Twistor

[jehovajah]

The bulk properties of space are deformation or transformation . Transformation occurs in chemical reactions and phase changes, that is wherever energy  or Newtonian motive as a concept clearly generates a new structural arrangement.

On the other hand, deformation occurs where energy or Newtonian motive swaps between 2 or more structural arrangements from a disturbed inertial or equilibrium position.. In keeping with Newton equilibrium is either static( relative to the reference frame or observer) or dynamic( but in this case exhibiting no net motive that can be extracted or absorbed externally). In such situations we state " conservation" laws and find general solutions without having to analyse too deeply.

However, wave Mrchanics is not so! While we have notationlly simplified the expression of the structures involved, and this from the classical considerations of men who did not have the benefits of modern technology,  we have found that over simplification, while instructive is misleading.

I have a text by Sadd in front of me in which in the first chapter he discusses the form of many general methods to " solve" the classical wave equation. At one stage he expounds that we choose separation constants to give us the Expdcted Harmonic Form! Later in the same chapter he notes this Harmonic form is way to simple to describe real empirical observations!

The notion of a wave equation is in fact conceptually misleading, despite the classical forms beautiful notation! The classical form is here not written in " vector" notation, and represents therefore a horrible reduction of Geometry and dynamics to a data set of numbers that fulfill the overlying referenced forms. Frequently Sadd refers to direction cosines, which is ancient maths speak for the underlying vector notions embedded in the complex number forms of Argand and Cauchy,

Later I will relate all of this to rotation as expressed by Grassmanns Twistors.
http://www.animations.physics.unsw.edu.au/jw/waves_superposition_reflection.htm

[jehovajah]

The following chapter in sadds readable treatment shows the development of the strain tensor concepts of wave motion.

http://www.mcise.uri.edu/sadd/mce565/Ch10.pdf

I mention it only in relation to the fact that the curl of a vector field is now traditionally associated with Rotation, and this more complex treatment allows for rotation to be isolated as a wave mode.

The treatment is complex and not a little off putting! At the end of the process only approximate solutions are given, and many of the distinctions become coupled in triads which are not intuitively obvious. The various types of bulk modulii or Hookes law relationships have to be put in by hand from empirical data and show approximate correspondence to measured velocities , but not much better than the classical treatment..

At the end of the day the" wave" speed and behaviour are deformations and disturbances in the bulk medium, either dilationl( expanding and contracting longitudinally) or volumetric( twisting and torsional transverse ). The additional complexity is mainly to allow these types of behaviours to be isolated.

The bulk modulus reveals a lot about Newtons measures.

Starting withmArchimedes Newton defines density as a balanced lever between a material on one side and volumes of water on the other. This balance is a dual . The ratio of this dual is the density ratio. Looking at the volumes of this dual in the ratio gives one the sense of how the two objects are materially different. One cannot squeeze water into a smaller volume, therefore the other matter must be proportionately dense.

Now we have a ratio that can be apprehended as the density to be assigned to an object, we also should note that density implies a concentration of force or rather acceleration. Objects or regions of different density naturally force or accelerate differently and do not mix. This property of density is all around us and we mis describe it as immiscibility. Density is a zero sum force phenomenon which obscures the internal dynamics of a material.  In outer space density cannot be determined by weighing. Centrifugal and centripetal and tangential accelerative dynamics have to be used. The " inertia" of space ie space in a dynamic system, differentiates density by acceleration differences in restorative systems.

The role of density then is to distribute  energy or motive in space such that acceleration can be aggregated in terms of Intensity rather than quantitatatively.  The intensity of acceleration we call pressure and measure pressure as a density product with acceleration distributed spatially, that is volumetriclly. For many situations we assume a uniform density or intensity distribution, however rotation or rotating space does not behave with a uniform intensity distribution of acceleration, that is it doesn't have a uniform density distribution of its constituent materiality.

Thus when space is deformed we should expect a reaction in terms of accelerative intensity, regional density and rotational integrity. The last concept is the one we find hardest to accept as a Newtonian concept, but it is in the description of the was of motion of a body in his reference to a spinning top. Newton could not explain rotational integrity physically and seems to have accepted magnetism as a defacto force holding particles together, at least he dismisses every other explanatory model based on occult fluids. I restore the fluid aspect of Newtons ideas in the term Newtonisn Fluid motive.

Newton certainly was not against either aether or fluids as states of the natural order, but he was careful to avoid any charge of occultism, especially as he was secretly engaged in Alchmical research. However, he also required consistent empirically evidence which lent itself to measurable processes and constructive reasoning along the principles go Mechanical philosophy and astrological geometry . He was fortunate to have access to Euclids Stoikeia as translated by Wallis as well as the Works of Apollonius as taught by Barrow, both of which he absorbed with great understnding, and whose principles he employed in his Principa.

To these he added the revered ideas of Aristotle critically reviewed by Gilbert in regard to his research into De Magnete, from which Newton divined his concepts of motive and celerity along the lines of ideas expressed by Galileo. Consequently he opined that philosophy of nature ought to be based on measures, through which certain intangibles might become apprehensible in a way wordy definition could not. That by measures used as in the mechanical geometries of the forefathers certain Dynmical behaviours might become measurable indirectly by such differences as they made to comprehendable measures and so by these means we might apprehend that which until his ideas had been shrouded in muddled thought and opinion.

This approach to natural philosophy was the main goal of his Principia. Cotes in his prologue refers to this as a sublime philosophy of Quantity! To Newton then we owe the mathematisation of natural philosophy , and the use of measures as the description of concepts otherwise defined by sensibilities rather than quantities.

As an example we use force to,describe actions and activities that affect our surroundings ad others physically and mentally. But that concept is defined as mass times acceleration. Most of us recognise that interpretation of the word force as foundational to its meaning. Thus we cn hardly read the historical accounts without wondering why they seem so " ignorant" of the basic idea! This is because it is not a basic idea, it is a Newtonian idea and principle! It took some 150 years to become accepted as the scientific and philosophical norm!

Within thevNewtonian philosophy of quantity is the quadratic proportionality. Thus force is defined by 2 terms which are nominally independent, mass and acceleration. Prior to Newton it was defined by Hooke's Law of a constant and an extension. Newtons formulation therefore had to explain Hookes law, and this required force dynamic interactions hard to describe. The principle of action and reaction thus obscures many details in favour of a simple principle. The more complex ideas come out in laws 4,5,6 and 7, which. Are rarely taught .

The deformable nature of mass is hidden away by teachers who do not establish density first as an Archimedin principle . Thn mass is seen as deriving from fluid dynamics and is curiously deformable! By excluding fluids and restricting his discussion to particles Newton established a clear approach. His second volume considers fluids as resistive media whose behaviours he hoped to establish by suitable measures. This he found harder than he could do! He left the problem to succeeding generations who have eventually derived a computational fluid dynamics.

It was thus premature for others to conclude that he had overturned Cartesian vortical theory or hypothesis.  He had made such progress in his thinking to realise he could not see beyond the computational load his fluid dynamics presented him with. Thus he could not establish his own vortex theory based on cylindrical rotation, and thus could not confirm or refute Descartes. However his material theory explained most very thing astrologers wanted to know about the motion of planets, so they did not care who was right about vortices!

The fluid origins of density show remarkably how powerful forces are locked in different materials, for to this day no one can compress a fluid into the volume of a denser material, accept that fluid be a gas. And certainly liquid water is deemed virtually incompressible. So the variation in density of water is so tiny that deformations are virtually transmissions of fluid displacements more than density variation. Consequently a deformation results in water level displacements which seem to flow over the water surface in a boar " wave". However surface tension effects modify the behaviour at the surface especially for deformations that are noml to the surface level.

The role of density variation in propagation is therefore one of the main constraints. Therefor the se of the sine wave model for wave description leads to many constants being determined to make as best a fit as possible. But it has to be pointed out that the sine function despite its adaptability is not a wave function, it derives from circular " rotation" and as the descriptor of rotation by quarter turns. It requires the sine and cosine to model even a part rotation. Now a rotation is not a bulk compression or density variation.

[jehovajah]

It is hard to explain how misleading the sine function is in describing physical phenomena, but how fundamental it is to our formal models of the world. Euler constructed the sine wave function by using the degree or radian measure as an axis. Newton decomposed circular and spiral motion into 3 forces that are orthogonal in the following way: 2 are radial and contra and the third is orthogonal and tangential. These accelerations are instantaneous impulses that sum to zero except in the tangential direction or tangent plane.

We cannot understand how this works without a concept of iteration. The iteration may be said to be instantaneous, but it is in fact time dependent in the sense of sequenced progression but only in our formal model. For a circlular path the centripetal force dominates the centrifugal force enough to move an object tangentially to where the centrifugal force dominates to move the object tangentially to where the centripetal dominates. The net momentum is thus changing direction but the net force sums to 0

It is perhaps difficult to realise that this is not necessarily how circular motion behaves but this is a Newtonian analytical explanation based on the method of resolving forces to orthogonal axes. In fluid motion Newton realised that tangential interaction between circular cylinders actually involves a spiral interaction of force exchange, but his cylindrical structure did not allow for vortices in counter rotation between the cylinders.

The structure with vortices ameliorates the transmission of accelerations in a non obvious way . In fact the transmission in the fluid is a vorticular oscillation with a kind of damping. Which leads to a block steady state body rotation eventually.

In a solid with greater viscosity this vorticular wave of deformation passes quickly to the edge where it breaks the viscosity limit or shear modulus for the material and the solid fractures at thst point.

In a Plasma fluid situation the vortices may continue as independent bubbles for some time at different scales, but each vortex circulates by its own circular motive which we can only model by Newtonian like analyses that resolve the behaviour into orthogonal orces.

Where Grassmann moves to a clarification over usual Newtonian teachers is in allowing for different kinds of line segments. To describe curved motion Grassmanns trig line segments have to be used rather than ordinary line segments. This means that Grassmann posits trig forces in the case of rotational motion. With trig forces acting on a body orthogonally we no longer draw straight lines for AC in AB + BC= AC we draw twistor arcs.

[jehovajah]

The acceleration required to squeeze a volume of material into a smaller volume is involved in a measure called the bulk modulus. This is a measur of the viscosity of a material which includes its elasticity and its inertial space properties, and highlights the nature and role of density.

The mass of a material is confined and compressed until the compressor is balanced by the material restorative reaction. The change in volume is compared to the initial volume (dL)³/L³

Acceleration a= v–u/t = dl₂ – dl₁ /t²
l₂ = l₁+ n * dl₁

Because we can actually measure the displacements of an accelerating body  I can choose hich scale I will record it in. In this case I have chosen to record it in the scale of l₁ so I can factor the displacement.

This gives a= d(dl₁)*n/t²

In this form one can see that acceleration has a displacement in and of space embedded. Thus the bulk modulus gives without further modelling or distraction by the sine curve all that we need to apprehend the characteristic speed of a compression!

The realisation needed is Newtons inertial space concept, that is laws 4 to 7.. Equilibrium is a zero force sum. Thus any compression will only achieve equilibrium when restorative forces equal instantaneously applied force. At that time the material will be compressed , but only in a differential of its entire bulk dB. The size of dB depends on the bulk modulus because that record how compressed the material has to be to achieve a given force. The compression takes the time required to achieve that displacement. That can be a very small time!

But now the material is compressed into a smaller volume. In practice several of the sides compressed in the bulk modulus measure are not compressed. This gives those sides degrees of freedom which amount to a directional deformation. We know this deformation is a change in volume but not mass so we should expect to see volume and thus density change in the material. This volume and density change represents an accelerative force in a given direction, but because of restorative forces the acceleration is countered leaving a dynamic volume and density change that spreads through the medium at a velocity of equilibrium.

The progression of the compression actually is a developing compression in the whole of the material form. If the material cannot spring back then the result is a compressed deformed whole. Oe ever if the material can spring back, the volume and density change proceeds through the material at the equilibrium velocity, but behind it a second restoration change follows. The speed of this restoration does not depend on the bulk modulus! The restorative forces, when the initial force is released typically will not have any resistance to overcome, so that energy might well be released dramatically and the material behaves like a spring. Consequently , in addition to a density nd volume change progressing through the material followed by a restorative change, the dramatic extension at the now free end may be fas enough propel the whole form. The speed or velocity of this reaction will be very much less than the deformation equilibrium speed .

There are other effects of this volume and density deformation I won't go into, but suffice it to say that the sine wave has not been necessary to derive this understanding.

What is the sine wave model for? Leaving aside its role of giving mathematicians something to do, it is a useful hook to organise ones thinking only after one has thought the physics through. The sine wave itself is non physical! The deformation does not look like a sine wave as it travels like a bulge through solid rock .

A seismograph recording of a deformation also uses the pen line to record information in a form like that produced by a rotating or swivelling arm but these are not sine waves.

However any curve can be transformed into a sum of sine nd coine functions called a Fourier transform. This is an intetpolative procedure. It just so happens that it highlight the super positioning of the sine and cosine functions. It was this property of the functions that Fresnel found so useful in modelling physicl behaviour.of diffracted light.
http://www.lightandmatter.com/html_books/0sn/ch06/ch06.html

[jehovajah]

A paper to note.

http://www.uv.es/gpoei/articulos/(2000)_OE_39_From_Fresnel_to_FRT.pdf

The history shows the interaction of fresnel and Fourier in particular with the scientific orthodoxy. Fourier had utilised the principles of the Fourier transform prior to 1807  in his theory of heat . Fresnl, indeed engle had used the principles of sine and cosine in super position in his work prior to 1815. Thus it is clear that this behaviour of the trig or circular functions was known. Prior to its use to model wave motion.
http://books.google.co.uk/books?id=gkJn6ciwYZsC&pg=PT251&lpg=PT251&dq=joseph+fourier+and+++fresnel&source=bl&ots=jNV8u4ITEi&sig=HCCbMfTuW5WVNri7nVTW5FJwJUU&hl=en&sa=X&ei=fnVoU63CIoXQOYzHgJgN&ved=0CDsQ6AEwBg#v=onepage&q=joseph%20fourier%20and%20%20%20fresnel&f=false

In fact it was used first by Fourier to model heat conduction, and possibly by Lagrange to analyse tidal behaviours. In any case Fresnel had access to the mathematical method which he employed to explain diffraction persuasively. Thus we find that the association of sine to wave behaviour is a degraded version of the principle of transforms used in sine interpolations of polynomials.

At this stage I am not sufficiently interested to trace this back to Euler and Lagrange by Documentary evidence, but anyone willing to do so please contribute and correct where necessary.
It appears to be a concept of LaPlace, expounded upon by Fourier and called the principle of superposition by later mathematicians.

http://www.instinct.org/texts/shufflebrain/shufflebrain-book07.html

My point is the general point that the sine "wave" is a misleading model of physical behaviour of transmission of volumetric and density changes in space.

Heir essential use is in modelling the shape of the envelope of the deformation, and in progressing the deformation in an animation at the equilibrium velocity for the material. The deformation shape is universally a bulge or bubble depending on constraints, and it's equilibrium velocity is described by the bulk modulus of the material of space.

The application to light requires a measure of the bulk properties of the electro Thermo magneto complexes of plasmid save.

[jehovajah]

Comparing these 2 Grassmann Twistor forms at bailout 400

z=(cosh(0.5*x#)+sinh(2*x#))*exp(z)+1*c
and

z=(cosh(0.5*x#)+sinh(2*x#)*i)*exp(z)+1*c

Both are mandy type iterations in Quasz.

[jehovajah]

z=(cosh(0.5*x#)+sinh(2*x#)*i)*exp(z)+1*c

[jehovajah]

Notes on the study of the principle of uperposition.

http://www.naturalspublishing.com/files/published/mbk5wx7r245f58.pdf

These researchers point out something obscured by traditional presentations, lingered by the view that the sine wave is a physical wave model. In that regard they emphasise the mplitude modulation, or the spatial shape of the wave, certainly frequency modulation is known and studied so don't get me wrong, but the principle of superposition is taught in terms of amplitude variation without the equally present frequency variation. In fact frequency variation is eliminated by choosing monochromatic light sources, and tuned sensors. The most famous phenomenon of frequency modulation is the red and blue Doppler shift.

However a better understanding of phase modulation is the phenomenon of rotation reversal often seen in older films when car wheels appear to be rotating backwards. Phase and phase angle are the same concept. The rotation is referenced from different arcs of the circle. If the rotation speed is varied the eye can be deceived into picking the different arcs as relatively stationary or moving relative to each other.

The phenomenon of standing waves requires this phase modulation to be exact at l cycles for the reflecting and progressing rotation, at the same time higher frequencies of rotation may be brought into standing wave positions due to this phenomenon of rotation.

[jehovajah]

Penrose spacetime Twistors

<a href="http://www.youtube.com/v/hAWyex1GKRU&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/hAWyex1GKRU&rel=1&fs=1&hd=1</a>

Note how the geometry comes from and involves spheres and rotations. The interesting knot or twist in space time or twistor space, was imagined by Hamilton, Tait, McCullagh and the Irish school of thought that followed Hamilton.

[jehovajah]

This image is a slight modification of the uploaded examples.
In Quasz i can rotate the figure in 3d to get a better sense of it. What i could not until now was understand the rotation method. The rotation is a rotation of the plane not the axis. By this i mean that the orientation of the axes do not describe the rotation , but 3 reference planes do . the y plane or slice is in the screen. This slice rotates the image as if you were turning the screen counter clockwise/
The x slice i a plane horizontal but going into the screen while the z slice is  plane that is vertical going into the screen. You must imagine seeing these 2 planes only edge on. Principal rotation is by convention anti clockwise.

The object carved by this single twistor and the bailout at 400  is a spiral galaxy or a hurricane analogue.

Current research indicates that such a form is not attainable by gravitational collapse at all, rather an electromagnetic driving force is required.

My claim is simple. Electro thermo magneto behaviours are analogous to rotational models that rely on dynamic geometry of Grassmann twistor forms. The geometry has to be dynamic and fractal to make sense of our dynamic interaction with space

[jehovajah]

Algebra is said to be derived from the Arabic description of a calculation method devised by the Indian mathematicians. To the Arabs it became a byword for a mind numbing form of convolution of thought that rendered all but the most rhetorically competent terrified! And yet the Indian notion was inherently simple.

Shunya in the Sanskrit means fullness of possibility. Thus the astrologers and philosophers of India held to a Vedic tradition of wholeness. This was expressed by cycles embedded within cycles. Gradually the Indian savants teased out a number system based on clocks. Like our old dil meters and hand driven calculating machines each dial was 10 times greater than the one to the left of it by observation.

Thus within the very number system we have inherited from the Indians iteration has structured our symbolic notation in terms of modulo 10 numerals and Place /position value concatenating what is fundamentally a power series, an added up sequence whose main skeletal structure is a power series.

Within this iteration the common rules and notions of algebra are derived.

Within this derivation the cyclical or rotational nature of Shunya is  both mechanically and engineeringly to do with rotation, and therein covers over the iteration or recursion of human space interaction.

<a href="http://www.youtube.com/v/Mv9NEXX1VHc&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/Mv9NEXX1VHc&rel=1&fs=1&hd=1</a>

But now it is clear that there are 2 aspects of rotation : that of a rigid form, and that of a fluid, Viscous form. That which is rigid rotates about a central rotating region or point with all line segments in rigid relation to each other. In the second, the points move cyclically as if around the forms plane surface segments edges this means that line segments vary periodically in relative orientation, a function of relative rotation, ( that is relative to the principal orientation); in principal direction quantity of displacement ( on that relatively oriented line segment); and finally in quantity of principal rotation direction to the relative line segment orientations.

It is not surprising then that we find general motions of fluids somewhat obscure because it takes a complex process to describe any instantaneous status. However the role of music in describing physical status intuitively has never really been fully explained thus some classical mathematicians through love of music found it possible to cogently encode these fluid behaviours of rotation, while others of course were mystified by the whole subjective process.

The analogy is with the progression of time in a piece of written music. Each element of the orchestra takes its time cue from the music sheet and behaves simultaneously alongside other orchestral elements. The whole ensemble of course is represented on the page as a combinatorial aggregation. This is in other words a lineal combination of lineal combinations.

The analogy with a musical score, and the more intricate of Grassmanns notation is apt, but the necessity of time as a combinatorial factor has been known by all musicians and physicians since Pythagoras.
Before passing to the modern video resolution of these issues, it is interesting to note that the paradoxes of Zeno and Parmenides are only paradoxes due to clever misdirection with regard to time.

Before they figured it out the Greek philosophers were inspired by Zeno's ideas to sort out distance/ displacement and duration. It was clear, once pointed out, that the tortoise and the athlete were being compared by distance travelled dependently, and in fact the analysis was the interdependence of their motions. However, once you reference their motions against an independent measure of displacement motion, a third parameter of motion independent of each of their parameters of motion, then the " paradox" disappears and the trick that was played becomes clear. Parmenides focuses the attention away from all the parameters the mind uses to analyse the motion in the world!

Why Parmenides in particular did this he explains: men ought to think rationally for themselves, not according to the direction of others, however reasonable they may seem!

Newton learned this lesson well and utilised it in his development of the Fluxions and their infinite series counterparts . Infinit process, which is what ad infinitum means, was never to be confused with a finite count!. It always had its place in the dynamisms of the mechanical world, in which it stood as an analytical tool for determining approximately the form of any spatial region. It was this form that gave spaciometric sense to motion in space. By use of these parametric descriptions, the observer is able to construct as desired some putative form in a spatial reference frame. The third independent parameter actually determines the form of this construction, and is abdeterner therefore of spatial position. It is a short step of the mind to the realisation that this third parameter can describe motion of the form in space.

Returning to Zeno's aforementioned paradox, the third parameter allows the observer to describe the independent positions of the athlete and the tortoise, and thus it is clearly seen that the athlete passes the tortoise quite easily as measured by this third independent parameter.

Now I have Pre supposed "time" or rather duration in this discussion, but really as Einstein stated, this parameter is just another dimension, or parameter of measurement. To assign to this parameter the characteristic of Newtonian Abdolute time leads to mental difficulties. We avoid Parmenides and enter into our own logical quicksand!

Again Zeno and Parmenides are instructive. If we limit our focus of attention we will always arrive at some paradox. The key is not to be drawn into limited thinking. This is what Einstein learned and lived by. But then how do we interpret the results and processes of calculation? The answer is surprisingly simple. We keep track of the spaciometric significance of each manipulation. We relate it to the ultimate sphericity of our conscious processing, we regard this as Shunya, that is everything, and we concentrate only on what is locally significant.

This is not the same as discarding certain results as " not real"! In fact, as we have found the unfortunate use of the term imaginary has lead us to discard perfectly valid calculations! Looking locally also does not discard the impact of those non local quantities or those independent parameters not represented in the display reference frame. These so called hidden parameters arise as a consequence of our inability to keep track and our fondness of agglomeration! That is we collapse counts into a single value that has no spaciometric sense,sic 2 apples and 3 oranges makes 5!

The importance of dimensional analysis in physics cannot be over stated, yet the underlying fault is not in physics but in the bad habits of mathematicians who have fallen in love with the numeral the symbol, rather than the Arithmoi, that which the symbol represents,

[jehovajah]

Commentary
Returning now to the rise of the technology for producing  photographic still and barely on its heels animated images, this created a paradigm shift in the understanding of the dynamics of our experiential reality. Yet it was down to enthusiasts rather than academia to advance the technique and methodology to a sufficient remarkable expertise before academia would deign to consider it.

Similarly, if you will, that which could be known of the magnet and it's behaviours was left to the few interested parties who took time to learn and investigate. But when one so learned as Gilbert should produce a remarkable folio in which all such knowledge was collected together and by demonstration and argument shown to reveal a remarkable law of space , it received such recognition by the less stuffy in academia as ought to have propelled it and Gilbert into the public consciousness evermore!. That this did not happen is evident, and betrays the darkness of the human heart, in that those that followed would rather step on Gilbert as lift him up, that their own names should receive the praise!.

In any case that which is and should be known about photography lies now in the vaults of many commercial enterprises who guard it as intellectual property pertinent to the carrying on of their business in the competitive marketplace. What little is known by academia is hardly worth knowing, and those that excel,in this area, the Walt Disneys, the Kodak families , the Sony entertainment megaliths hold in repositories such expertise that we are regularly forced to exclaim " Magic". Even though we scarce believe it.

The principles are the same as in the orchestration of music, except the recording medium being light sensitive is able to record a scene almost instantly. It is this speed of absorption of the chemical which forces us to quantise light. This we do by various shutters and various apertures, and various focal length settings.

Having employed some expertise in this set up, the protected film is sent to be expertly developed. Again this is such a process of wonder and skill that we can hardly recount it.

The final image is set and when presented, to our eyes faithfully records what was before the camera lens. Of course we should suspect any claims of fidelity after such a detailed process has been involved! Nevertheless we are persuaded by those that wish to sell us these expertises that the camera never lies!

Thus it is with some considerable sophistication that we view a still photo, and even greater that we view a film animation. Essentially a flat plane is made to record all the movement of the world. But in reality we know that it is a sequence of such flat plane still images, projected separately, that is in a separated fashion, rapidly onto a flat screen which forces our mind to create an impression of movement.

It is this realisation that underpins the modern idea of experiential reality. Using this notion Einstein creates the concept of space-time, which is essentially a 3d block of 2d film frames of an event. However prior to this conception, the motion of dynamic variations were interpreted in the. Physical experience as motions, rotations and fluctuations of deformation. This required a deformable medium which was de facto considered to be the Aether. Einsteins spacetime conception, mathematically and geometrically replaced the Aether. In point of fact it preserved the aether until such time as the hostile climate Einstein worked in could accept the terminology without the implication of Nazism or communism.

The video stack in a computer memory system now replaces the miles of physical film that attended the development of these paradigms. Now also we have multiple ways to interpret the same event and a way to compress space into a plane, now called the holographic surface. Animation in this surface follows the same basic rules of animation. Create a clear image of the changed position, fade out the old and emphasise the new.

This is basic iteration 101. In symbolic notation this is z= z+c. The new image on the left comes from the old on the tight with just a small alteration on the right. The process blends the right side into the new left side image.

I started by looking at the Indian contribution to this iterative process in our apprehension, and I finish by looking at Brnoit Mandelbrots contribution. The rise of fractal generators, with surface plotters has through modern technology put into the amateurs hands the technology to create worlds that model our detailed experience. While this is patently obvious, the fundamental nature of Grassmanns contribution to this is overlooked

AC = AB+BC is the fundamental product equation that ties all this history together and that includes the famous Mandelbrot equation for fractal dynamic,
Let B:= A then
AC = A² + AC  which reveals that the new AC is similar to the old but altered by a little bit. That is Mandelbrots almost self similarity, an absolutely fundamental aspect of his fractal geometry.

Yes the form is different and it will develop a different pattern, but essentially it is the same dynamic as z= z+c. The more interesting variation is z= z² +c but the connection to the product of line segments is more complicated to write down for little additional effective surprise!

Underpinning this formulation is the product of points . Within the product of points there is no rotation , but within the product of line segments there is an inherent rotation. This rotation is not that of a rigid body, but that of a fluid body. This rotation within the product of line segments requires the trig line segments and the inner product. These constraints are found precisely in the ellipsoidal strain measure used in fluid mechanics for streamline flow with a velocity gradient.

There is much more to observe regarding this cyclical rotation as it appertains to fluid dynamics, but suffice it to say that Grassmanns Schwenkunglehre is all about fluid Dynamics, and only secondarily about rigid body motion.

[jehovajah]

Commentary

It is a mistake in my view to exclude rotation from the product of points, just as it is a mistake to say that the product aa=0 where a is a general line segment in the grassmann algebra.

This is not grassmanns mistake because he has 2 classes or types of line segments within the same notation. the ordinary and the trig. The products for each type are different which he is at pains to state. Today we cleaerly separate these 2 types of products without separating the 2 types of line segments. The introduction of the trig ratios is not suffcient to explain the difference between the 2, but rather obscures grassmanns conception.

I have spent some considerable time restoring what i think is the foundational difference in mindset. Without it we might well consider this exposition to be some variant of trigonometry. Grassmann is attempting to give the philosophical and metaphysical background to our conception of trigonometry based on a fundamental primitive caleed a line segment. In the course of doing so he draws in another fundamental primitive that of the circle and is half and quarter perimeters. Because of this he must necessarily rely on Thales theorem which states that all angles subtended at the perimeter by the diameter are orthogonal. In essence Grassman has recounted the fundamental demands or Aitemai of Euclid. His development then follows Euclid identically. This is not surprising because his teachers would have impressed upon him the methodology of LeGendre.

Algebra as a symbolic discipline replaces rhetoric with precise symbol, and in a real sense this is what Legengre, Descarte and Grassmann did. It was thus not uncommon. Where Grassmann differed was in his idiosyncratic mindset that developed this formula as a general law
AB+BC=AC, allied with AB=-BA.

As simple as these are , only Mobiius had a similar algebraic notation, and that was from a different mindset.

Having explored these ordinary line segments in this way he realised that vertauschen and umgekehrt were related by the sign conventions. It took me some trial and error to realise he was directly referring to cyclical interchange of factors. This cyclical interchange thus encapsulates the notion of directed numbers as described by Wessel, but in a much more subtle and useful geometrical form . In short Grassmann had encapsulated rotation into his algebra by means of the alternation of sign.

None of this is new to those studying trigonometry. But those studying it will admit that at some stage it becomes o confusing as to make your head hurt. Thus the use of trig line segment notation naturally is employed to minimise the iteration involved in look up.Both types of lin segments are known and utilised by mathematicians from Descartes onwards. What grassmann did was to make these 2 types fundamental primitives in his LINEAL algebra, and to construct over time and with much wit and insight an algebra of Algebras based on these primitives and the 2 fundamental laws, and the concept of cyclical rotation.

While rigid rotation is best described by the exterior product, fluid rotation is best described by the quotient product. It is this quotient product that i have defined as a Grassmann Twistor. In addition it is quite general in its extension by combinatorial sum, and as a consequence general twistor rotation will be represented by trochoids, or alternatively by Fourier transforms, which are general trochoids or roulletes, as some now denote them.

While of course the Ausdehnungslehre emphasises the line segment, Grassmann explained his intention to produce a second volume devoted to the trig line segments that is to those line segments tha presuppose rotation . I term this the Schwenkungslehre. In point of historical fact he never followed this putative plan, Instead, with the help of his Brother robert he fully completed what he regarded as an imperfect work in !844, and set it on he most rigorous grounds. In so doing much of what was intended for the second volume went into the new format of !1862 plus some new material beyond that. The Schwenkungslehre is thus within the pages of the !862 version, and Grassmann helpfully anntates where both in the reprint of the !844 version and in an insightful document deposited in the Gronier Archives.

So for me the story is complete. The Ausdehnungslehre 1862 contains his perfected format of his mindset, but it also was a full outworking of the seminal ideas in the 1844 version. The 1844 version continued to inspir him to new insights as it did others, so the 1862 version is by no means the last word on his conception.

In this light various other authors, but more particularly Robert continued to develop the ideas put with such ardour in the 1844 version. Today we can trace the presentation of the wholw of modern mathematical physics back to this seminal work in 1844. Whil it seems hardly fair, even the great work of Hamilton eventually pales into an insignificant but nevertheless important sub position in the Grassmann algebras, a point that was not lost on Hamilton at the time! despite his best efforts he was never able to regain the fundamental ground held by is acknowledged master Hermann Grassmann
 

[jehovajah]

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[jehovajah]

Rigid rotation

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