Bombelli operator

[jehovajah]

Kujonai also introduced the mod(n) categorisation which means that the logarithmic additions are mod(n) clock arithmetic. This means that the actions of the unary operators on themselves are added mod(n) and thus we can think of them as acting in a kind of multiplicative way. So a unary operator acting on another unary operator is a product mod(n).
For example sign acts on the real numeral 2: sign⁰2 =+2 and sign¹2=-2
sign¹sign⁰2=sign¹⁺⁰2=sign¹2=-2
sign¹sign¹2=sign¹⁺¹2=sign⁰2=+2.

The indices to sign i have called signals. They look and act like powers mod(2). In polynomials they may also get referred to as degree. These notational references speak of the history of notation more than anything else, but it is important to be clear from the outset that these indices refer to the action of an operator on an appropriate operator not to numeral manipulation,numeral products or numeral multiplication. Therefore i retain the term signal. This means that if sign is taken as the nomial polynomials in sign will all reduce to signal 1 polynomials mod(2)  
        sign⁰2+sign¹2+ sign²2+sign³2
      =2sign⁰2+2sign¹2

signal 0 is defined as the identity signal and as we have seen the identity signal on sign means that a positive sign is symbolically attached.

Now i can use a consistent geometrical representation of a unary operator as long as i clearly define what that is. For this discussion i am going to define unary operators as acting in a plane only. I am implying that i can define them as acting in geometrical space but it is a bit of a tort to do so without establishing the definition in the plane first.

To have any geometry we need a specific orientation first. This orientation is essentially not definable by any geometrical reference frame i construct with it. So to be literal it is ground 0. Because of this every reference frame is relative to the observer, and it is only by agreement we construct a common reference frame .This orientation is the axiomatic orientation. However it helps me to see that a general property of space is orientation. i have a special capacity to fixate on a specific orientation and reference from that orientation.

Another fundamental property of geometrical space is extension . Orientation and extension are logically and practically inseperable and a sensory synaesthesia. The third fundamental of geometrical space for our discussion is rotation, and again this is inseperable from orientation and extension. However, customarily we ignore the sensory synaesthesia because we are not reference free, we live in a gravitational geometrical space and that determines our orientation fundamentally and frictional forces determine our rotation.

Nevertheless we need three fundamentals to establish a geometrical space : orientation, rotation, and extension. Of the three orientation is the ground 0 and cannot be defined,and extension can then be defined as being in a certain orientation (direction) and rotation defined as around a certain orientation (axis). I am going to use the notion of axis and axes to refer to an orientation different to and from  the axiomatic orientation, and to give axes therefore there own extension and rotation. Because of this an axis will have a direction in the plane referenced from the  axiomatic orientation.

I will define as a scale  a division iteration such as: divide into 10 parts a unit length. This iterated will produce the real numeral scale along the axes. From the axiomatic orientation i define unary operator cycle mod(1) to be a rotation about any axis that returns to the axiomatic orientation . I define the unit length as sweeping out a circle radius 1 circumference 2π and the axis of rotation as being always right to its direction which in a mod(4) unary operator scheme will be identified as π/2, and the axes under cycle mod(1) i define as sweeping out a plane, which makes a normal to a plane (the axis of rotation under cycle mod(1)) the definition of that plane. AS the mod(n) n increases i can define a scale by division iteration: divide 2π into n parts. However we normally use: divide the unit (radius) into 10 parts divide the cicumference into parts that are equal to or fractions of the unit (radius). By this more complex iteration we construct a radian measure of rotation.

It is worth noting that geometrical space has all these attributes but we only distinguish them under some operation,and the vector notion ought really to include rotation as axiomatic and within its definition, thus a vector has magnitude direction and rotation.

As you can read i was struggling to get to grips with this topic of "imaginary numbers" based on inadequate information and limiting assumptions which i was keen to shake off but i did not know how.

today i can look again from a freer vantage point.

Greek mathematics was based on and derived from  dynamic geometry.

Greek mathematics therefore represented a powerful mixture of pragmatism and analysis of empitical data. it was the mathematikos the learning and science of the greeks, and represented greek thinking about the cosmos. The greeks were lovers of philosophy of knowledge for its edifying nature,and studied and read widely in the Sophia of other cultures and languages, but their own wisdom was produced by mathematikos, scientific thinking and analysis and manipulation.

Thus mathematic
late 14c. as singular, replaced by early 17c. by mathematics, from L. mathematica (pl.), from Gk. mathematike tekhne "mathematical science," fem. sing. of mathematikos (adj.) "relating to mathematics, scientific," from mathema (gen. mathematos) "science, knowledge, mathematical knowledge," related to manthanein "to learn," from PIE base *mn-/*men-/*mon- "to think, have one's mind aroused" (cf. Gk. menthere "to care," Lith. mandras "wide-awake," O.C.S. madru "wise, sage," Goth. mundonsis "to look at," Ger. munter "awake, lively").
math.
online etymology dictionary

Etymology of the French word mathématique
the French word mathématique
derived from the Latin word mathematicus (mathematical; astrological; mathematician; astrologer)
derived from the Greek word mathematikos, μαθηματικός
derived from the Greek word mathema, μάθημα
derived from the Greek word manthanein, μανθάνω (to learn (in any way))
derived from the Proto-Indo-European root *mendh-
 myEtymology


As is established mathematiics derived from greek science based on geometry: a Geometrical science.

We are more familiar with Academic geometry than Greek science because Learning required travel and empire. Euclid was made famous by the arab empire, and his geometrical methods filtered slowly to the west by islamic influence and an education in the classics which required travel. But al kwarzim's work Al Jibr promoted indian numberr solutions and this defined algebra in the early days as Arithmetic, that is to do with "a number as reckoned up" .

What this means is only just becoming apparent to me. We would call it a scalar number, the result of a division of a proportion,a quadratic number! It is not a "number" it is a ratio a proportioning of different portions of space. Arithmoi represented proportions of geometric figures in particular scalar values proportioned against a unity.

This fine distinction was not easy to grasp and is why confusion arose over negative "numbers" from india and arithmoi from greece. The square root of negative numbers were always going to cause a problem because of the clash of these two systems or "mathesis", doctrine of how to calculate,or as it is now called algebra.

People learned in both systems had to have a classical education and had to travel to read the sources. Therefore information was spread with a great deal of invention and interpretation, and a rigid protocol of academia.

Bombelli was fortunate to be able to travel in his work and to get sight of original papers. Bombelli also wanted to popularise algebra for italian engineers. thus over many years e wrote "L'Algebra the major part of the art of arithmetic". In this book Bombelli set out his mathesis

www.math.uoc.gr/~ictm2/Proceedings/pap285.pdf


What Bombelli did was have a mad idea and this was to lead to the birth of modern Algebra and to predate Descartes.

www.math.unipa.it/~grim/Pre-mod_algebraQuad11.PDF

About the time of Bombelli Napier developed his logarithm of sines, again by dynamic geometry of proportions, movement of points and lengths. Both men dealt with what the greeks called proto arithmoi, ratios that had no reduction and were pure scalars of one. These arithmoi lead to numbers that could not be represented by ratios of integers through the application of methods of mean proportioning. These methods of mean proportioning Napier accomplished by his "bones" and skill at calculations and Bombelli by his use of neusis in geoemtrical constructions, but until Euler R.  was used to refer to this method .

Euler however was preceded by  Roger Cotes.

The use of the root sign was promoted by arabic and german influence, so it is clear that in the west the word itself clearly conveyed the fact tha an algorithmic operation was being called .

Where Bombelli made a real contribution is by linking the  R.Algorithm to the Elements of Euclid,rather than just a numerical estimation of a repeated fraction.

Bombelli therefore linked square rooting to geometrical manipulations called neusis thus showing that the Greeks had solutions to these questions by geometrical means. He therefore popularised Euclidean geometry.


What Bombelli showed was  that you could find square root  by not doing the operation of square rooting negative numbers,but rather treating them as an operation to be canceled out  when they were being squared or compared, otherwise they were allowed to just hang about!They were not part of the normal numbers they were adjugate to them yoked until needed.

So from the greeks Bombelli had a geometrical solution that gave him radicles of a certain magnitude but multiplied by these adjugate numbers, while from Cardano's input he had an algebraic analysis to guide his neusis. All he had to do was keep his nerve and notice how the sign rules worked under multiplication

Piu via piu di meno, fa piu di meno.
Meno via piu di meno, fa meno di meno.
Piu via meno di meno, fa meno di meno.
Meno via meno di meno, fa piu di meno.
Piu di meno via piu di meno, fa meno.
Piu di meno via meno di meno, fa piu .
Meno di meno via piu di meno, fa piu .
Meno di meno via meno di meno fa meno.

[jehovajah]

Piu via piu di meno, fa piu di meno.
Meno via piu di meno, fa meno di meno.
Piu via meno di meno, fa meno di meno.
Meno via meno di meno, fa piu di meno.
Piu di meno via piu di meno, fa meno.
Piu di meno via meno di meno, fa piu .
Meno di meno via piu di meno, fa piu .
Meno di meno via meno di meno fa meno.

The issue has never been about finding the roots of negative numbers, but about what was the most understandable and consistent way to find roots: numerical or geometrical? Bombelli says , use the above rules of sign and you can use both!

The rules of sign is Bombellis operator, and encapsulate the neusis of the greek geometry. Bombelli could  see the geometrical application of his sign rules, but no one was looking for a geometrical representaion in algebra until its methods clashed with the Greek method, or rather the indian method clashed with the greek.

The operators are: sign   piu/meno  ±
            and sign sqrt sign    piu/meno sqrt piu/meno ±√±

These operators were immediately generalised to ±^p√± where p is any root but in particular prime roots.

So sign is a (mod 2) logarithm arithmetic for mutual action so

sign⁰sign⁰= sign⁰⁺⁰

                                                   

sign¹sign⁰= sign¹⁺⁰

sign¹sign¹= sign¹⁺¹ = sign ⁰

So now if we make this sign (mod real2) and define sqrt as sign^1/2, pth root=sign^1/p then

Bombellis operator becomes
 

sign¹sign^1/2sign⁰

sign¹sign^1/2sign¹

sign⁰sign^1/2sign⁰

sign⁰sign^1/2sign¹

Now define the sign operator as rotating the measure anticlockwise through π then sign^1/2 is a rotation through π/2

sign^1/p is a rotation of π/p.

The measure is any system of measurement we place on a region to measure dynamic changes.It is used to give a measure to an object which we may call its magnitude,but it is just a scaled unit measure.

The measure is rotated relative to the object and  it also slips and slides, so some additional operators are implicit in Bombelli's operator, which are the affine transformations.

Numerically however we would use an iteration algorithm with a difference test and generate a continuous fraction/proportioning.

Thanks to Bombelli the rules of combining these approaches were made clear.

[jehovajah]

I have to admit being underwhelmed by Bombelli's operator.

I think this is because i have looked for it for a while and the idea is simplicity itself. A lot of myths have been laid to rest along the way, and i find hat it is a story as usual of apathy and disinterest not puzzlement and perplexity! What is there to be puzzled about? From the indians to the greeks to the early italians the introduction of negative numbers was a kind of nonsensical whimsy. Debt did not need to be recorded with negative numbers the chinese showed that, and it made sense not to call them numbers any way.Debts they were and as debts they had a utility for merchants and accountants. Even in india it was a fancy to go beyond debt calculations even if you could intellectually show an interesting result. NEgative results to solving quadratic or cubic equations were universally pooh poohed as meaningless. That was the opinion from Baghdad to Rome, Athens and beyond.

Because of this i suppose Bombelli toyed with the way forward for solving unsolvable equations and was willing to try anything. The problem was to find a way past what was everywhere being taken as a signal of "this far and no further"; " this way is fruitless and a waste of time!".

So i think Bombelli just pioneered and burst through what turned out to be not a wall but a tissue of conceits, lack of nerve, peer pressure and lack of confidence in exploring all ways forward.

Bombelli like all scientists had a mathesis, a praxis, but of course no one else had to follow it unless he became a great celebrity.
Bombelli had a passion to communicate Algebra to the engineers of his time, and that and his simple but effective solutions to polynomial equations of degree 5 and under did make him a celebrity in his local area of influence and beyond. But like my reaction his peers were underwhelmed by his operator rules, despite the advance in confidence they gave. Unfortunaely the distaste for negative numbers was too great to be overcome in a generation. It took time for scientists to find a use for negative numbers and that time takes one up to John Wallis via Descartes.

In the meantime Bombelli devoted the rest of his life to his operators, solving all sorts of equations through them.

The question of what these operators mean is therefore a later question, Descartes named them derisively imaginary,Cotes found their logarithmic nature intriguing and linked them to Napier's logs( of sines and cosines) and the logarithmic spiral, and Euler showed that the exponential function is linked to the trigonometric functions through them. He also renamed them i meaning imaginary. Gauss proved the fundamental theorem of polynomials using them, and Newton started to toy with the notion of vectors at around the time he invented the binomial power series that lead to Taylor.s theorem which Euler used in his identity.

A sea change therefore was necessary to reveal how fundamental Bombelli's operators really were.

The question what are they is a late question, not even properly considered in Bombellis time and requiring the great Algebraist Hamilton to give them a algebraic meaning as a group property of a fundamental nature.

[jehovajah]

Lets have a bit of fun,
Kasner represents pretty much a popular view among lay mathematicians of the triumphal march of mathematics into "modernity". He is a good read actually but he still manages to comment in such a way as to make one feel that we are somehow better tan our ancestors!

In doing so he creates the impression that people were mathematicians by denotation, rather than what they were scientists and engineers and artisans and prodigies. The science of the day was mostly greek, The arithmetic mostly indian, the algebra mostly arabic , so to this mix was added numerology, astrology, and religious mysticism.

No one therefore had any love of or understanding of "less than nothing" or " cut from nothing", because every one expected "something"  from the gods even in the afterlife!

So scientists faced with this incredulity really had to tread warily about advancing an idea of "numbers" with an attribute of cut out of nothing. Even the indians with their veneration of the void had a hard time selling the idea of numbers cut off from zero, which return to the void as soon as they meet, that cannot be aggregated normally.

In india a rich imagination attributed colour, flavour,sensation and scent and even pressure or weight of fortune to numbers. The Chinese naturally attributed colour to their "number" rods, sticks, bones.

As you can see attributes are reflecting sensory meshes within the human animate. It makes no sense to attribute to nothing, never mind to something cut away from nothing!

The visual mesh gives magnitude and motion and equilibrium as attributes, and a system relativity and linkage. There are many other attributes,like system balance polarity,bi polarity etc. So how come we end up with something as dry as sign or mark?

Why can't we have different flavoured numbers?, or different coloured and scented numbers?

There is an "autistic" prodigy who has a different relationship with each number up to thousands of numbers in his imagination.
 
So can i smell a number? Of course! ijust take a unit volume of some odiferous substance and bingo i have a smelly unity that i can scale!

So why did we get rid of the smell of our "numbers"?  Because we wanted to abstract and emphasise one attribute that applied to all spatial objects that were in motion?   Rather we habituated ourselves to ritualistic motions and points of view that "fit" most of the experiences we got involved in in an activity. Builders noticed lengths of movement, chemists noticed smells and textures, with those , physicists noticed mass and density and equilibrium along with the builders,and relative motions,etc. Each individual abstracts his/her own sense of "number" which is related to a dynamic motion space of motile elements, substances and sensations.

We make sense of it, but how? And When did we start to do this as individuals and cultures?

[jehovajah]

The Bombelli operator is a version of the Brahmagupta shunaya operator. The hatred of negative numbers stems from there astrological significance of bad luck. In addition greek thought about unity meant that it was nonsensical to describe an element even a debt as cut of from "nothing". This misrepresentation of Brahmagupta's shunaya is as old as it is persistent. Shunaya means the infinite potential of the void.

So Bombelli's operator is an example of the powerful and mysterious workings of the void.

[jehovajah]

 So now if we make this sign (mod real2) and define sqrt as sqr^1/2, pth root=pwr ^1/p then

Bombellis operator becomes
 

sign¹sqr^1/2sign⁰

sign¹sqr^1/2sign¹

sign⁰sqr^1/2sign⁰

sign⁰sqr^1/2sign¹

Now define the sign operator as rotating the measure anticlockwise through π then sign^1/2 is a rotation through π/2

sign^1/p is a rotation of π/p and sqr^1/2 as {extending(translation)(a,b)bisection(a+b)circle(centre: bisection)translation(a) sign^1/2extend (to intersect circle)} where a*b is the magnitude i want the square root of.

Sqr¹ is defined as {extending(translation)(a,a)bisection(2a)circle(centre: bisection)translation(a) sign^1/2extend (to intersect circle)translate this length via this length to generate square area} where a is the magnitude i want the square  of.

The set of operations described is not tight notationally but it is clear that it is an algorithm which contains translations rotations and constructions like bisections which are probably formally equivalent to reflections. Of interest is the multiplication algorithm as a motion of a measure via (in) a certain direction for a certain length(not time!).

Bombellis operator rules can now be written

sign⁰sign⁰{sqr^1/2sign¹}=sign⁰{sqr^1/2sign¹}

Piu via piu di meno, fa piu di meno.

sign¹sign⁰{sqr^1/2sign¹}=sign¹{sqr^1/2sign¹}

Meno via piu di meno, fa meno di meno.

sign⁰sign¹{sqr^1/2sign¹}=sign¹sqr^1/2sign¹

Piu via meno di meno, fa meno di meno

sign¹sign¹sqr^1/2sign¹=sign⁰{sqr^1/2sign¹}

Meno via meno di meno, fa piu di meno.
 

sign⁰{sqr^1/2sign¹}sign⁰{sqr^1/2sign¹}=sign¹

Piu di meno via piu di meno, fa meno

sign⁰{sqr^1/2sign¹}sign¹{sqr^1/2sign¹}=sign⁰

Piu di meno via meno di meno, fa piu
 

sign¹{sqr^1/2sign¹}sign⁰{sqr^1/2sign¹}=sign⁰

Meno di meno via piu di meno, fa piu

sign¹{sqr^1/2sign¹}sign¹{sqr^1/2sign¹}=sign¹

Meno di meno via meno di meno fa meno.

From which i can define  {sqr^1/2sign¹}= sign^1/2 or sign ^3/2
 

And Brahmaguptas shunaya operator can be written in more general terms

equilibrium¹transformation^1/pequilibrium⁰

equilibrium¹transformation^1/pequilibrium¹

equilibrium⁰transformation^1/pequilibrium⁰

equilibrium⁰transformation^1/pequilibrium¹

Where equilibrium^1,0 refers to the balancing attributes in the void which facilitate dissolving and resolving into and out of the void,inverse attributes or condensing and sublimating attributes into and out of the void.

Whether i have to stick to a bipolar arrangement i do not at present know but for the time being the structure is that of 2 balancing opposites, requiring only a (mod2) arithmetic.

And where  transformation^1/p is an algorithm based on iterations of affine transformations with actual measurements being taken where necessary to construct or read off the measurement for construction. Therefore it relates to a constructive spaciometry and not an abstract one, although analogies should be applicable.

Since construction depends on measurement we see that unity is fundamental to this idea of transformation, and by contrast the void is fundamental to this idea of equilibrium.

The BG shunaya operator has some more rules which i will add later.

[jehovajah]

I am in the process of continuing with this thread but am in flux between some insights and their impact on the topic. Chiefly aggregation rhythms and the algorithms associated with them, their relation to logarithms and the structure of them, plus the dynamic magnitude paradigm and its relation to the iterator- convolution structure of algorithmic transformation.

The notion of recursive definition which has to be made explicit with an iterative counter the structure denoted by : convolution. Therefore i need to signify the three terms and their relationship, which is unavoidably "recursive", or some may think tautological.

The Logos Response to shunaya in all sensory meshes and particularly the proprioceptive mesh.

A clearer exploration and meditation on the "radial expansion/ spaciometric rotation"  relation and whether this is indeed a fundamental relation which provides a tie for all measurement  of dynamic unit magnitudes, through the relationship to roots of unity.

And whether a more general transformation group theory can be delineated which contains within it a theory of algorithmic measurement that will structure the units of measurement at all scales and across all situations.

In the process of which i hope to lay the ghost of "imaginary" firmly to rest!

[jehovajah]

Following up on the flip algorithm which is an algorithm we apply when measuring using a measure of choice: condition; flip: assign orientation marker; carry out aggregation rules.

I was thinking of a more general set of orientation markers(refer to polysigns for the notion) and the idea of a radial came to mind.

The marker is which is a bit long winded  but not as long as sign , and a bit more flexible than Tim Goldens  coordinate suggestion

Thus   +  = 0

  +  =  +

are  distinguishers for  roots of unity.

When Kujonai and Tim Golden introduced the polysign idea i had no knowledge of roots of unity. So at last i can say that the polysign notion is the exploration of roots of unity and beyond.
   is the distinguisher for the 4th root of unity

Thus ( )⁴ = ⁴ =

and ( )⁴ is the same as i⁴

This distinguisher separates the magnitude from the "sign" again and helps to give a clear link to the geometrical forms underlying all this.

It also i hope gives a clear link to De Moivre's formula for the roots of unity and might generalise to the sphere( horribly i think)

So any comments or further thoughts?

I really would welcome some input here. 

I have moved it here because it is the same issue, and the other thread is more general.

The issue is that the root unit identities i define later have an action, but it is not multiplication. To define rules for this action i need to understand the De Moivre formula.

It is important not to make the same mistake i initially made in polynomial rotations which was not to realise that orientation and rotation though linked are not the same.

So the distinguisher is separate from the magnitude because the distinguisher is an orientation. Like sign the have no magnitude or action they just indicate/

I have chosen them to indicate the roots of unity.

Geometrically i have to construct the roots of unity and name them with these markers to distinguish them. So all the action takes place in the De Moivre formula and this is where i have to derive the action rules.

I have to do this circumspectly because everything else depends on it.

The firsr issue is that De Moivre already has in his equation, so i have to clarify if this is tautologically the same in which case why bother, or is it different and more fundamental?

So some help please would be appreciated. Tinab!

[jehovajah]

Here is a version of roots of unity to help us out.

Here is a set of propositions to guide the fundamamental understanding of the properties of the "complex" group of magnitudes.

Now notice how the presentation sidelines the polar version favouring the algebraic, placing algebra over geometry.

The polar form is the geometric representation chained and wrapped around with algebra so closely you struggle to see the simplicity of the geometry and the absolute trigonometry of it all!

De Moivre establishes the actions i am interested in here.

So using this material i hope some one will help sort out the root s of unity distinguishers and then look at whether combining these orientations with unit magnitude creates any procedural problems.

[jehovajah]

√-1 and the Bombelli operators are necessary and sufficient to define all other  aggregations. So these are the rules of actions of the flip from one form of unity or unit to another. What is the conditiob for the flip?

Well in bombelli's day it was a stage in the calculation of a root which had a - under the √ bracket or rather the rhetoric stated R. -number.

There does not appear to have been any other flip algorithm for about a century when De Moivre uses it in reverse to flip from the new unity to the normal integer unity. Not a new thing,just an application of the Bombelli operators in a trigonometric setting and From the new units!

So the condition for the flip was the instances in the bombelli operator specifically dealing with the new units, and flip was being called constantly in this setting.

How come he chose the unit circle trig? It seems that his concerns with probability theory and with infinite series and roots  arithmetic and geometric series. He worked on Cotes paper on the nth roots of 1, and picked up from Cotes the notions involving trig functions and the √-1. Cotes in turn picked up his intense interest in the trig functions From Napier , through his Logarithms .

De Moivre was able through this education to work comfortably in a new "unity" along with Cotes using Bombelli's operator to direct flipping from the new to the old.

I am not able to say what De Moivre was thinking, but Cotes was definitely thinking of harmonising all measures, and did not tremble to include the so called "imaginary magnitudes" in Logarithms. Cotes was an astronomer and was keenly working to map the heavens. A reference frame and a a "calculator" was of great importance to him.

At this time sequences and series seemed to become prevalent. Where did they come from? Repeated fractions? Logarithms? Root formulae?

Also at this time Leibniz was exploring his monad theory. The 16th century in the west was truly a renaissance.

There was a zeitgeist seeking a more fundamental harmony based on more fundamental unit revealed by differential calculus and conservation of energy, and momentum results in dynamic systems.

It seemed and seems inevitable that it would be found, and such rapid progress was being made towards its realisation that many felt justified in venturing the most crazy ideas, just like Bombelli. Soon science spawned an incredible technology for measuring and building mechanisms, and electronics. John Dalton updated the greek notion of the dynamic atom and the solution it appeared had been found.

But the initial explosion of invention and discovery was curtailed by calls and concerns for "mathematical" rigour, and the imaginary magnitudes were squeezed out onto a shaky limb and gradually replaced by vectors.The spherical trig of the 16th to 18th century went on the back burner until Quaternions which flared again briefly until vectors and tensors covered their potential glory.

So the way forward i think was laid out by De Moivre and Cotes, and in a sense buried in abstraction by Lambert.

I would start then from the work and method of De Moivre and Cotes who made √-1 the new fundamental unity and trigonometry and the circle the structure of this measure

When that is accepted then suddenly their are an infinite , dense collection of unities which we can accept from roots of the negative magnitudes which all are distinguishable by orientation.

[jehovajah]

Kujonai, Tim Gold, Fracmonk. Three members i now understand as exploring this area of roots of unity in the same vein as Cotes and De Moivre.

Because of this , i am now able to tackle Kujonai's system. And i do it in the way Fracmonk outlines in his paper.

Kujonai could have done this i guess, just as Fracmonk has done, and maybe he should. All i can say was i was not able before to go in their direction, but their doggedness and persistence i am going to use as a guide.

My contribution is to try to free up the notion of sign, not the mark of sign or the notation of sign but the notion of sign.

The flip algorithm gives me an algorithmic basis to understand the difference we are distinguishing. Kujonai's use of mod() was so innovative it started me n this road of enquiry, Tim Golds explorations provided me with some pretty pictures that gave me a handle on the distinctions and rules. Fracmonk showed me how to go from different roots to programme them into Quasz and get a 3d picture. and Lazarus Plath has not got back to me yet, but his programme apps show me the tangible trochoidal effects of these roots of unity when used as the controls of relative spinning magnitudes.

Yes i am pursuing a strange new world, and i guess i want some help from any one who would care to join in, or comment . After all Tinab!

[jehovajah]

One of the things i have explored recntly with quasz is orientation assignment. Terry has  some controls he calls slice in the parameter section. Some of my recent gallery items have utilised this.

Doug  points out that Hamilton did not use quaternions the way at least i have in quasz and the way Terry has in his mandlebulb built in functions. Hamilton actually used a vector form with a unit vector for the reals and a unit i vector for the x, unit j for the y and a unit k for the z ordinates of the cartesian coordinate measure. However there is another level of confusion brought in by the Euler equation where x may refer to the x axis , geometrically it refers to an arc on the surface of a unit sphere, or in the planar case just an arc of the unit circle.

This is not really confusing, it is more that i have not been taught spherical trig or spherical geometry before!

Well yes i have done navigation on a spherical  earth in terms of eastings and northings but nothing of the rich significance of this area of math to our fundamental view and measurement of the universe!

Any way, that aside , Terry has made it possible to assign the cartesian ordinates(orientations!) to different roots of unity i, j, k!

He only programmed for the usual 3 so that's cool, but if i can get this orientation figured out this may be a possible extension he can make. These will then become generalised coordinate systems.

Laz Plath i think has figured out how to do this in his apps, so i would like to talk with him about it. 

[jehovajah]

With a triplex  a + qb + pc   ←→  (a,b,c)   with a,b,c belong to z (complex)

      q^1=+q      q^2=$p      q^3=$1
      q^4=$q      q^5=#p      q^6=#1
      q^7=#q      q^8=+p      q^9=+1

and  $1=(-1+i*sqrt(3))/2     #1=(-1-i*sqrt(3))/2 

..example for the simple iteration F(n+1)=F(n)^3+k 
with F(0)=0 and k belongs to z ....

I am going to work through this equation of Kujonai and illustrate the roots of unity, the flip algorithm, and the Kujonai operator!

[jehovajah]

Kujonai and Tim Golding have been working in the new fields of yoked magnitudes "discovered" by De Moivre and Cotes.

All these guys have started with the new units as the basis of their exploration, and they all take i as a unit magnitude with a set of aggregation rules that are what i call a flip algorithm.

Without the flip algorithm you can not relate your new units to each other , their yoked units or to the unit of natural scalars.

The first flip algorithm was i think the Brahmagupta- Bombelli operator for signed "numbers" or precisely : signed scalars.
Then the Bombelli operator for the square root of signed scalars which introduced the notion of imaginary magnitudes.

Utilising these flip algorithms Cotes began to explore trigonometrically these imaginary magnitudes as a more fundamental measure of magnitude. He did so with  Logarithms, and attracted the attention of De Moivre. Both were in  sense collaborators and friends and students of Newton, who shared time with them both.

De Moivre in collaboration with Cotes came up with the De Moivre Formula, in particular to solve a problem Cotes was tackling on infinite roots of unity. Cotes died before he could complete his work, and De Moivre finished for him. On the strength of his work with Cotes on infinite roots De Moivre was elected to the Royal Society.

De Moivre had added another rule of aggregation to the Bombelli operator. In effect he has generalised the Bombelli operator and so the flip algorithm. De moivre has revolutionised the notion of sign, but he starts from an established sign called men and piu, negative and positive eventually notated - and +. This last step is unfortunate, because + and - became symbols for aggregation, and the potential for confusion was realised right at that moment of adopting this convention.

Aggregation and sign are to different algorithms now, or we could define aggregation as an activity/operation and sign as an operation modifier. However my analysis places sign and aggregation in an algorithm i call flip, that is there is a process of parsing data and assigning attribute based on conditional or contingent sequences of actions. These attributes then are utilised by an aggregation algorithm to control or direct aggregation.

Flip tests a condition and assigns an attribute according to the condition tree. Once this assignment takes place measurements are flipped accordingly, and the aggregation structures start at that moment. The flip yokes/ relates the aggregation structures
to one another.

The initial attributes were men and piu and they came from aggregation calculations to find roots. Bombrlli cut through the confusion and pointed us back to the geometry of which the calculation was a measurement process. Men an piu had straight forward geometric reference: orientation relative to the measurement instrument.

Thus i venture that the fkip algorithm assigns orientation, and that is where we have to begin.

So lets play for a bit.
The chinese had red and black rods. Colour is entirely appropriate for a solid object distinguisher. So red was annihilated by black was the aggregation rule.
But measurement involves dynamic magnitudes! a tape measure, a moveable rod, etc and measurements are not static, so we utilise a static unit, static relative to us, and allow it to plethorate in any direction that needs to be measured.The aggregation rules are then set according to the direction and applied according to the project in hand.

So our everyday use of measurement is vectorial and orientation is assigned by a flip algorithm as we go about our measuring. We even write it down on sheets in orientation terms!( length, width, depth, height!)

These orientations are the attributes assigned by the flip algorithm and the aggregation is done accordingly. Bombelli's operator assigns the orientation height, the orientation width and the orientation length etc, but it assumes you and i know that bit! Bombelli focused on the relative opposite motion in measurement, which he knew was obtained by π radian rotation from his work involving neusis. What he did not know was his rhetoric would be changed to notation in such a way that much of what he said was "lost" or ignored!

Thus Bombellis operator was assigning orientation to measurement, and he assumed students would know that. owever i o not think that Bombelli knew the full significance of his orientation assignment, but he was getting there. He it was who used the terms adjugate and conjugate etc.

Think our "numbers" could have been called north and south numbers or east and west numbers! 

This links me directly back to trigonometry, where distinguishing  the orientation of measurements was taken as a basic requirement. A requirement without which surveying, land measurement, wall construction, astronomy etc could not have been done. Tis lead to the codification of oriented measurements, by the application of ratios in a right angled triangle, the sine ratio.

The sine ratio in particular linked three oriented measurements to the orientation of the hypotenuse. This orientation was called an angle, and the angle measure and the sine ratio developed together, indivisibly. Thus he angle measurement is the orientation of the hypotenuse, and our measurements flipping from one orientation to another are distinguished by the angle measure between them.

The flip algorithm therefore could and should replace + and- by radian measure related to the sin ratio primarily but the trig ratios generally. The sin is the primary ratio historically best developed.

Thus we could have 0^c numbers and π^c numbers!

[jehovajah]

Omg it is so wonderful

De Moivre and Cotes realised that the trigonometric ratios as used in spherical trig and circular trig and Logarithms behaved in the same way as Bombelli's operator. They knew that the Greeks and the Arabic mathematicians had been using this operator for centuries, but not in the form Bombeeli and Brahma gupta put it in. They were the first to see the connection and to use it , but even they did not realise what a big deal it was, because nobody had invented the number line concept we use today to define numbers! so they were not excluded from a measure concept, and never have been!

The Number concept is what caused all the difficulty with the so called imaginary magnitudes! For cotes and DE moivre they were clearly trigonometrc magnitudes in the everyday sense and were a way of riting cown compactly trig calculations!

They never got a chance to fully explore what they had discovered!