The Fundamantal basis of all unity are the trig Ratios and Logarithms of trig ratios.

Originally posted by author:

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

Euler decomposed a solid into faces edges and vertices.

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

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

Thus a cube aggregation would be

[center]1S+6F+12E+8v[/center]

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

This gives say

[center]1 xyz+2(xy+xz+yz)+4(x+y+z)+8(0) [/center]

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

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

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

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

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

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

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

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

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

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

[img]http://nocache-nocookies.digitalgott.com/gallery/5/410_14_01_11_9_31_48_0.png

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

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

[img]http://nocache-nocookies.digitalgott.com/gallery/5/410_14_01_11_9_31_48_1.png

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

[img]http://nocache-nocookies.digitalgott.com/gallery/5/410_14_01_11_9_31_48_2.png

Originally posted by author:

[center]1S+6F+12E+8v[/center]

If i wanted to extend this i would do it through a modulo arithmetic aggregation structure; thus :

[center]1hS+6hF+12hE+8hv+1S+6F+12E+8v[/center]

where h stands for hyper, or at least it means some scale which subsumes all to the right of it.

We use these structures all the time, and the reason is that "reality" is not based on straight lines but on rotational motion. Thus shunya is the link which takes us up or down to a higher or lower level of description through relative rotational motion. Relativistic motion creates through self assembly, self organisation structures that have vertices edges and faces, and structures with a more curvaceous form. All of these are trochoids, lately called roulettes. All these forms are related to a higher vertex around which they rotate and their forms self assemble organise into higher trochoids etc.

This higher vertex is in fact shunya, so in 3d space we have a convolution, an endless looping back on itself through shunya linking higher and higher forms of trochoidal organisation and assembly.

This is truely fractal, and at what level our world and set of experiences and laws comes into being , i cannot tell, but it implies similar universes at higher organisational levels and at lower assembly levels.

This is based on rotations around and within shunya with different phase ratios called roots of unity. The result is a dynamic magnitude of fractal vector,versor patterns , a kaleidoscope of beautiful, fascinating arrangements, some of which are extremely disruptive but only because phase ratios are not in "harmony" providing stability.

We can look at the elemental phase changes in the same way: solid to liquid to gas by changing rotational phase relationships in the relativistic rotations within the form.

Spin relationships are fundamental to physical properties.

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

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

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

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

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

Originally posted by author:

Extremely fast.

Although i have issues with time i have no issues with motion , and the motion in a reaction can be "filmed". If rhey can pu tenough frames together we can have a movie, which will show how truely fast our sensory processing meshes are!

Originally posted by author:

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

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

Originally posted by author:

[center][tex]cos2theta =cos^2 theta – sin^2 theta[/tex][/center]

so if [tex]theta =frac pi 2[/tex]

[tex]sin^2 (frac pi 2) =cos^2(frac pi 2) – cos(2*frac pi 2) [/tex]

=[tex]frac {cos(pi ) + 1} 2 – cos(pi)[/tex]
=[tex]frac {1 – cos(pi )} 2 [/tex]

Squaring is linked to doubling angle, changing phase (sin to cos in this case) changing proportioning,AND changing sign, but is it modulo?

Phase refers to direction of increase, decrease or rotation or travel. so it is a relativity measure of direction.

In this idenity the – is an aggregate gate. and providing the ratios are written in the same aggregate structure and units they may be summed.

I am trying to differentiate sign from this exposition, so i am distinguishing the ratios as unsigned scalars. I hope to show that sin and cos are adequate to the role of sign if we allow angle rotation as a control in place of sign.

Flip is a great term for a sudden change of direction. Flip and sign are linked not just by rotation but by a decomposition of direction into binary distinctions.

Binary distinctions can be eliminated only by avoiding them. However we cannot avoid them all and flip seems to be one of those. If i define a mark as a flip point i can move around that mark but never reach it again: no flip

But if i reach it again the flip signifies i havs started again, and may define direction of movement in and around the flip mark.

So i can flip by rotation, translation and reflection.

Does flip define sign?  If sign is general direction then no but if sign is either or then yes.

Flip is necessary but not sufficient. Flip needs further defining to each situation.

So flip 180 may be necessary and sufficient to define sign. but we still need the Bombelli  operatot to  know how to aggregate.

Really then sign and the Bombelli operator form an indivisible structure for constructing aggregations, therefore sign will need a similar comprehensive structure to replace it.

If i was going to use π i would have to define relations like

sinπ/2 +sin3π/2 =0
sinπ/2 – sin3π/2 = 2*sinπ/2
sin3π/2 – sinπ/2 = 2*sin3π/2

etc.
It just occurs to me that Bombelli wrote his operator rules under the influence of accountants not navigators or spherical geometrists. Thus i suspect these artisans have a version of Bombellis operators in terms of rotations

Indian and greek

I think i might be confusing myself here, between the sign function, with its aggregation algorithm and the basic distinctions i make, which are binary and form a boundary condition. This is: any way i use to define something starting or finishing and the direction of motion varying in relation to the measuring tool.

It is quite a complex entanglement to free oneself from, but flip seems to be necessary to describe it, even if the rate of flip is very slow!

Any one care to help me out here?
Tbc

Originally posted by author:

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

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

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

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

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

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

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

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

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

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

We do the flip for a stays dependent purpose.

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

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

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

Once flip has been operated certain actions are associated.

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

How does this apply to negative numbers?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Because of the loathing of negative numbers it took a while for people to tackle the issue, and by then some confusion between the sign and the magnitude had been established. Thus i popularised by Euler was a mixture of sign and magnitude. The sign here represents measurements flipped [tex]pi/2[/tex] from the initial measurement direction. But the sign also represents magnitude, so it is not a pure sign like -. Once this was realised they made – into a mixed impure sign: a -1.

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

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

Originally posted by author:

I guess the flip analysis i have explored enables me to see that algorithms are at the basis of these notations and marks we make. therefore a notation implies an algorithm is a reasonable assumption. Notation is therefore dense because it means the underlying algorithm has to be understood, or grasped. This means that those who are familiar with a field of study, have in the back of their "minds" a translation programme running that applies the algorithms  almost automatically using the sign or mark as a cue.

Imagine now how you would feel if you did not have such a translating programme in place?

This is the problem with conciseness in mathematics and the move away from rhetoric to notation. Rhetoric is bad enough, but notation has got to be even more alienating.

So how can something so bad for you be hailed as something so good for mathematics?

I have to put it down to intellectual snobbery.

Of course there are conveniences; but if it alienates all people who may have an interest if shown, that cannot be good or outweigh the harm it causes others.

These dynamic magnitudes  have taught me a lot about where our mathematics has come from.  From my exploration above there is a relation between sign and orientation. From work i have done in another thread i know that orientation and rotation are different parts of the same process: i rotate to orientate, and orientation is a resultant of rotation.

These distinctions are simple , but cause profound confusion if one is not made aware of them or where they apply.

Flip has helped me to see how all these simple foundational distinction relate and work together. They help me to relate magnitudes that were once called imaginary in a derisive tone to real life everyday ratio distinctions of magnitude and orientation.

When we dig deeper we find more simpler structures that we do not understand . When we understand these the rest becomes clearer.

My next post will be about this thread.

Any comments or critiques are very welcome here. After all it is not a blog,now is it?

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

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

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

Sorry once again and see you soon!

Originally posted by author:

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 [tex]r_theta[/tex] which is a bit long winded  but not as long as sign , and a bit more flexible than Tim Goldens  coordinate suggestion

Thus [tex]r_03[/tex]  +[tex]r_pi3[/tex]  = 0

[tex]r_03[/tex]  +[tex]r_{fracpi 2}3[/tex]  =[tex]r_{fracpi 2}3[/tex]  +[tex]r_03[/tex]

[tex]r_theta[/tex] 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.
[tex]r_{fracpi 2}[/tex]  is the distinguisher for the 4th root of unity

Thus ([tex]r_{fracpi 2}3[/tex] )4 =[tex]r_0 3[/tex] 4 = [tex]r_0 81[/tex]

and ([tex]r_{fracpi 2}1[/tex] )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?

Originally posted by author:

On further reflection i have a modification to the root of unity distinguisher.

To make it less clumsy and to link back to some earlier exploration i did in another thread i wondered about using signal instead of distinguisher?  

However that loses the fact that the sign does pick out or identify the specific root of unity so what about a root of unity identity?

To avoid confusion with r or R for radius what about [tex]rho[/tex] and [tex]Upsilon[/tex] ?

In line, perhaps, with the name identity i think it is more useful to recombine the distinguisher with the unit magnitude thus making

[tex]rho_theta[/tex] = [tex]Upsilon_theta[/tex] = unity in magnitude or

|[tex]rho_theta[/tex]| =| [tex]Upsilon_theta[/tex]| = |1|
an unsigned magnitude

with [tex]rho_theta[/tex],[tex]Upsilon_theta[/tex] being "signed" or signal magnitudes which i will name in full as root unity identies. Does that work?

Got any suggestions to improve?

Extending it to the unit sphere looks not to bad now:
|[tex]rho_{theta , phi}[/tex]| =| [tex]Upsilon_{theta , phi}[/tex]| = |1|
an unsigned magnitude, and [tex]{theta , phi}[/tex] are the radian arc measures on the surface of the unit sphere which use the great circle, circle of latitude scheme common in spherical trig, and spherical geometry.

So the root unity identies can now be assigned by the flip algorithm, and the
action ([tex]rho_{alpha , beta}[/tex][tex]rho_{theta , phi}[/tex]) can be called rotation and distinguished as it must be from multiplication.

I think the rotation rues would look like
action([tex]rho_{alpha , beta}[/tex][tex]rho_{theta , phi}[/tex]) =[tex]rho_{alpha+theta , beta+phi}[/tex]|1|2

But i am not sure yet.

Any body help out here?

So for n rotation actions i would expect a logarithmic effect on the magnitude but an arithmetic effect on the root unit identity.

This is to be expected in spherical trig, and is the basis of Napier inventing his Logos Arithmoi.

Incidentally Napier had no word for it then, hence logarithms, but as has been pointed out this would directly equate logos to geometric series and arithmoi to arithmetic series and the tables being function tables enabling a look up of related terms to aid in calculation. Logarithms truely are a wonder of the world.

So i have chosen [tex]rho[/tex] and [tex]Upsilon[/tex] to enable a relativity measure, which consists of at least two unit spheres touching each other [tex]Delta_2 [/tex], [tex]Delta_1 [/tex] where [tex]Upsilon[/tex] is the root identity in [tex]Delta_1 [/tex], and is to be considered the basis sphere and [tex]rho[/tex] is the root identity in [tex]Delta_2 [/tex], and is to be considered the relative sphere.

From these i can adopt a minimum 2 tuple as a tensor description of relativity.
([tex]rho_{theta , phi}[/tex], [tex]Upsilon_{theta , phi}[/tex])

And i will just need to specify the aggregation rules.

If this in any way produces geodesic curves and links them to 3d trochoids i will be happy!

Any one want to check this?

After all, this is not a blog. :rotfl:

Originally posted by author:

From Brahmagupta to Sophus Lie there is a fascinating and wonderfully exhilarating set of connections to the study of Shunya.
Today Lisi ,Garret Lisi is continuing in that vein of exploration, studying the fundamental unity of Shunya [1,0)!

Many Great scientists have drawn from this rich vein, but my favourites are Napier,Bombelli, Newton,Cotes, De Moivre, Hamilton and Dirac and Feynman. I guess Einstein if his wife and Ricci, and  Levi are included with him.

quote author=jehovajah link=topic=1163.msg26755#msg26755 date=1295533596]
Boy am i going to love reading this!

Shunyasutras are dynamic forms of magnitude from shunya. The arithmoi are a special group of shunyasutras with rectilinear form.

So standing in a supermarket brought home the existence of shunyasutras to me. Every stacked shelf was full of them. But they were in static equilibrium. The only ones in dynamic equilibrium were the people and trolleys, and tru to form they had direction magnitude and rotation. Then I realised that this rotation comes out of shunya with the form thus an object has motion and maintains motion because of dynamic equilibrium.

I then noticed that shunya coming put of the unit circle expand while objects moving into the unit circle contract. Depending on the rate of rotation is the size of the objects that come out and expand. The slow rotations produce bigger shunyasutras and the forms break up as they rotate faster, and the continue to expand as they move away from the source of origin.

Shunyasutras of all shapes and sizes explode out of the unit sphere and expand until their spin breaks them up into smaller pieces.

Then I thought about the yoked sunayasutras have, no mater how far they spin away from each other the yoke cannot be broken but only annihilated when they spin across each other again.

Then I thought a root 3 unit yoke cannot be annihilated by a root 2 unit yoke, but they will still interact! Thus the root unit yokes mean that bonding takes place between shunyasutras and we have all forms of attractions from gravity to nuclear forces to spin!

I then thought gravity would be a root unit 2 yoke, which in terms of bonding means opposites attract and nothing is repelled or spun away? All othe root units spin other root units in various ways only root unit 2 attracts all in some way.

I enjoyed that because it seemed as if I had a full description of the behaviours of all shunyasutras in my reality.

Although not surprising, the idea of a polynomial developed in the west quickly once Tagliaterri, Cardano Viele and Bombelli published and disseminated their works, Although the precise time when "poly" was chosen is a matter for research, certainl by Newtons time it was accepted to refer to the terms as Nomials after the bi nomial expression. In this heady time after Descartes and Bombelli, the discovery of the compound interest formula in he west and the introduction of repeated fractions from indian influenced arabic sources, with Newton's infinfite binomial series! which he invented lead to the fashion of infinity and multi. So De Moivre was able to deliver a paper to the Royal Society headed"…. infinite roots for multinomial equations".

Around this time also is when analytical methods started to undermine the older synthetic geometrical demonstrations, and the gentleman's agreement thai if it had been shown so plainly that it was obvious by inspection, then that was proof enough. Now arguments were flaring up about who was right, who was prior , and who actually had demonstrated something!

This is why besides his autism Newton was loathe to publish anything that would leave him open to argument which could also lead to duels! Galois is a famous example of a "light" unnecessarily lost in this way.

Many people questioned established "sages" and so schools of defenders sprung up. Newton's happened to be in part Cotes and De Moivre; and research shows that alliances were about more than philosophical argument, they extended to politics and religion too.

Anybody know where sequences and series come from?

Originally posted by author:

Now that i have identifed the flip algorithm, it puts periodicity into perspective, and i can now experience another measurement called "duration" or some say "time". But i experience duration as a "change in state of tension". it is associated with motion, or observation. Its a kind of "bated breath" and "expectant state" a waiting- ready to react when something happens!

It is very wearing, but it seems to be a proprioceptive response to measurement, like  balancing on a rope.

This "duration" is event dependent, and a sense of quickness or hyersensitivity to detail accompanies it. Therefore there is a sense in which the events seem to take longer to happen. or pass by very quickly. Tus duration is subjective and not suitable as a standard such as time is. This is why duration is allied to periodicity to form the full conception of "time".

Time therefore is completely dependent on the motion of objects, but without "duration"we would not sense periodicity, and know when to apply the flip algorithm.

Of course you might think we would notice when the sun rises! But that does not seem to be a time marker! It is the single most visible phase change which catalyses and coordinates many chemical change reactions, and thus acts as a kickstarter to activity, but it is the change of state , the increase in tension that signals duration. Thus we may sleep with no sense of time passing, relying on periodic  chemical reactions some reliant on the sun to wake us up to activity and duration. We use the sun and any other periodic motion to atach our sense of duration to, or to attribute a sense of duration to and so measure time by motion.

This leads to strange experiences of time, which can be explained in this way. So when you appear to be craling allong at 30 when previously travelling at 60, this is due to a hypersensitive adjustment of the sense of "duration" bought about by a habituation to travelling at speeds requiring fast reaction times!

So in my view time travel backwards is not a realistic possibility, but different rates of time are to be expected, because time is based on motion and a reference periodic motion.

Does duration extend with a slowing down of time? Only if each and every aspect of space is affected by motion, thus slowing down chemical and nuclear events. If chemical and nuclear events are slowed down by motion, it follows that every aspect of space is in relative motion, and thus space is a motion field.

the Lorentz transformation is therefore a key measurement of the relativity in a a motion field.

One unfortunate result of the Lorentz transformation, at least according to the current view of it, is the result that not only does time slow down but molecular interaction and quantum interaction. What this means is that the closed the speed gets to light speed the more fragile the bonds within any molecular conglomeration become. Eventually ceasing to be at light speed. A molecular body would thus disintegrate if it even went one zillionth of a billionth of a quintillionth faster than light speed!

Very unstable!

However there is the symmetrical consideration of the Lorentz Transform to consider. Whatever happens to observer A should by relativity happen to traveller B.

To explain what i am thinking, and it does not originate with me, i am going to explore symmetry primarily and centre of relativity == centre of gravity as a consequence.

Anybody know what symmetry is fundamentally?

Originally posted by author:

It seems clear to me that either Wallis, or Newton, or Cotes calculated the Logarithms to the base e and introduced the term Naperian logarithms to distinguish them from Briggs base 10 logarthm. Either of the three were certainly capable of computing the value, but of the 3 Wallis and Cotes were more  to naturally inclined do it, just for fun.

It is also clear that Wallis in dealing with the squaring of the area of a circle was able to introduce a more algebraic formulation of the problem based on the solving of rational roots of equations. This would inevitably mean that Wallis was probably first or among the first to link the trig functions to the unit circle, before Bernoulli or Euler or Leibniz.

This gave Newton, and subsequently De Moivre and Cotes a tremendous insight into the analytical power of Wallis, and trigonometric functions in general. Newton stopped short of what became known as the De Moivre Cotes Equation, and certainly did no more than intimate the Cotes-Euler Equation. At this time Newton was busy in other spheres developing the calculus of Fluxions.

Cotes was able also to describe the roots of unity as depending on "ratios and angles in a progressive series" based on his share insight of Newton's work with De Moivre.

There is no doubt that these 4 gentlemen, in secret, formed and defended the foundations of Modern Mathematical science.

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