuring tool I have no measure for rotation, but I will add one and call it radian and give it a unit of direction called i and a parameter of rotation from that direction called [tex]theta[/tex] and now we are done.

Originally posted by author:

Originally posted by author:

hello jehovaja, really the positive and negative signs are simply arithmetic operators, or if not they dont work with numbers, my proposition, like you said it, it s a operator generalize ¨of a way¨ the concept of signs with the help of mod operator, where, the 2 signs is a particular case (mod =2), giving de option of to work with numbers or signs depending when it will be more easy to use.

I think that using numbers at least in 3d, for obtain in a simple way some kind of 3d fractal, giving 2 options:

the last that i propose, it was to use a 4-signed arithmetic that is represented by a tetrahedral tilling in 3d ( in mod 4), and it is of the form  ..i]a + [j]b + [k]c = M, 3 coordenates because of it is 3d space. and later to work it to generate fractals.

the first that i propose, it was use a 3-signed arithmetic, that is representated by a triangular tilling in 2d, of a form ..i]a + [j]b = S ( that result to be a isomorphism if complex plane), and later to extend it to obtain a of their posible analog complex( but for 3 signs , in six real dimensions ) and later to work it with this algebra to generate fractals ( iterating)

some other questions?

regards

Rereading Kujonai i realise that he proposed a tetrahedral tiling, not just a triangular one. I think this has been done in the tetrahedral folding of the mandelbrot thread.

In any case i reread Kujonai in the light of yoked decompositions of Shunya.

Originally posted by author:

Why i not y?
Not why? but y.
Way aye, mon,
Aye!
I eye i
Sign, sigh!
O Shunya?
Hi!

more word origin
At Jeff Miller's web site on the first use of math symbols I found the following discussion on the origin of the symbol "i" for the square root of negative one:
i for the imaginary unit was first used by Leonhard Euler (1707-1783) in a memoir presented in 1777 but not published until 1794 in his "Institutes of the integral calculus."
On May 5, 1777, Euler addressed to the 'Academiae' the paper  To The 'Academy' the paper "On  Angular differentials  by a formula which nevertheless  mostly results in purely irrational  logarithms, and  integrates the circular arcs  pertaining thereto," which was published posthumously in his "Institutionum calculi integralis," second ed., vol. 4, pp. 183-194, Impensis Academiae Imperialis Scientiarum, Petropoli, 1794.

"For me at least, there is yet another way,  not obvious  to guarantee that this is [it…the correct way], except by proceeding through the imaginary [quantity] , the notational form  which I will designate as i hereafter, so that it may be ii = -1  and thus  1 / i =-i. "
According to Cajori, the next appearance of i in print is by Gauss in 1801 in the Disquisitiones Arithmeticae. Carl Boyer believes that Gauss' adoption of i made it the standard. By 1821, when Cauchy published Cours d'Analyse, the use of i was rather standard, and Cauchy defines i as "as if was a real quantity whose square is equal to -1."
Throughout his Introductio, Euler consistently writes ," denoting by i the infinitely large number of" . Nonetheless, there are a very few occasions where Euler chose i with a different meaning. Thus, chapter XXI (volume 2) of Euler's Introductio contains the first appearance of i as quantitas imaginaria:
For when  negative numbers are logarithmed, results are imaginary  ……. log (-n) will be  the quantity  imaginary, which is equal to i. The citation above is from "Introductio in analysin infinitorum," Lausannae, Apud Marcum-Michaelem Bousquet & socios, M.DCC.XLVIII (1748). Please note that, in this fascinating passage about logarithms, Euler does not introduce the symbol i such that i^2 = -1.
[This entry was contributed by Julio Gonz�lez Cabill�n.]

And translated using google translate to help.

Here we have the terms number and quantity used differently and i think after the greek. Quantity is magnitude of a geometrical form and number is a scalar of unity.

To the greeks Arithmoi were dynamic geometrical forms that scaled and transformed and moved and spread and covered as units. Thus by plethora the units measured as arithmoi, scalars. The idea of number in other cultures was less dynamic, more staid,and scalars were looked on as and for results, Quantity was the same as magnitude, but not as dynamic as greek magnitude, they did not multiply as greek magnitudes do by motion in orthogonal directions by certain lengths. it was more of a rote learning of number bonds.

Cuisenaire rods illustrate greek arithmoi.

So negative scalars in logarithms Euler found could be treated as quantities, arithmoi.but only as imaginary ones following rules, and related to log of trig functions.

Euler did not need Cartesian coordinates because he had trig ratios and triangles as his measure, and by relating to De Moivre and a unit circle he was able to tie all 3 together in a way that he says is not obvious, or guaranteed to be correct.He used differentials of De Moivre theorem to obtain infinite Taylor expansions
Cotes had done something similar through studying the log of trig functions nearly a half a century earlier.

Euler treated i tentatively as a unit quantity and therefore applied scalar math to it under the given rules of Bombelli, but as he says in his own notational form, with an innovation of division.

By cauchy it was treated as a real quantity, and by Wessel, Argand and De Moivre   it was given a richer and geometric interpretation. Gauss popularized it , and surveyors and scientists latched onto the geometric cartesian form,

The cartesian measure was recognised as a model for whatever measurement tools were required, thus a measurement tool for the adjugate form was made by Wessel and Wallis and Argand. in which y was replaced by i but associated with y

This tangled mess no one seemed able to satisfactorily sort out, and so piecemeal ad hoc rules grew up into the field as it stands today.

[tex]itheta[/tex] is a real quantity obtained by multiplying a quantity by a quantity, a magnitude by a magnitude but under a set of algorithmic rules. However mathematicians have never stopped teaching students that they are imaginary and complex and they are numbers.

It is time to stop. They are not imaginary,  and they are not numbers, but i is   complex. They are magnitudes and they are units of measurement and they exist in spaciometry as real dynamic aggregations.

Originally posted by author:

When one has looked at the struggle of scientists and artists over the ages to measure shunya it  is a blessed release to see all their work come to fruition in some elegant and useful form. It is a thrill to look back over the tortured mutterings of feverish minds and gay abandon of playful minds, to look a chancres and dancers, and singers and scientists, at astrologers and philosophers, at artisans and housewives at those in plenty and those in penury mendicant and monastic, egregious and generous,in short all and any human engaged in human survival, and see that they all have contributed to the formulation of the notion of the roots of unity as a measuring tool.

Thus it comes to this.

[tex]i * theta[/tex] is rate of rotation quantity x radian arc length quantity

And all are unit and scalars of units and yoked decompositions of shunya into roots of unity based on the unit sphere, the dynamic unit sphere,moving relative to any other spatial form.

We need one element for this tool and that is a equilibrium reference from which we can construct all other equilibrium references of relative motion. This I have called a pole as in a pole or a pole star or an pole of orientation. This is a particular radial  of equilibrium in the measure or in the form being relatively measured.

The rate of rotation quantity is the root of unity quantity and governs the rotational symmetries of the dynamic sphere; in regards of unit: is to measure dynamic equilibrium rotational states. As with all unities  it is a set of nested yoked unities, but scalar ratios do not apply, these are roots of unity and so by definition are not scalable.

They can however carry a radial scalar that scales only radially, thus providing axial extensions and axial motion and taking orientation from the root of unity.

They also carry a special scalar of sorts called a radian, and this serves to transform one root of unity into another. The radian scalar is motion scalar dividing the rate of motion into smaller unities that relate by rotational symmetry fractal scales to the standard rotational magnitude.

De Moivre  formula gives us the roots of unity in terms of i  because all roots we may choose can ve written in terms of each other, because they are roots of unity.

So defining √-1 as +And- 1 and denoting it by i and laying out clear rules for it's usage we come to the quintessential algebra of spaciometry, to the realm of Wessel and Hamilton, the realm of quaternion algebras..

Many have gone on beyond this, but as pioneers not knowing where or how they would be going, not knowing that they were dissolving into the mists of shunya.