Shunya is Everything!

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Trapaaing has worked hard to communicate the philosophy of Brahmagupta  with regard to his teaching on fortune and misfortune in our daily lives.

The Sanskrit concept Shunya is a very ancient philosophical concept of the Brahmans or Vedic cultures. It means everything in its entirety and the symbol is O or the circle. 

This circle is representative of the infinite sphere of everything. 

Because Brahmagupta was an astrologer his words carried significance, when he mentioned misfortune, his listeners recoiled in horror and consternation. That same fear and dread haunts us when we are first introduced to so called negative numbers.

Hegel and his school considered this philosophy very deeply . While his exposition is cast into Germanic western form it is pure Vedic and upanishad conception. His deconstruction of the human condition into an angst that denies half itself is very cogent. 

Not all are able to embrace the knowledge of good and evil, the knowledge of fortune and misfortune, and yet it is placed before us in all the ancient wisdoms and philosophies. 

The solution is placed before us also as the life or active being and becoming, never standing still, always surviving.

In the edenic myths the clear advice is to choose life first! Psychologically this makes the individual a better adjusted entity that can cope with the knowledge of good and bad, fortune and misfortune.

As a constructed identity you must unravel and reconstruct many times. To do so you must choose life! The plain and simple choice is between life and not life. Choosing not life does not condemn you to death , but to many shades of grey and misery , before the eventual blackness of death!

You shall not die, but instead you will live  a half life, a fraction of your potential because you did not choose to fully live in Shunya!

Shunya being everything naturally included this choice alongside the fortune and misfortune of life. That it is yin and Yang reflects the traditional Chinese philosophical explanation of yin and yang: what is fortune in one situation may turn out to be misfortune in another, and further that misfortune May indeed be fortune at a later event! And so it goes around?

Thus we come to see that the misfortune of misfortune may indeed be fortune!

The clearest statement of this symbolically using – to represent not only misfortune but the turn of events through 180 degrees to its opposite; not only debt but the rotation of fortune into debt is this 
N. – N = 0 that is Shunya

Thus N +{–N} =0

Thus N  = –{–N}

If we identify –n as distinct misfortune  and N as fortune  we see that the swings and roundabouts of life can add and subtract fortune or add and subtract misfortune. In subtracting misfortune we experience that as fortune.

This fundamental notion or equation underpins our analysis of being alive and interacting with space

Bombelli was able to adduce that the quarter turn of events is also a significant part of the equation. Life inShunya is not just about fortune and misfortune! In fact it is all about constantly and continuously rotating change!

Shunya is every thing!

On the notion of Number 2

Copied from my thread on the Ausdehnungslehre at Fractalforums.com just google “jehovajah whatever”

Commentary

While I cannot quite frame it yet I have the notion of the logarithm of rotations. By this I mean that rotations may be described as exponents of the exponential function, and in fact must therefore be complex or quaternionic logarithms. Whether these can be extended to ndimensional Tensors or reference frames I do not know.

The structure of Grassmanns algebras allows for sums and products of these exponential forms as well as quotients. While Grassmann was at a loss until he researched Hamiltons Qaternions( only then realising he had solved his “looseness” problem for swivelling arms in 3d without realising) he later set a task for himself to do the incumbent processing to continue his planned development. However he died before he could make much more headway

The tantalising snippet from his proposed “Schwenkunglehre” whets the appetite and the imagination for more detail. The logarithm of rotations is precisely his idea of the log of a quotient. There he considers it to be the point of intersection of 2 lines forming the angle of swing. My notion is more related to Napier’s logarithms using the sines.

By the way, it seems clear that this name is misleading. These proportions are found in the sine tables, but they are true geometric terms, the angle is insignificant to the calculation in fact  the geometric mean between sin90 and sin30 is sin 45 not sin60.

The logarithms do not index angles uniformly, nor indeed the tangent line. The concordance however is fairly accurate up to tan45, by then over 4 million calculations of the proportion had been done. To get to tan60 so as to calculate sin30 2 million more calculations needed to be done.

It is not necessary to perform the calculations with these constraints as Brigg showed, but in doing so Napier reveals how logarithms can be shaped to any scale or form, as they are only indices not measures. Setting them out in a measured way allows a calculation process to trace out a form by the logarithms.

As an example, placing the logarithms uniformly on a circle drives a rotation around the circle by an infinite iteration,

binomiallogarithms

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Napier  also placed the proportions on a line parallel to the logarithm line. He could have placed them on the radius of a circle, starting at the centre. As the sine decreased so the angle of the radius decreases and the point on the radius traces out a small semi ctcle, if it is rotated by the sine itself. But if it is rotated by the evenly spaced logarithms of the calculations it will trace out an inward spiral.

The relationship between the trig ratios is extremely convoluted. It requires the subjective process to account for orientation, direction and direction of rotation. It has multiple concordances between ratios and many surprising and convoluted algorithmic identities. Accordingly it is a rich field for meditative exploration, and a school for concepts of rotation, reflection , translation, rotational and reflective symmetries and computation.

It is the computation or arithmetic which is the odd one out! The introduction of quantity into a magnitude is simply so we can make a song and dance about it, literally. The experience of magnitude is entirely subjective, to communicate to an external other we have to specify and bound a region, this quantifies it and we can then communicate that specific region to smother. Depending on the conscious process of that other, they may understand the quantity on face value or as a label for the experience of magnitude .

The quantification of a magnitude always introduces a difficulty. The form or magnitude in a form is as is. The quantity we introduce is totally subjective and arbitrary. Thus as we compare,count , distinguish  thus generating a logos or language model of the activity of comparison, the dynamics of it, we have no way of predicting if the comparison will be artios or perisos , perfect fit or approximate fit ( even or odd, which I hope you can see is now inane!) . Consequently we have subjectively moved from an indeterminate whole without anxiety to an indeterminate multiple form with anxiety! Will the quantity specified fit?

These specific quantities are called Metria a single ( Monas) one (en) is called a Metron. The idea of singling “one” out ( ekateros) is fundamental to book 7 of the Stoikeia of Euclid. This Monas becomes the standard monad or unit for a process of covering (sugkeime) which is done by placing the monad down(kata) onto the form/ magnitude to be measured/ quantified/ compared, and counting( Katametresee). This count is literally a cultural song and dance, by which we interact with and order space.

The form so covered by contiguous( edge joined) Metrons as monads are experienced as multiple forms( pollaplasios). But in fact they are also experienced as epipedoi or floor coverings. We came to call these things mosaics . Archeologists finding these patterns on the floors in Mousaion, houses for the Muses coined this term.

The mathematical significance of Mosaics is a fundamental and continuing nalysis of the Pythsgorean school of philosophy. Indeed no Pythagorean astrologer could qualify as an Astrologer( Mathematikos) without a deep muse inspired intuition of these forms.

These mosaics did not consist of standard Metrons, ie a cube tile or a hexagonal tile, but of a mixture of tile or block forms that continuously tiled or blocked the space being compared. Thus the spaces were Topologically described and counted. Area as a standard concept of counting only standard monads is a much later idea and of a different school of thought.
Mosaics were aesthetically designed to inspire, and thus often depicted scenes as well as just abstracted patterns. Such patterns were often traces of shadow dynamics throughout the yearly cycles.

By introducing standardised Metrons, a standardised approach to topology was introduced before we came to realise how limiting that was. Also anxiety was increased because one form as a Metron does not fit all!. The proclivity for perfect fitting forms drives aspects of mathematics today, but it is perisos or approximate fits that these mathematicians see as monsters! These standardised multiple forms are called Arithmoi. Thus all Arithmoi are mosaics but not all mosaics are Arithmoi. The counting of these standard forms eventually became confusingly modified into the notion of number.

Engineers and architects however love these perisoi! These approximate fits are pragmatically engineered to construct or sculpt real objects and structures. Pragmatics chooses the best approximation for the task, nd filling snd dmoothing gives the final fom. It is artisans and engineers who apply forms iteratively in construction projects which are grand mosaics! We live and have our conscious bring in these grand mosaical structures of our own hands and minds. And we continue to process the experiences around and in us in this way.

As much as this is formal and subjective it is also our experience that magnitude is regionalised. The very deepest mening of this we encapsulate in the perfected magnitude, a formal creation, called the sphere.
There are 2 other formal creations hich result from the deep processing subjectively of the sphere itself, these are the plane and the later straight line. Neither exist as magnitudes in our experience. We formally construct these notions from regions, that is from plane segments or line segments. In fact it is clear that the sphere is a formal construction from an iterative process of construction requiring infinitesimal regions.

The complexity of the notion has fascinated ever since it was first conceived nd continues to this day. The sphere encapsulates all our notions of analysis and synthesis, all our methods or processes of calculus both differential,integral and logarithmic. All our conceptions of topology and finally all our concept of spatial mosaics.

Because we quantify and thus introduce perisos anxiety it is not surprising, after do long a time of philosophising about it that we should find some counts of seemingly unit magnitudes should involve an endless process. In fact Zeno and Parmenides drew pointed attention to this. The pragmatist had no problem identifying the solution, as do engineers. You embrace approximation!

At some stage you simply decide enough is enough! This is essentially the principle of Exhaustion! Motivating such a principle is not only tiredness but also a notion of cyclical count. This count, as a record of planetary positions became known as Time and is dynamically measured, by dynamically cyclical objects in motion. Such measures are called Metronomes!

As you can see the Metron concept underpins all our measurement, including dynamic ones.

Dynamic measures answer the Zeno Parmenides conundrum. An infinite subjective process of analysis occurs like everything else within a dynamic cycle. Thus unless we actually extend the analytical process into infinite cycles, we can stop at any cycle by design or exhaustion! In particular. We can note that Dynmical systems traverse thes infinite process measurements in cyclically finite ways! That is to say I can count a number of cycles and while thus distracted a dynamic object would have traversed a magnitude I was unable to determine by an infinite process!

The issue therefore is pragmatic. Is the infinite process necessary ?  The answer is no, but to be able to be as ” accurate ” as desired or needed is necessary. The use of the term accuracy and exact is misleading. Simply we can choose the form as the standard and then it is exactly and accurately 1 !

The underlying process is a comparison. The count is to determine a ratio. The ratio is to be reapplied pragmatically and iteratively in some construction or synthesis process. We only need to be using ratios that do not ” fall over”, crumble or shatter under stress and vibration! In addition, if we can we want to use ratios that are aesthetically pleasing. And we want to do most of this within the dynamic cycles of a lifetime! Pragmatics and aesthetics govern many of our most fundamental processes.

Before I finish, it is good to observe that iteration of the pragmatically generated ideal forms is fundamental to our experience of change. Having devised these forms and reapplied them to interacting with space we are reminded of why we had to devise them in the first place! Everything moves! Panta Rhei!! Our anxieties drive us to try to keep things still, but in so doing we lose contact with real life experience. We Leo often kill the thing that caught our interest and so inspired us in the first place. However a compromise is to abstract by analogy a form and then use it iteratively to identify the dynamic experience. This is precisely how are neural networks work!

A good example is found in film or video capture. Each frame captures an analogous form to the real life obje t. As the cycles continue the analogous foms change and we thereby capture change by iterative analogous forms synthesised into a contiguous mosaic.of frames.

Grassmann in his analysis and synthesis intuitively understood that these forms found our notions of everything, and their mosaic combinations are the stuff of our Musings. He therefore worked very hard to establish a labelling system that made this very clear, and rediscovered a deep and abiding connection to the philosophical enquiries, observations and formulations of the Pythagoreans!

The heuristic, mnemonic and whimsical approach is actually psychologically consistent with the way we subjectively process our interactions with space.

The modern number concepts, devoid of this rich association actually obscure the natural human processes involved in the logos analogos response: how we language our experience of real life , and thus synthesise a language model of our subjective Kosmos!

On the notion of numbers

The average mathematician believes in numbers. They are real objects to them. Most of the rest of us follow suit. However the concept of a number is a notion fraught with tautology and not a little leger de main!

In the west the concept takes form over many centuries of intellectual turmoil, reflecting the business of Moines! War, overthrow, pillage and resettlement each contributed to the eventual meaning of the notion. The biggest and most obvious factor is translation between languages and mistranslation.

The western philosophers who are most influential in examining the notion were the Pythagorean school in southern Italy . They represented even at that time a cosmopolitan apprehension of the subject, drawing together influences from all the major cultures and civilisations around the world. And yet no other civilisations quite put it together as did the Pythagoreans!

The essential idea is that humans respond to the environment with a logos response. This is a complex of all types of signal responses in the human frame associated specifically to one type of channel: the vocalisation of sound.

Already this is an inadequate concept, because humans ” think” sub vocally, without making a sound also! The muscles involved in the speech act still fire but no major muscle group responds with the required intensity to make the air vibrate audibly.

The reflexive nature of the early languages captures this dichotomy, the action can be external and interactive or internal and reflexive.

This distinction is really rather fundamental. It means that tautology has a function. It is able to support many distinctions. In this case we can use the Logos response to represent the external speech act and an analogos response to represent the reflexive internal speech act.

Both of these are experiences, but one I might distinguish as objective, the other as subjective.

In short when I respond to an object and name it ” one” (1) I alo name an associated internal experience ” one”! It is this subjective experience that I can associate to every similar experience regardless of the actual form of the object responded to, even if it is an entirely subjective form!

Similarly I have an experience of multiple form which allows me o associate multiple form to any single or complex object.

The Greek concepts I will outline next.

Why It Is Possible to Fly

The World As Computation

  • To those who fear flying, it is probably disconcerting that physicists and aeronautical engineers still passionately debate the fundamental issue underlying this endeavor: what keeps planes in the air?(Kenneth Chang, New York Times, Dec 9, 2003)

The Wright Flyer 1903: the first sustained powered heavier-than-air flight.

The material of this knol is developed in further detail in the new book The Secret of Flight.

The Mystery of Gliding Flight

The problem of explaining why it is possible to fly in the air using wings has haunted scientists since the birth of mathematical sciences. To fly, an upward force on the wing, referred to as liftL, has to be generated from the flow of air around the wing, while the air resistance to motion or dragD, is small. The mystery is how a large ratio L/D can be created (see video of model airplane).

In the gliding…

View original post 3,891 more words

On the Origin of Trailing Vortices

Claes Johnson demonstrates how the potential solution to fluid flow leads to a contradictory result and how therefore, after zD’Alambert pointed this out fluid dynamics has been unable to satisfactorily explain aero dynamic behaviour. Various advances by Helmholtz and Kelvin and Prandtl  lead to a solution of sorts, but this was not empirically accurate,  the non slip boundary layer issue was the stumbling block which was pragmatically resolved by allowing slip at the boundary by Prandtl. However then the potential flow around an object gave a result that was non “physical” because slip means no pressure can be exerted.

The problem is the Eulerian differential equations slve ith a zero solution, implying reactionary pressures on a circular cross sectioned object at the 4 axial directions. The solution was to add in an overall circulation which is an encompassing vortex. This provides the pressure differentials for lift and drag but is purely a mathematical device. No such circulation is seen about an aerofoil.

The solution came to Claes through Computstional fluid dynamics. The Numerical solutions showed that the trailing edge could have trailing vortices which had a zero potential evaluation!. Thus the non slip boundary was able to generate an unstable point as it left the aerofoil and this would rotate as a vortex generating vortex plumes.. The overall solution thus avoided D’Alamberts conclusion of stability due to equally opposing potentials while giving a zero potential at the trailing edge. Drag was thus preserved and the pressure variation above the wing becomes physical. Prandtl had assumed the separation of the boundary layer at the top of the aerofoil, which was not observed except in stal, when no lift is generated.
http://claesjohnsonmathscience.wordpress.com/2012/02/22/why-it-is-possible-to-fly-yvfu3xg7d7wt-18/

The retention of the boundary layer means that pressure can act on the aerofoil, and the Bernoulii principle and venturri effects can be added into the hydrodynamic effects of displacement of fluid mass. Thus lift is a consequence of several effects resulting in the aerofoil being pushed and sucked into the air while being impeded by the forward motion creating the energetic interplay.

Note the gyrodynamics not elaborated in ths explanation:
Electro Thermo magneto complexes are ignored in this purely mechanical explanation, because the scientists did not choose to include them, but also because they did not know how to include them.

The kelvin water generator explains shear thinning in terms of ” electrostatics!”


The implications for how magnetism flows in a wire and electricity ( electric magnetism) around the surface of a wire are visible in the gyro dynamic heap that forms.

Reflection and refraction and diffraction of magnetic gyre is alo illustrated.

Now we can add a detail which reflects the role of viscosity and rotation in the whole dynamic. I have borrowed a term called gyrodynamics to label this. In fact, research into DeBroglies Pilot wave analogy. And of Course John Strutt’s ground breaking notes on periodic motions in material such as vibrations, and waves  underpins the observed behaviours. The kinematics of vorticity have been modified by Claes Johnson to accommodate the new findings , and thus remove non physical assumptions. The following text and video is a note I made on a relevant phenomenon. As you will see the behaviour of a vorticular streamline  with viscosity is not just to flow around an object. The streamline that is normally incident undergoes viscous deformations which allow reflection and refraction of the streamline. I do not observe diffraction as one is taught to expect. Rather the streamline takes a preferred direction. In aero dynamics this means that the boundary layer in it non slip region generates these vorticular streams from the area of incidence. These vortex streams are pressed along the boundary by the enfolding flow and thus leave the trailing edge as intact vortices.

The vortices themselves represent a fibrillation of the initial “diffracted ” or spreading of the streamline into a viscous ” heap”. This spreading is the result of rotational reaction to the regional boundary caused by the aerofoil, the internal viscosity and the enfolding flow of other streams. The separation vortices thus receive their energy from the impeding pressure area at the front of the aerofoils motion , and their instability is a function of this pressure area. The greater the pressure the greater the vorticular energy in these rotational fib rules. Thus as they leave the trailing edge they form distinct plumes rather than a turbulent sheet.

The nature of vortex plumes is observed in the behaviour of bubbles. They collect into dominant vortices with a fine structure and other surrounding vortices that are independent but contribute to the overal turbulent behaviour of a given system. Thus vortices do not all coalesce into one giant vortex, nor hold they . Vortices exist in pairs as vortex and anti vortex. These pairs depend on the viscosity of the material they are in. The higher viscous materials do not tend to generate small vortices or show vortex shedding, the lower viscous material generate versmall short lived vortex behaviours which do not complete the rotational formation of a vortex. In these cases the medium appears to simply displace from one rotational direction to its opposite. We might call this a ” wave” if it was not do misleading in this context. The best and safest description is to refer to all these rotation types as Trochoidal.. These low viscosity trochoids do indeed have a sine function description , but it is not one based on the rotation of a circle but rather the interaction of 2 or more arc motions acting from different rotational centres. Such motions are short lived and the rotational centres thus appear to decay or come into existence spontaneously. They are artefacts of rotational behaviour in our formal description systems

Norman’s ” rant” on pi

You may not know that Norman Wildberger is one of the most forward thinking Mathematicians on the Internet and youTube today. So I put the wrd rant in quotes more to connect it to Viharts tite than anything else.

Check out his video below.


Pi has a very important role to play in mathematics as the premier ratio! Just as the sphere and thus the circle are the premier forms of ancient Spaciometry. The process to these platonic forms , at least in the Stoikeia of Euclid, go through Eudoxus. In books 5 and 6 in particular Eudoxus sets out the Logos Analogos calculus. Later it is claimed that HippRchus another prominent Pythagorean set down the Trigonometric calculus. These are both a landmark philosophical forms and to be distinguished completely from the circular functions of the Indian mathematicians, the Newtonian triad of Newton De Moivre and. Cotes, and the preeminent Euler who clearly laid down the radian or “Halbmesser Bogen” measure intimated by Cotes who died before publication of his Harmonium Mensurarum.

Having made ths distinction I can now turn to Napiers Logarithms with some clarity!

The calculation of the sines was a centuries long enterprise, perhaps beginning with Brahmagupta, but certainly the Indian Mathematicians, and then continuing through the work of the Islamic ( Arabic and Persian) scholars. Thus by Napier’s time, voluminous details, formulae, methods and tabulated results existed in multiple forms. While the angle was associated to this enterprise, it was of astronomical significance only. For most, the pure ratio was the significant part. In fact stretching back to Ptolemy, the spread of 2 rays, lines or line segments were recorded as the ratio of chord to the diameter of the circle. The sine is precisely half that ratio.

The arc subtended on the chord( there are2, so the smaller one) was not related to the sine ratio until Euler, who characteristically set it out very clearly. It was only later that scholars found that Cotes had been moving in precisely this direction, and had discussed this on a European scale with some other mathematicians besides Newton and De Moivre. In relation to the calculus of Fluxions, especially of elliptical functions.

So, at the time of Napier, the prostate resins that was a method of using the sine tables to do calculations, was not as is presented , based on angle Arcs or degrees, but on angle ratios. The difference is profound for a modern mind to understand. For the most part Napier worked directly with a set of tables of sines. These were interpretable as arc measures, but not especially so. They were and are the ratio of 2 orthogonal lengths. Thus when Napier declares that his method depends on the proportion of diminishing length by a unit as one length increases by a unit he is comparing these orthogonal lengths at a very small scale. Typically 10^-7.

In his actual introductory comments he uses the isosceles triangles in a small sector of a circle!

Where he differed from Tycho Braes method was in basing his selection of sines on a geometric progression. This meant that instead of using prostate resins as described in the ancient texts he modified it by Approximation. By his time the sine tabulation was sufficiently rich for him to choose an entry sufficiently close to his values calculated by tha geometric progression. Tabulated values were often written to 20 + digits!

Once he started with a geometric progression his work was relatively straight forward addition of the sine entries. A couple of calculation properties relating to factors were important for logarithms to be useful: thus 4×3 must have the same logarithm sum as 6×2, and this is a straightforward property to demonstrate in genera
Hence by a lot of work and careful annotation he was able to write the first table of calculating sines turning multiplication into addition and back again.

Briggs was much taken with this discovery and collaborated with Napier on many other logarithmic tables using different” bases” that is different ratios written in tabulated form.nby avoiding prostatharesus Napier had made a breakthrough in calculation speed accuracy and amen ability. Many around the world were very much relieved to have access to his tables for calculations .

Later, through Eulers work the circular arc measure could be related to logarithms, but are to be distinguished from Napierian or Natural logarithms based on the ” natural” sine ratio not the arc sine ratio!

Using. Napiers method Euler was able to derive the exponential constant e used by actuarial professions in calculating compound interest, work previously done and advised upon by Newton. Based on this work De Moivre had moved into the formulation of his work on probability. But it was Cotes, drawing on De moivre’s wrk on Multinomials and their relationship to the sine tables, something only he and of course Newton his mentor fully appreciated; it was Cotes who went on to tie together the trigonometric series to the logarithmic series using i and to suggest it use in the gravitational description of orbits!

In this regard I want to look at how pi becomes i !
I need the notion of Logarithm described by Napier.
http://www.maa.org/publications/periodicals/convergence/logarithms-the-early-history-of-a-familiar-function
A series of articles contextualising Napiers ideas and Burghi’s. However Burghi’s tables were more difficult to use for non mathematicians because of not properly clarifying the characteristic of the logaarithm. The mantissa behaved in the same way as it must.

http://www.maa.org/publications/periodicals/convergence/logarithms-the-early-history-of-a-familiar-function-before-logarithms-the-computational-demands-of
http://www.maa.org/publications/periodicals/convergence/logarithms-the-early-history-of-a-familiar-function-john-napier-introduces-logarithms