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I have just written a blog on my notion of a twistor and followed it on the notion of Twistorque
http://jehovajah.wordpress.com/jehovajah/blog/defining-the-twistor
The point here is some clarification on i and the _/-1

I am writing on a general notion of Pressure at the minute in which i will draw attention to a reformulation of Twistorque and Torque itself.

http://my.opera.com/jehovajah/blog/2013/05/05/the-problem-with-force-is-it-is-non

I have completed the general notion of pressure and have drawn attention to the Grassmann Algebra that i am investigating.
http://my.opera.com/jehovajah/blog/grassmanns-point-algebra-and-potential-scalar-fields
http://my.opera.com/jehovajah/blog/2013/05/13/associativity-anticommutativity-and

suppose i wrote

(1,ß)

and said this is a general polar coordinate, where ß is a magnitude of arc,

If i now made the magnitude of arc identical to the radius of 1

(1,1)

represents a unit twistor, or a radian on the circle radius 1.

If i scale this unit twistor by a multiplier t

(t,t)

is a unit twistor on a circle radius t.

The curvature of these unit twistors is different, so each unit twistor in fact also measures a magnitude called curvature.

A spiral is therefore a motion that engages many unit twistors in its description, a point noted by Archimedes who describes this continuous transformation in terms of radials increasing in some proportion to the arc traversed. By this I mean that the radial intersects with arcs whose unit twistors are directly proportional to it, at the same time as it moves instantaneously along each arc it intersects! Motion in a circle is always a quadratic proportion or some lineal combination of quadratic proportions.

The polar coordinate reference frame is the most direct and elegant way of rhetorically describing these Logos Analogos relationships or dualities.

Suppose i introduce the identity

e^iø

This identity with

cosø+isinø

has a meaning only in terms of a polar coordinate system which is being transformed to a Cartesian coordinate system in (x,y)

It is supposedly derived from an infinite series expansion, but in fact this is just some fancy symbolic manipulation for what is more readily apprehended as the trigonometric ratios in the unit circle.

The use of this identity lies in its encapsulation into algebraic symbols an elegant representation of the circle and its trig ratios(logoi), which neatly translate a polar coordinate into a cartesian one.

In fact, this identity is a parametrisation of the circle by the parameter of its arc lengths, that is its twistors.

t*e^iø

represents scaled circles from the unit circle. and though it is not obvious, the arc and the radius are both scaled by t as before. The simplest way to see this is by using the identity and considering

t*(cosø+isinø).

The scaling of the right angled triangle means also the scaling of the sector that contains that right angled triangle. For this to work ø cannot be a number or a length or any such notion. ø is a constant. In fact, this particular notation relies completely on the Logos Analogos framework. ß and ø are related, but ß is a magnitude of arc while cos ø is a ratio of arc to radius or some analogous ratio, where ø is some metric of the arc magnitude ( degrees, grad, radians) . Whatever metric is chosen a particular fixed or constant value has to be chosen to compare physical arced or curved  displacement.

Because of the confusion during the renaissance we lost sight of this simple correspondence.

Normans Treatment of Euler Rotations fundamentally illustrated the geometric notion of a Twistor.

The Algebra allows spheres that are in contact to define general rotations in a system of spheres. Of course the radius of the common sphere will vary thus making the resultant rotation lie on a varying radius of an "equator" great circle. This complicates the resultant calculation but only in terms of the great circle planes reflected in. The resultant twistor will be a projection onto the sphere's equator whose axis has just been determined.

http://www.youtube.com/watch?v=0_XoZc-A1HU

I had occasion to use the term gyre in a discussion on Magnetism. The way I use it is to eliminate the confusion between clockwise and anticlockwise.  Thus gyre is a local reference frame term. In that local frame gyre is always anti clockwise around the axis of rotation which is always up in 3 d. . Once the gyre is spinning the up is always the direction which makes the circular motion anticlockwise..

For a continuously spinning gyre this relationship holds regardless of the axis orientation in an external reference frame.

However should a gyre stop and thn start in the opposite direction the up axis has not flipped, but we do not know that unless we are party to the slow down ,stop and reverse spin up.. This is therefore a special case , but otherwise we will assume that gyre is constant and that any clockwise to anticlockwise flip is due to up axis rotatin in an external reference frame.

The twistors defined here therefore have to be carefully annotated, thus an axis flip has the – sign applied to the axis only. , while a spin down spin up will have the — sign applied to the arc measure..

The end result may look the same but in fact polarisation is very different for light undergoing these2 types of events.

Conservation of rotational momentum.
http://www.youtube.com/watch?v=yeR_24KZeqs
Compare with torque.

http://hyperphysics.phy-astr.gsu.edu/hbase/n2r.html#n2r

Defining the notion of Twistorque and it's measure.

This is a copy of a blog post mentioned above, but the opera blog site is closing.

I have migrated my blog posts to Wordpress.com, but have not updated the links in all my posts, yet.

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Rotation is a separate magnitude to translation. When a pressure is applied to a body it is within the context of an inertial equilibrium system holding the object in its static or dynamic situation. Thus an arbitrary pressure arising in such a system will generate arbitrary trochoidal motion .
When- sphere or a circular disc is analysed, often the inertial system is wrongly characterised. Firstly a rotation is observed and then assumed to have arisen by a tangential impulse acting in opposition either to another tangential impulse or an inertial central force.. These opposing force descriptions are defined as torque. They are also called a moment or a couple. By such a disposition all rotational motion is modelled, and indeed believed to be generated.

The generation of rotational motion is so varied that to assume it is a special case of moments and couples is misleading. This is especially clear when one leaves the cosy world of rigid body motion and enters a fully fluid domain.

In the inertial frame, when equilibrium is disturbed, then inertial pressures appear as if by magic to restore  or retain equilibrium. However, the effect is propagative and sequential and if equilibrium is restored it is often through oscillation or damped oscillation.. The stability of an equilibrium is indicated by these outcomes. However unstable systems reach a certain level and then run away. In such cases we might regard the inertial pressures as resistive.

For a rigid body rotation the resistive pressures may occur tangentially to a pressure which acts on the whole body. Because of this a torque or a moment acts relative to every other point in the body. However when the instantaneous torque has acted there is an instantaneous angular or momental  acceleration. This angular acceleration increases until the angular velocity is such that the rigid body outpaces the initiating pressure. At that stage the motion of the rigid body is purely due to rotation and the pressure gradient is no longer able to keep up . This is no longer torqued motion because there is no resistive force what is happening is dynamic angular momentum. If the angular velocity increases beyond the ability of the tangential drag force then slippage occurs.

If the slippage does not occur then the angular acceleration now uses the frictional forces to drive the rotating body forward. There are other resistive pressures that now occur attempting to maintain a larger equilibrium.

The discussion above regards torque as impulsive only. A rotational motive is engendered by some torque, but this is not the only way to start a rotational motion or induce a rotational motive. Rotation exists independently of tangential torques!.

Rotational motion like all motions can be resolved along different axes, but that does not make these resolutions motives of the rotational motion..

What is torque? It is officially defined as the stopping force. How much force is needed to stop a turning wheel?
http://en.wikipedia.org/wiki/Moment_of_inertia

It is clear that the faster the wheel is turning the more pressure. But it is also true that the further from the centre the pressure is appliedthe greater the instantaneous tangential force that is required!

The turning wheel is mounted on an axle and is virtually free of friction. To stop it we have to apply a normal pressure to generate a tangential resistive one. The angular deceleration times by the mass of the wheel must be proportional to the stopping pressure. So why the  use of I?

The technical problem is translating between 2 independent magnitudes, one rotational and the other translational.

If I take a frictionless pulley and attach a light unstretchable string to it for one revolution. To the free end I attach a mass. Then I assume a gravitational force field that is constant acceleration, then I can imagine an accelerating tension acting at the point of attachment. This tension is in fact wrapped around the pulley so a normal force of 2 times the tension acts on the pulley where the string is in contact. As the acceleration is applied the variation will create oscillations in the Bearing of the pulley. Wrapping the string round more than once will minimise this effect only if the strings mass acts as a damping mechanism.
Leaving these to one side I can assume a constant tension acting at the point of attachment with no frictional resistive forces contributing to the motion. The only resistive forces are compressive supporting the tension of the string around to the point of contact.

There are many assumptions made about this tangential tensile force. It can not be eassumed to act on all the surface of the pulley or resistive forces will need to be considered all the way round and a constant accelerative tangential tension will mean a different action is being considered. It is like having one person pushing a roundabout compared to a million pushing it all at the same time.

There are other considerations, too, for example how the accelerating tension, transmits the motive around the curve, whether this is instantaneous or propagative.

So finally we make this approximating model and it shows that the radial distance of the point of attachment has an astonishing effect: the nearer to the centre this tangential acceleration is applied the greater the angular acceleration of the whole!. Thus the principle of the lever is seen to apply in the case of tangential  and so instantaneous pressure if constant.
http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html#mi

I can now propose that tangential acceleration is related to rotational or angular acceleration by the radius of application,

Firstly I wii define Twistorque as mass times angular acceleration.
€ = mæ
Then I can define  tangential force as
F_t = €r

Finally traditional torque is defined as
T =F_tr

Traditional torque therefore has only an instantaneous application to a pulley, but it can be mechanically engineered to produce constant acceleration, continuously or discretely as in the gear chain systems in a grandfather clock. Again a geared action is entirely different to a tension action., but a pendulum gear control can metronome it out.

Traditional torque is best used in oscillating systems, where the forces are aimed at being balanced. To use traditional torque to define stopping force for a freely rotating system is misleading , so to hide this they rewrite F_tr as Iæ

There is no concept of Twistorque until my analysis last year(2012), and even then I was feeling my way.

Now I can use a Twistorque vector to represent angular force, especially using radians

The unit circle angular acceleration is
exp( iæ)
 and the arc acceleration is

Rexp(iæ).

This can be shown to be an  identity with the instantaneous  tangential acceleration in a tension system, with the above caveats . And we can relate this to traditional torque acceleration by

rRexp(iæ)

where r is the radius of application of traditional torque acceleration.

To use the equations or identities, the idea is to match R and r, that is to set R to r in the formula

The angular acceleration is always defined in radians by an arc length on the surface of a sphere of unit radius. However, the arc does not have to be a great circle arc, and in fact could be a spiral. This arc length is procedurally a differential displacemrent length and is not constant for acceleration.

The notion of Angle in space is to be generalised to the area of a shape cut out on a spheres surface, forming a " cone" at the centre of the sphere.

Defining the Twistor

A copy of a blog post mentioned above. My blog site is now jehovajah.wordpress.com.

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SpaceMatter has trochoidal motion as its arbitrary motion, and the motive for this motion is arbitrary spherical pressure. The pressure acts through volume , a perceived form in space and impinges on the surfaces of forms to transfer the motive for motion, in consequence the celerity of sn entire form. The point or surface of contact for this pressure, becomes itself a source of spherical Lpressure transmission..

The curvature of the surface on which pressure acts alters the effective resultant of the pressure. The representation of these lines of motion by lines is the basis of what has become known as the vector representation.

Grassmann in calling them Strecken, perhaps reckoned that remembering precisely what they are is more helpful than assuming the lines are the quantities. However following Newton and Lagrange, after an inspiration from his father Justus, Grassmann realised that he could use a parallelogram to combine these line representations to identify a unique line as representation of the resultant or a resultant line of motion. In point of fact Grassmann never called the identified line a resultant, rather Grassmann called the 2 lines Strecken if they were active, and let the two end Points represent the beginning and end of the action. Gradually the line through these 2 points came to represent the action, and finally under Gibbs the line between these points became another Strecken and considered as a resultant of the other 2

While many may not get the subtlety of Grassmanns approach it nevertheless meant that points were his fundamental elements, and magnitudes such as orientation or displacement derived from the relationships of these points with additional rules of relationship.

How were points to be Compounded or combined? The substance of his analysis was that the synthesis of points in combination or compounding structures developed all forms and formal relationships! Each of these forms had an associated algebra which was a group algebra or a ring algebra, but each was contained within a "higher" group or ring algebra and so on.

This amazing structural relationship was remarkable in its formal relationship to addition. At the time he had no similar formal link for multiplication. That is when he realised that the Euclidean parallel relationships for area formed a distributive group rule representing multiplication.

It has taken me a while to understand that Justus, Hermann and Robert were early contributors to fundamental ring and group theory, just as Hamilton was. It is probably fair to say that European group and ring theory were fundamentally grounded by the works of Grassmann and Hamilton. Only later, when group or ring theory took up its name as a subject did the fundamentals become reinvented. It is perhaps only in this millennia that Grassmann and Hamilton have been properly recognised. Even now the subtlety of Grassmanns thought eludes many.

The work of Euler is akin to that of Grassmann who drew inspiration from Lagrange who in turn drew inspiration from Newton. Euler however drew inspiration from all but far advanced beyond all . In this particular area of cyclic classes, modulo arithmetics, equivalence classes etc, he laid out the limits of combinatorial sequencing, and what essentially was represented by computation. In regarding the arc as represented by a marker he called i, meaning both infinitely large but constant and the imaginary magnitude he showed as did Newton, that taking this symbol as a representation of a sort of magnitude , like surds and irrational quantities, like one could do combinatorial computations that made sense algebraically. Gauss particularly formalised this. However it became apparent, that by representing these distinctions by another device called an axis graphical representations could be translated into these kinds of magnitudes.

What was going on? What was being conceived? No one clearly knew! But Grassmann and Hamilton figured it out, just as Bolyai figured out another algebra of forms. All these forms, as Hamilton seemed to realise immediately were equivalent to the ancient spherical geometries!, where Grassmann exceeded his peers was like Euler he considered the most general structures! In terms of distinctions this would be n, in terms of aies this would be n, in terms of magnitudes this would be n!

This posed no problem to Hermann Grassmann who simply saw clearly the heuristic development of rules from the lower orders upto the higher orders. Others were conceptually blown away, nd remain so to this day. The main reason being an unfounded belief in 3 dimensions!  Grassmann wanted people to realis we live in a multidimensional space Die Raume! The credo that it is 3 dimensional is based on the orthogonally property of straight lines or rather radials of a sphere. There are for a given radial  5 others that are orthogonal, 4 orthogonal to the given one while being orthogonal to at least 3 others. This complex relationship is simplified to the term 3 dimensions!

In addition, the word dimension is misunderstood. It means magnitude in an arbitrary orientation.. What is an orientation? It is an indication specified by a point or a mark, relative to a reference mark or point. What is a point?

The point is where we begin our synthesis, and the Grassmanns laboured in this field quite extensively, attempting to get it straight, to set it right! Of course their efforts were against a background of certain rigid beliefs, which have turned out to be non universal. Nevertheless, the work they did helped to clarify the road ahead, sometimes it lead to blinding new insights st other times to turgid dead ends, such as Russell and Whiteheads Principia.

Essentially then the group theory and ring theory aspects of their work have been recreated and extended by other workers, as Hermann hoped and prayed for. The general group and ring theoretical analysis of natural philosophy has occurred, but not as Grassmann envisaged it, and his own exemplar was misunderstood and twisted by Gibbs for his own purposes. Nevertheless, modern physics, at it's base is a Grassmann creation! In fact Modern mechanics bears the names Lagrange, Hamilton And Grassmann. Gauss, and Euler, for all their genius are of lesser fame.

I pointed out that Newton nd Gibb went down the road of compounding 2 Strecken to get a resultant , whereas Grassmann did not. Consequently Grassmanns approach related the compounding of 2 Strecken or the combinatorial of 2 Strecken to a form, that is a parallelogram. We slip from the form to another construct we call area. This concept called area is a recognition of the multiple forms that form a mosaic in a larger form. We slip from this multiple form to a word or a symbol or a concept or a sound we identify as a number. This is simply a name in a culturally accepted sequence we metronomic ally sound out as we grow older.

To focus multiplication on a single form is therefore very mysterious. However, the more Herrmann progressed his analysis, he more he dug into the Eudoxian philosophy of proportionality. It is not clear whether he recognised this, and doubtful that he did. Despite being a linguist, the notion that Euclids Stoikeioon was merely about geometry made it very difficult to see where mechanic had defined geometry rather than Euclid. The Stoikeioon is not a book on geometry. However mechanics had always used these forms found in astrology on the earth bound projects. Thus the work of star surveyors and land surveyors was extremely valuable to mechanical engineers. These Tekne have found inspiration in the Stoikeioon, evn though it was a philosophical text book! So I do not think Herrmann grasped how much he was tapping into ancient Greek thought in his Formenlehre., Robert also was not fully aware, but both argued eloquently for science to be based on a more rigorous foundation than religious or clerical authority! Ultimately they wanted science to be accountable to reason, empirical method and consistent logic or dialectic. The only way to ensure that was to synthesise it that way. This was the belief of there father and the family business thereafter.

With such an awesome responsibility it is no wonder Justus displayed rigid structural thinking. He wanted his work to be sure, solid and dependable. His reasoning though rigid was somewhat inconsistent over time, but only in a developmental sense. As his research and work progressed, his understanding of the impact nd consequences grew, and he revised his earlier opinions to accommodate.. Stubbornly, when he came to a logical impasse he would not give in or give up in his attempts to find a solution.

Justus work shows these trademarks of the struggle to forge a consistent theory, but it was Hermann who corrected his Fathers mistakes, without losing the rigid adherence to form and formal rules. This is why his work is so subtle. He had to apply inflexible rules creatively to get the correct results! Thus a TEM that may have had a fixed meaning in the past suddenly was forced to embody a more abstract notion.

The work of Hermann is said to be obscure, but this is not the case. His use of familiar words in unfamiliar relationships is dizzying and mind blowing! It is not obscure just a fantastic liberating trip!. However, after reading once you feel completely lost. You have to hold it (1844 version) like a bible or a manual to guide you through a wonderland of creative inventiveness.

Thev1862 version is Roberts toned down more mathematical version strictly for mathematicians, not natural philosophers. It lacks the depth of ideology and imagination, the heuristic discovery, the mental fluidity of thought connections that make all the terminology throw a light into every dark corner. Consequently Hermann was at pains to draw the readers back to the Ursprung!

Because Hermann clearly did not recognise the almost identical Eudoxian philosophy, he did not realise that his dream of applying his anslytical method to the circle was already done. His conceptual difficulty was that the Strecken were straight lines, and these he took as primitives. In fact no straight line is a primitive. The only primitives are points and spheres.

I have shown repeatedly that even prior to the sphere one may take the arbitrary trochoid as primitive, making a spiral or vorticular form an essential precursor to the cone snd the sphere. In fact all the conical forms..

Given that, the question is how can I compound them or combine them?

The solution is straight out of Grassmanns development ? For any 3 points through which a trochoid passes we can denote that trochoid or 2 trochoids by its end points. However if we wish to assign a magnitude to a trochoid we have to take note of its curvature and where the circles that define it have their centres. The lengths of trochoids are therefore summed obe the curvatures of its instantaneous parts.
What about multiplication. Again we do not use a magnitude, we use a common point for 2 trochoids to" originate"  from, that is technically a" join", and thn we use the idea of parallel trochoids., to form a trochoidal parallelogram. The easiest way to achieve this is to join a straight line to the end points of the Trochoids and draw a rectilinear form, and then copy the orthogonal displacements from those lines to form the " translated trochoid.

Now we can see immediately that the multiple form will be the same, so Escher diagrams and pictures transform the multiple form into wonderful pictures of the same area conceptually. Finding the length of a trochoid in standard units or rather being able to convert between different basis trochoids is all that is required, the combinatorics is the same.

With that in mind we can look at the arc as a unit basis "vector". That is the arc is going to be considered as a curved line in motion having a curved motion of a certain magnitude. If I have another curved line it must have a different centre of curvature. Now I can represent that curved line , if it is a circle by a parameterisation  ¢ relative to the scheme for parameterisation . Using a set of cartesian cross axes  and a  a right angled triangle formed between 2 radii we can use the cos and sin ratios as we vary the triangle. or the tan ratio of a line that uses the diameter and chords as it varies round the circle intersection with it.
http://www.youtube.com/watch?v=OqxYLyGLqcs&feature=youtube_gdata_player
What this means is that we can change bases from a point on the arc that is a join of 2 arcs to any other such pairs of arcs or to any other rectilineal set of axes of a generalised coordinate system.or a vector coordinate system.

There are 2 main systems we use therefore, the polar coordinate and the Cartesian coordinate. Before we could generalise the Cartesian coordinate into the vector  coordinate system, and it did not matter whether the vectors were orthogonal or not. Independence of axis was inherent in orientation, not orthogonally. This allowed Grassmann to face n axes ith equanimity. However it was preferred to keep contra axes clearly identified, because this gave the sense of simplifying some of the terms. This is a mere psychological benefit. In addition keeping track of the sign becomes a problem of its own, particularly when it conjuncted with a subtraction process.

The real value of Grassmanns analysis, is the way it heuristic ally solves the difficulties by lifting the mind out of the particular into the general, thus clarifying the particular. Of course the difficulty is the generating of new terms that support the synthesis and the generality, indicating when specifics need to be applied..

Let us now look at the circular arc If we parameterisation it from its centre we can use e^ix  exp( ix) to characterise its parameterisation. It is an identity with cost + isinx. This is a procedural statement in a vector reference frame  using orthogonal unit vectors 1 and i.
However exp(ix) is an arc vector when x is a radian measure, the i serving only to distinguish this radian vector. If x is measured in degrees, then the identity between the two sides does not hold. The exp(ix) in its entirety is a label for a vector summation of 2 unit vectors given in parametric form.

The radian measure works for both the radial and the diameter parametisations., but the positioning is different. The radial one is a pure polarisation, a rather special form of curvilinear axes. The rational parameterisation allows for a curvilinear reference frame through a join to be developed.

Do we need the i?
In point of fact it now serves as a distinction. But what of it's other meaning as the ? To avoid confusion this meaning should be highlighted as it used to be, so that when one form is used it clearly signals when a surf calculation is intended or when a vector procedure is intended.

Grassmann for example derives a vector or lineal form of the Eulerian exponential sin and cos relationships in 1844. Clearly showing the sign i has only a distinguishing role in its derivation. The careful and strict observance of the formalists gives the same or analogous result.

The exp(ix) where x is a radian I will call a twistor vector . Rexp( ix) is a twistor with radius R and can only be combined with like twistors. For twistors to be like they must have the same R and the same argument. If they have any difference in argument  that is not multiplicative they represent different twistors.

If they do not have the same R  the may form a frame by the rational parametrization, but not by the cos sine parameterisation, because they will be concentric.

So for the cos sin twistors to form some basis they must be of the same R but have a different argument..

All of this is predicated on the existence of the trochoidal curve of motion and the motive that follows or drives this motion instantaneously.

The General notion of Force

((((((((((((((((((((((((((((((((((((((((((((((((((((

The problem with force is it is non specific, even today. Newton defined a measure he called vis which was a differential form of Hooke's law. Hookes law is specific to springs, but force is a more general vector type.

The metaphysics of acceleration required a cause Newton called motive. This, like celerity entered a body to hasten it. But when Boyle  et al. studied fluids they found a motive disbursed throughout the material they called pressure. The concept of pressure and motive are identical, but the measure of pressure was counterbalancing: a force against the area it acted upon against a pressure and the area it acted on.

It is clear that pressure is a more specific notion than force, being MULTI directional and appreciated by its action on a surface. This in fact closely matches Newtons description of an action on a body that produces a vectored acceleration. The notion of pressure is a better more satisfying notion than force. We can accommodate the so called four forces into it.

If we have a pressure, we really do not know the cause. It could be an electric motive, or a magnetic motive a mechanical motive( including gravity and gas pressures) , a nuclear weak or a nuclear strong motive, not forgetting a thermodynamic motive, that is heat pressure(temperature) and expansive contractive motive.

 Since we do not know the distinguished motive we put them all in to the equations. Effectively they are weighted pressure terms, their proportional effect either guessed or discounted. In this way we determine a weight for the action of each motive by approximation and judgement of the observed behaviour.

What are we looking at in terms of pressure? It seems to be a kind of weighted mixture of motives,which we have distinguished into 4. They act radially , but seem to have a vectored maximal for the electric and magnetic motives. Others seem to be uni vectored but environmentally determined. Others seem to have their own innate vector action, and all are susceptible to scale, except the electromagnetic(fluid dynamic) descriptions.

Is Pressure a vector?

We find it difficult to apply the notion of a vector to a multioriented magnitude. We tend to call them scalar potential fields. Theirs is a theory of conservative fields that defines this precisely, but essentially it is simple. If something is everywhere and in every direction then it is a potential scalar field. We cannot vectorise it. our notion of vector as a line label does not apply and is misleading. However, in spaciometry i have called these types of fields compass multivector networks, and written a few posts on the topic. These are the basic or fundamental seemeioon algebra, that is a Grassmann point A;gebra.

So i am going to be looking at how a Grassmann point algebra compares with a conservative field theory. These types of fields rings or groups are looked at as topological spaces in which the usual way to measure is to use the real number measuring tape and pythagoras theorem. However more unusual "metrics", rules of measuring, can be invented to help by analogous reasoning in other areas of comparison or specification.

P = α pe + β pm + γps + δpw +  ζ pl +  θ pt + μ pi +  ς pd  


Which is electric, magnetic strong weak, lever, thermal inertial and deformation pressures weighted.

Now we are accustomed to thinking in terms of gravitational pressure, but i have deliberately left this out of the general pressure notion except in the sense of a balanced lever or a mechanical "force".

The fact that things fall with acceleration is unexplained. or inexplicable out of context, but within the context of a general pressure notion "gravity" may be interpreted.

Why do we include the others? Each one creates spatial motion of acceleration, some damped. Gravity therefore obscures the effects of each of these others. It also obscure the fact that at least 3 other pressures could be used to define relative density and so mass as a product concept of relative density and volume.


We have to acknowledge, as in the case of boiling water, our environment feeds into our local measurements. Gravity as it is used is a catch all because all our science has been defined against an assumed global characterisitic, and so in universal contexts we need to account for it. It is merely an accounting correction which we may be able to eliminate by more strictly defining these others

The empirical data suggests that pressure acts radially and spherically.

It is not possible to isolate a spherical action from a radial one, nor should we fall into that mistake. Therefore , using newtons resolution of his reference frame, my model must contain radial vectors and circular arc vectors or twistors acting in the surface of an expanding sphere. This is the fundamental structure from which I can resolve a tangential vector! In fact in 3d it will be a tangentially expanding circular plane which provides tangential vectors to the spherical surface relative to a given point, but also arc twistors in that circular plane. These can be resolved into tangential vectors to the circular plane in the plane

A spherical  pressure thereby exhibits a fractal functional relationship in detailing its likely vector structure. However this vector structure is not realised until a test particle is placed in a pressure surface, so the description is Potential! Because it is not a vector it is called a scalar potential, but this is not explained clearly, rather it is obscured behind symbolic relationships. Probably because no one really understood what it meant rhetorically, they could just give examples in definition.

For a scalar potently to be useful it must be measurable. So a scalar potential exists in a topological space.

A simple topological space is an  inelastic line. Now if I have another line that is elastic, I can compare the 2 and consequently recognise deformation. The elastic line is a topological space, but it is dynamic, which means I cannot describe its measured behaviour except relative to an Inelastic line. This means my visual sense is relying on my kinaesthetic sense to describe and distinguish an observed behaviour. If I did not make this comparison I would not be able to measure reliably, because I would be unaware of the elastic nature of my Metron.

Given this, I may now define a scalar potential for the elastic Metron. By making points on the elastic topological space I can refer to the extension of the Metron under different kinaesthetic pressures by noting the displacement of the distinguished points on to the inelastic space as a ratio. The inelastic ratio data is a scalar potential. It is a "measurement" associated with a point. Given the elastic Metron, and the correct kinaesthetic pressure I can read off a scalar definition by comparing distinguished points against the inelastic scale and noting the assigned ratio. We forget that all measurements are in fact ratios normalised.

Of course the inelastic line has its own orientation and so I have turned a potential reading into a representational vector in the elastic line. The vector is in the elastic line because I can orient this extended piece of elastic in any orientation on the surface of a sphere with the radius given by the topology of the in elastic line. It is the elastic line that has realised the vector potential of the topology.

For a pressure we realise the vector potential of a topological space by placing a surface that is translatable and orient able in it. The topological measure in the space is given by some function of the coordinate frame established appropriately in the space to provide a Metron, and in addition a Pythagoras rule for trianglesl

The Pythagoras rule is fundamental to our apprehension of how straight lines behave in space. It is not so much that it gives us a metric as it gives us a relationship between 3 points in a plane that is universally true for straight lines. The Euclidean notion of a good line goes beyond it bing straight. It defines a difficult but supporting concept that is a plane. While points can be defined by the first 2 given, a plane can be defined only by dual points and only then can a straight line of dual points be defined. Thus a straight line implies some plane and 3 dual points connected by straight lines specify it.

Of course dual points imply intersecting spherical surfaces, which is the fundamental superstructure or Hupostasis of Eudoxian and Euclidean ideas/ forms.

So now these forms or spaces can be specified by some reference frame. And some function based on this reference frame can specify a scalar potential in that space.

I have just used an example of Hookes law, let me now use an example of an inverse square law.

Specifying in polar coordinates makes this relatively simple. The scalar potential is (1/r^2 ,€) if the reference points are ( r,€). This is in the plane.

How do we now turn this circular potential into a vector field? We use newtons reference frame and resolve into vectors using newtons parallelogram rule, avoiding the mistake of giving primacy to the tangent. The tangent is a resolved vector in Newtons framework..

So now let us apply it to a spherical pressure potential.

The nature of potential is spherical so I can expect to see changes in potential radially therefore I can draw a vector radially to indicate a direction of potential change. Now I have a choice ofndrawingnanvector whose magnitude is the potential at the point or a vector which indicates the potential difference.

Placing an object in such a field of Vectors allows us to use the resolution of vectors . Thus we find that circular twistors counteract but tangent vectors do not for a spherical curved object. The material resists by Twistorque forces  that cancel, leaving the tangent force( derived) to combine with the normal forces to push the shape  or attract the shape. The force vectors for the potential field act as if the body was enclosing or embedding the field within its volume.  Thus we have to calculate the overall effect of a pressure field on a body from all the pressure effects, not just from the surface pressure effects.

Because of the spherical potential field the pressure vectors will act on a spherical surface differently to a flat planar surface.mthe resolution of the vector fields will be different.

Now, so far I have only considered the lever effect of a pressure field. There are other pressures within a pressure field. I need to know the potential field for the electric and magnetic pressures within a general pressure field. In addition the resistive or reaction pressures themselves differentiate the electric and magnetic pressure effects. Triboelectric and tribomagnetic pressures are resultants of a general pressure field. The other contributory pressures also require their potential idles so thir effect cn be considered.

Anyone of these many component pressures acting through a body surface, and throughout its volume could compound  to effect the dynamic stability of the combined system. The potential to rotate a body is therefore always present, and overwhelmingly so. The naturally resultant motion on any object under pressure would therefore be to follow an arbitrary trochoidal path. "Damping" of rotation or forcing of rotation may lead to a smaller or larger radius of curvature to the resultant motion.

 The ballistic description of motion often incorrectly identifies the resultant motion of a missile as parabolic. It is in fact elliptic, because the object would return to its starting point if not impeded. A missile would have to exceed the escape velocity of the earth to get anywhere near being parabolic. However, it is the collective experience of these elliptical paths, more generally trochoidal paths, that we call gravity.

As I have hinted at, these general trochoidal paths are the resultants of a spherical potential pressure field  consisting in many components.

The notion of a potential field from which we compose a vector field should not obscure the fact that these are topological models of a dynamic experience! In this the very topology is dynamic. It is exactly saying that Hookes law is dependent on the substance in which it is applied, and for how long that substance remains stable in its configuration, and in its position!

A pressure field varies dynamically. In fact from studying weather we know that we have to conceive of a system of pressure fields in dynamic relationships, and at different scales and levels all fractally entrained. The most energetic of these systems we call turbulence.

Turbulence is a matter of energy driving rotational regional motions a at all scales with fractal damping mechanisms back feeding through the complex system resulting in diffusion, dissipation and transformation.

The deformation of space matter involved in these turbulent conditions reveal the application of fractal damping, or inertia in maintaining some form of regionalised structure at each scale. We can only account for this by means of conservative actions. Conservative actions allow forces to react, dynamic situations to be in static equilibrium, inertial actions to be proposed, momenta to be maintained and opposite or contra actions in general to be expected?

How we frame our conservation laws defines our models of spacematter interactions, but in general it is reasonable to divide any magnitude into 2 contra magnitudes thus for any pressure field there is an anti pressure field. How that exhibits itself to our senses is not defined by conservation laws, but by experience.

When a spherical pressure potential acts on an area or in a volume it produces twistorque vectors as well as radial vectors. The acceleration radially is a(r)_r, the acceleration vorticularly is @(r)_r which indicates that the combined motion is a funcyion which is dependent on the vectors and the radial distance from the source of the pressure potential .

The resultant force for a given volume with crossesction A will be RAm+TAm where R is the radial accelerative motive. and T i the twistor accelerative motive when resolved and summed. This is for a given radial with a surface neighbourhood on the sphere with Area A, small enough to be approximated by the tangent circle at that radial.

The twistorque forces are usually not accounted for. If they are they are set to zero, implying perfectly elastic materials circilarly but perfectly rigid radially and tangentially! Since this can hardly be the general case, we should expect twistor vectors to be non zero and there to be net twistorque related to the viscosity of the medium under pressure and the wave propogation properties of the medium as a function of that viscosity as a tensile medium.

Finally the torque of the vorticular forces must be a function of the energy required to conserve matter in that action with that viscosity. The model is therefore fractally complex

Grassmann Point Algebra and Scakar Fields


Grasmmann point algebra
http://jmanton.wordpress.com/2012/09/03/introduction-to-the-grassmann-algebra-and-exterior-products/

basis

http://adsabs.harvard.edu/cgi-bin/nph-abs_connect?bibcode=1891Natur..44..105T&return_req=no_params&selfeedback=1&use_title=YES&use_kwds=YES&return_req=feedback

http://mathoverflow.net/questions/22247/geometrical-meaning-of-grassmann-algebra
http://www.stebla.pwp.blueyonder.co.uk/Whitehead.html
http://books.google.co.uk/books?id=bUEcbyfW55YC&pg=PA4&lpg=PA4&dq=Grassmann+point+algebra&source=bl&ots=Sp6kEc8vDw&sig=103dFwhOCuFoWLXJ-EkW3U2KXYE&hl=en&sa=X&ei=ysOMUd-UHcjFObT2gKgL&ved=0CDUQ6AEwATgK#v=onepage&q=Grassmann%20point%20algebra&f=false
http://mathoverflow.net/questions/102917/urge-reason-for-inventing-interior-product-grassmann-algebra
https://sites.google.com/site/grassmannalgebra/bookandpackageversions


So I start wit an undefined scatter of points. I distinguish two points A,B. they are completely arbitrary, except the tool I use for a synthesis process imposes some limitations.
The first construction action I define is to use a pair of dividers and fix them on A  and B. this gives me an instrumental  copy of something  I will call displacement §
§AB is an algorithm or method. It does not affect the points per se, it affects the observer and the tool used.
Now does it make sense to use a divider? Only in the plane or in contact with an in elastic surface, which nevertheless is markable. Already you can see how this method of analysis/ synthesis sets certain constraints to be achievable or pragmatic.

§AB makes sense only in a certain set of circumstances.

Confining the observer and actioner to those circumstances enables me to write
§AB = §BA.

But in fact this says nothing about points. It says that the measuring instrument ends up in the same fixed configuration. Thus we immediately fall upon the notion of an exterior algebra!

Leaving the points to one side, and concentrating on the dividers I can compare gape and set up an additive Alebra of gape. This is an exterior age ra, a prior one necessary before I can develop a concept of displacement as gape, and the practice of measuring using gape.

So dosplacent is an exterior algebra associated to the scatter points. It does not synthesise anything from points entirely of points. This observation is at the heart of the notion of an interior algebra and it is the notion of closure. This idea is that if we are talking points then everything should be about points? Later we will see that the reatriction is even stricter.

So a second synthesis I could do is construct aset of points from the scatter set which are centred around A with the same displacement §AB. This forms a spherical surface of points. However,nab spherical surface is not a point, so now I have constructed a new object that is not a point. It is an exterior topology even though it is embedded in the scatter points.

There is an algebra constructive on this surface, but we tend to call it a hyper geometry. I just call it a Spaciometry. However it cannot be constructed without some prior Spaciometry of these spherical shells,
¢AB is not the same as ¢BA.

However the two do intersect and form a new collection of points called a circle which is the identical form for both
So €AB = €BA.
We can construct an exterior algebra on this circle  between the points, which is again embedded within the scatter points but does not include  A  and/or  B .

The first interior algebra, one which is closed for all it's elements, which is set talk for it includes A and B  is the set of points on any curve that goes through both A and B.

However, since we call this a curve or a line it is not a point, and so it is called an exterior  construction, even though it is really interior!
So what is an interior construction  for A and B ? Currently it is defined as the midpoint , a single point on the exterior product straight line between A and B.

This confusion is not uncommon in a developing subject. At the level of a point algebra it seems a moot point about which construction is exterior and which is interior, but the notion of multiple and extensive use of a Metron underpins it. A line is clearly this kind of extensive multiple form made of points..

To distinguish the two the + sign is used
So p(A+B)=(A+B)/2 = p(B+A)

This construction in the plane is the standard line bisection, but in 3d space it is a lot more complex, but still true..

The upshot is that most construction actions in Grassmann analysis produce or invoke exterior algebras. A few with stringent conditions create interior algebras.
Now I have to discuss the use of the contra notion and the – signal.

Associativity, Anticommutativity and Antisymmetry in Grassmann Algebras

http://www.stebla.pwp.blueyonder.co.uk/papers/Euclid.pdf
Normans Wildlinear Algebra series is a great introduction to Grassmann algebras. But his projective geometry is also a Grassmann algebra, something Robert Grassmann particularly emphasised to Hermann in his 1862 revision of Ausdehnungslehre.
Finally the hyperbolic geometries are also Grassmann algebras.

There is  an expression revealing Grassmann  claimed to have realised this in 1844, and it is clear in 1862 that he had hoped for collaborative researchers to have advanced his ideas into a new root and branch formulation of the house or tree of Mathematics and the sciences. By and large. Peano and Whitehead, Gibbs and Clifford have managed to do this, with Dirac being the jewel in the crown.

 I have struggled to appreciate what he was thinking, much of which is inspirational and subtle in his German originals. The notions he struggled with are well put and far reaching, as expressed in his Vorreden. One of the notions is the notion of "contra".

Hamilton uses this notion to build his model of conjugate functions, otherwise called a science of pure time. He struggled also to break free from Cartesian coordinate thinking, and created much of the vector terminology. He acknowledged that Hermann was ahead of him in this game and redoubled his efforts to catch up.

The notion of contra underpins the sign conventions. We can start with Brahmagupta who introduced his fortunate and misfortunate numerical and algebraic and also aphoristic designations.
I can go next to Bombelli, and. Napier and Stevins for bringing it into European thought, against opposition by Hellenistic protagonists.
Then to Descartes Newton and Leibniz for making it de rigour against considerable antipathy , and then I think to the Grassmann's and Hamilton for developing the ring and group theoretical superstructure to apprehend it.

Contra, as a notion applies not to the elements of objects things, etc, but to the subjective processing of the observer/actioner. It is that individuals point of view or perspective or even apprehension that is being modelled by contra..

Often I read some researcher stating that there is no negative numbers in our experience of reality, or space. While that seems prima facia to be a truism, in fact it is a misapprehension of such thorough obscurity that one cannot explain it to one who is of that opinion. One simply has to learn that our models are only a small eclectic choice of all possible and impossible models! And then one has to apprehend the group or ring structures of all these different possible approaches and models, and how that affects ones appreciation of any supposed difficulties.

Eventually one may learn that our models programme our abilities, behaviours and reactions and viewpoints.

On a purely utilitarian criterion, some models are more useful than others. Adding the contra notion has proved to be extremely useful in all sorts of comparisons and judgements.

Sometimes we get stuck in one way of thinking, so contra means only one thing. But usually, by analogical thinking we recognise contra in many descriptive experiences. In this blog I am looking at contra in terms of spatial orientation and spatial symmetry. Because it is a spatial comparison I need the notions of associativity and within associative relations the notion of commutativity of the elements of the relations.. It is this commutativity which invokes the analogy of symmetry, but not just as a metaphor: it is a cognitive apprehension of the spatial relationship of the elements! In other word we have to quit being objective and examine our subjective processing of the actions, notions and relations , the ennoia going on with our mental engagement at the very least.

Symmetry derives from summetria, a fundamental Greek notion of group structure! The term seems to have its origin and weight in architectural concerns, where it's principal notion means a " common" or" gathering" measuring scheme. Thus a Metron , a single measure is only part of a summetria. A gathering of Metrons/ Metria.

The underpinning concept of a summetria is a beautiful or harmonious form. The standard example is the human form. This is not a conceit, but an important design principle. Humans feel more comfortable if their surroundings do not jar with their familiar Metrons. Put another way, architecture has to have human dimensions!( Metria).

The analysis of what is judged aesthetically pleasing reveals a " collection" of Metria, different scales for different regions of a form.nfor the human we have phalange size, gfnger size, hand size arm size, abdominal size head size etc. these sizes or Metria are tabulated for the aesthetic ideal. You will note Da Vincis man in a circle and square as a geometrical exemplar of this.

During the renaissance, artisans studied these summetria, that is collections of measurements to learn how to compose the aesthetically ideal forms. Such forms therefore were designated as being  " of summetria", that is derived from symmetrical considerations and meditations.

Much, much more can be said about the proportions derived from the summetria, not the least being the golden ratio, but the thing I want to note is the fundamental model a summetria presents of a group structure.

The elements , the closure, the combinatorial actions are all exemplified in this concept. For example, any building designed by these principles will scale, so the ratios and proportions are fundamental distinguishes of the Metria within the group, and to what the combinatorial resultants must conform! This means that the group has a quadratic action on its members when deigns are considered and elements combined. The golden ratio forms give a simplified superstructure to get designs " right".

I use this derived notion of a group distinction process: a common bond, or binding or boundary defines the elements of a group. It also defines the essential meaning of a summetria.

So how can a group be anti symmetric? It can't! What is anti symmetric is the action required to keep resultants within a group.

A symmetric action always produces resultants within the group. All closed groups are therefore potentially symmetric. An anti symmetric action produces resultants that require a contra element in the group. Thus a group has an extension which is contra to the group and these are combined to form a new group which now is symmetric, but only through including anti symmetric resultants. The group can be partitioned with no overlap!
The action that produces an anti symmetric result is often referred to as symmetry breaking. What this means is that sometimes our models are forced by us to be symmetrical, when the reality we live in is antithetical to that constraint. We have therefore to submit to empirical data and modify our model, no matter how beautifully aesthetic it is!


Before we can start to synthesise we must have analysed the "finished" or goal form. When we have that basic knowledge as default information we can use Grassman's synthesis algebras.  His algebras only have to specify the elements and how they combine. The first aspect of combination is " pros" that is placing in a relationship or association by putting an element relatively befor another element.

In space no one can hear you scream: "what is 'before' something else!" the answer is not spatial but sequential, and it is sequential in process. So when I describe a as bing before b I can interpret that spatially only if I establish a sequence of processing. The spatial notion of "before" or " pros" derives from this sequential action. This sequencing is totally subjective.

From this notion we also derive the temporal notion of before in time. In fact Hamilton's synthesis of algebraic arithmetic is based purely on this notion of " progressive" sequences, which he identifies with pure time .

So associativity is about sequencing the elements. Because we start with 2 elements we mistake this point. The 2 elements are associated in a sequence. Because of this we can actually have as many elements as we want in an associative sequence.

Commutativity comes when we commute or change two elements in the sequence. In fact it is any two elements in an associative sequence whether adjacent or at different places in the sequence. Most Group structures follow Euclid and deal only with the two sequence and the three sequence. When we look at commutativity in the three sequence we introduce brackets to emphasise what are commuting. The bracket creates a new object, it is not an element it is a synthetic structure made up of several elements in associative sequence. Beyond that the bracket is left unspecified.

However we can now isolate which elements we want to direct our focus onto. These elements cn then be synthesised indecently, and then the resultants combined in a synthesis. This creates a problem. Because if my combinatorial action is specified on points say and produces lines, can my action apply to lines?

The answer is no if you take an anal approach and yes if you take a goal oriented approach. If the goal is to construct a parallelogram, then we must use the points and the synthetic products of points to do just that!
Anally this is horrible, because as a computer we have not given it enough information! It takes a demonstration of what to do, what to vary etc to specify the construction! However for a trained monkey this is quite doable. The difference is that the monkey has analysis or analytical skills to help it learn nd try heuristic ally to reach the goal for a banana!

The use of these additional subjective processes in this way are what mathematics and science had ignored until the Grassmanns.  They did not exclude the observer from their prt in the construction process.

Effectively the notation takes on the role of human software, or a set of construction instructions like those found in an IKEA flat pack!

Now, we can make sense of Antisymmetry and anticommutativity. They represent the very real differences that occur when you attempt to construct something in the real world. If A and B are processes then sequencing the processes AB does not give the same result as sequencing them BA in general . It gets worse! If a,b are elements in process A then regardless of elements in process B changing the sequence of elements in process A will change the resultant of AB in general.

Thus we se that the commutation of associative sequences is fundamentally anticommutative .

This is not true of representatives however. Representatives can always be commuted. This gives us a general distinction between representative terminologies and fundamental constructive element terminologies.

In general associativity as defined is not going to necessarily give the same resultants, because the bracketing actually implies a different process order within an unchanged sequence order. The reverse polish notation highlights this very effectively, because it shows how the brackets actually change the process sequence in a computation. The BODMAS rules are an attempt to "code" between the two, so as to retain some aesthetically pleasing notation. This is why Grassmann analysis was so fundamental. Educated people were putting aesthetic form before constructive function and getting everyone in a constructional muddle.

It was not easy for the Grassmanns to see through this aesthetic fog, and Justus made some fundamental mistakes because he could not see where he needed to go! But by continued diligent effort and the rise of computer coding we have made the breakthrough, and the Aesthetes have even started to pretty it all up again!

Grassmann defined AB = –BA because he found that the sign portrayed this difference in orientation symmetry.. The difference in orientation is subtle . It is not as simple as  negative or opposite. It involves not only orientation, but also rotation of the observers point of view! It is non intuitive therefore, and took a while to apprehend.

Hamilton had given the sign more thought than Grassmann and came up with he word contra to deal with its subtleties. However, by the same token Hermann explicitly laid out its subtle effects but used the same word Zeichenwechsel to convey its subtleties. So the notion was lost between the two. Lost in translation. Both Hamilton and Grassmann are not referring to the rules of sign, they are referring to behaviour from which we derive the rule of sign. Thi behaviour is that of the trigonometric ratios of the radius of a circle as it is projected onto the diameter of the circle.

But contra also refers to reversal of process as well as reversal of orientation, so the changing of "sign " is not just about flipping from one sign to another, it is more complex than that! The reversal of sign is about the orientation of a form in 3d space relative to an observer.. While there are no negative measurements in space, there are oriented measurements. Sign is the rudimentary mark we use to alert us to orientation in space, both ours and the objects relative to each other.

The anti commutativity and anti symmetry in Grassmanns algebra relies on one fundamental process. Designation.
Designation is the same as distinguishing, so if you have  two points you can designate the first one A or B, which determines the second one .mthis automatically means that A,B is not not distinguished as a pair!. However, the relationship between the 2 designations is that  Streke AB = –{BA} Streke.

Most of us miss this distinction, because we are waffled around it by our teachers. The sequence position is sacrosanct so if I associate  2 points in the sequence 1,2, then designating them A,B is different to designating them B,A. Thus when Grassmann writes in the vorrede to 1844 that he had been meditating on the negative and noted that the Streke AB is the negative of The Streke BA, he does not mean that we are looking at a fixed Streke in space. The Streke AB  becomes the Streke BA very simply. We redesignate the points. But this simple redesignation has a profound spatial effect; it reorients the observer! Alternatively it flips the line round ! It does this either in the plane or out of the plane, around a centre of symmetry placed anywhere in space. In fact the flip can be specified, but it is usually left unspecified?

This is obscured by the modern vector treatment of Grassmann. A point is designated by the observer. Justus set some ground rules. Points should be designated in alphabetical sequence so 1st 2nd 3rd maps onto A, B, C, and direction follows alphabetical order, and construction is done in the same order, and crucially angle measure is done in the same order!

Changing one of these rules changed the resultant construction. Everything was specified, the only free thing was the placing of the first 2 points and the drawing of the lines. Points therefore had no specific position on the page and a line had no specific orientation or direction  or length until it was designated. This by cycling the designations different constructions could.be done.

Suppose for points A,B,C two lines are produced . Designate AB as a and BC as b then producing a parallelogram from a and from b is the same as producing it from b and a. However AB means more than that. It specifies the sequence of construction , the position of the elements relative to each other, the orientation between them, the direction the line is drawn, the direction the angle is measure in construction . BA therefore contains a specific instruction to put b in the first position. That first position has an orientation relative to the observer. Thus b has to be put in this orrentation, and thus flips the construction, because the angle construction has to be performed in a certain direction, that is clockwise or counterclockwise. Justus specified all angle measure construction to. E done counterclockwise. We still have that tradition, but it is the missed out elementary factor in the explanation of anti commutativity and antisymmetry.

So the notation AB means construct a parallelogram using a,b with b rotated counterclockwise from a. Thus ba dictates that a is rotated counterclockwise from b. most vector treatments ignore this fundamental instruction. Instead they place before the student the bald axiom AB =  –BA  and do a bit of sophisticated algebraic manipulation to produce the result.

Grassmann was actually stunned for a long time when he arrived at this result. He was stunned because the notation was telling him that he was missing something in his understanding about space. It was simply that synthesis or construction has a sequence and a specification. To achieve a consistent resultant everything must be specified. When this is done as Justus had insisted throughout the primary schools in Sczeczin(Stetin) you reveal a fundamental process superstructure which Grassmann felt was embedded in 3d space. It shows itself over and over, because in constructing the parallelogram the same anti commutativity stands out!  It stands out because of the precise, rigorous, anal., autistic specification of everything! Thus I think it is a consequence of our subjective processing rather than space. Grassmann's " Lebe und " Seele" metaphor is in my opinion the actual foundation of this curious antisymmetry and anticommutativity.
.
Without this superstructure most of the anti commutativity  notion collapses, and with it antisymmetry.

What is confused about rotational motion?

Compare this clear exposition of angular displacement
http://hyperphysics.phy-astr.gsu.edu/hbase/rotq.html#rq

With this seemingly clear exposition of angular velocity.

http://hyperphysics.phy-astr.gsu.edu/hbase/rotq.html#avel

And then notice we somehow end up with tangential acceleration and tangential velocity!

This always puzzled me and no one really could or would explain how it all fits together.

Firstly, do not do as I did for most of my life, think we can only measure with a straight ruler accurately.  Somehow I  down graded a tape measure to approximate , substandard  as a method of scientific measurement! Also , degrees were marks on a scale on a protractor and different from measuring length. You " could not measue the length of an arc" you could only approximate it!

This childish version of metric theory, childish because I adopted it as a child, obscures what is in plain site: metrication is comparisons of like things , thereby forming a ratio or logos by counting!

With the right mechanical tool and method I can establish a measurement system for any object by comparison.

So by forming a circular disc nd rolling it , without slipping on a straight edged marked ruler I can measur its perimeter as accurately as i want.
Because I m using the same ruler I can compare the measurement of the arc with that of the radius.

We can now form the ratio S/r and this gives us a metric for the perimeter that is specific to the spread of two radii. We do this because of similar figures. This makes the spread measure independent of which circle we measure it in.

All our fundamental measures are ratios.

The reason for this use of ratios is onvrnience of scrap. Tools for measuring and processes of measuring need to be as straight forward as possible. The tool for measuring spread is a protractor or a set of triangles called "set squares" or carpenter's squares. .

The secret of ratios is the magnitudes they address.  So a ratio of 3 magnitudes has 3 parts order. The system of  pairs of magnitudes drawn solely from these 3 introduces sequence and combinatorics and increases utility.

These 2 magnitudes are also related by the process of rolling a disk without slipping. This enables the perimeter to be counted in multiples of the radius. It also connects the centre of the disc uniquely to the distance rolled!nthis centre traces out a parallel line to the surface rolled on, providing it is kept in a fixed angle relationship to the surface?

The movement of this centre point is quite revealing. It is dependent on the geometry of the space the disc is in contact with. Thus a flat plain means this point copies the path the circles perimeter traverses exactly, but an undulating path creates what looks like a copy but in fact traverses a shorter or greater path distance , and in physical terms results in deceleration or acceleration of this centre point? SImilarly, if the disc wobbles then this point descends in height relative to the surface, relative to the highest height. And tends to travel in a circle of smaller radius , than any point on the perimeter.

The paths traced out by distinguished points , including the centre are called trochoids or roullettes. In a physical sense, these paths, taken as a whole collection trace out the complex dynamics of vortex surfaces and volumes.. Thus the behaviour of rotational momentum or even rotational velocity of a physical body is poorly understood and confusingly described.

I have spent years meditating on Laz Plaths trochoid app to come to this view. In a very real sense it has changed my apprehension of space and reality as i experience it.

The trochoids have helped me develop an intuitive appreciation of fractal apps and what they sculpt. They help me to understand what relative rotational motion in space can do and be as a dynamic system. They help me to understand gyre in its most abstract sense. And they are always surprising, beautiful and intuitive even at their most complex.

I just needed Grassmann to awaken me to the fact that all geometries are formal, that is generated in my head as perceptions, as definitions, as ideals ; so that I could realise that all geometries are so we can interact with space to count it . We can count it by fixed forms , and we can count it by dynamic forms, but the ultimate dynamic form is the expanding and contracting sphere. Such a concept encapsulates all possible forms, and I call these forms the Shunyasutras! Some of these Shunyasutras are living organisms by our reckoning. Others are invisible turbulences in the space around and in us. All can be modelled by this amazing trochoidal dynamic we have glimpsed in this dynamic spaciometric conception the living sphere, or vortex.

So I now accept that there are rotational force fields in space, whatever density that space has physically. And these rotational force fields are studied as vortices in fluid dynamics . These vortices exist independent of external torque. Torque is a model of interaction between these vortices and lineal space, in that the energy in a vortex can be tapped as lineal kinetic energy or as trochoidal kinetic energy.

The important concept for rotational force fields is the rotational kinetic energy they contain. This can then be dispersed as trochoidal kinetic energy.

My starting point is therefore rotational relative motion of and in space. From this motion all other motions can be derived. The miraculous thing is, this rotational motion is fractal: it is at all a scales..

So I nearly forgot about the instantaneous tangent! Lol!

Really , this tangent is the point of contact with the disc as it rolls! In reality, a wheel has a flat surface that contacts the road, but that just makes the applied disc smaller, and leaves extra annular ring mass as disposable kinetic energy for driving the wheel forward.

All the instantaneous tangents act in opposition to the "frictional " forces transmitting a frictional forc of equilibrium in the annular ring through the rotational force field in the disc to the centre. This is " action at a distance"!

Even though it does not obviously seem the same, this is precisely what happens in the vortices that form rotational force fields around every object in space, and which are mistakenly identified as gravity. Because space is something, and therefore not empty, this something is consytituted of these vortices at all scales , and properly account for field effects and field behaviours.

Newton investigated Descartes idea of a vorticular cosmos, but could not do the math! By this I mean his brilliant attempt in book 2 of the Principia to explain the planetary motions by fluid dynamics failed to be any where as accurate as his purely abstract model, based on immaterial and non physical point masses. There was something about the geometry in real space thst he was missing, and it was the existence of physical vortices. His model vortex based on cylindrical sheer forces treated as vanishingly small cylinders accounted for some but not all experimental observation. What he missed is what we call vortex shedding. These shed vortices carry some of the rotational kinetic energy away from a system, but also they store " hidden" rotstional kinetic energy within a rotating system. That is, there are vortices even within a vortex wall or flow.

Today we have the empirical data, the computational power and the visual modelling or CGI to reveal these effects of vorticular motion.. Evenso, many scientists still puzzle over how it works because they use the complex rotational momentum as their model, instead of the rotational kinetic energy.

Centrifugal, centripetal and Coriolis force.

http://hyperphysics.phy-astr.gsu.edu/hbase/corf.html
The assumption that an object in isolated motion moves in a straight line is one of those fundamental notions that affect everything!

An object in dynamic equilibrium either moves in a straight line or moves in a not straight line! There is only one straight line for any given orientation, but an infinite number of not straight lines!

As a first approximation it is easy to see why a straight line was chosen, because a straight line is in fact a tangent to any curve!

It is time to wake up!

Newton applied calculus like ideas, that is the principle of exhaustion in all his fundamental ideas. Like all Pythagoreans he believed the ordained and thus natural motion of any object was in the surface of a sphere. Thus his first law is introductory! By the third and fourth laws he has introduced what is called an inertial frame of reference. This means an equilibrium law governs all motions in that type of frame, either static or dynamic.in accordance with physical relativity, there is no way of knowing if an object is truly static. Thus every notion must hold for the dynamic situation also.

There was only one way to set out laws in this way and that was by a process of iteration! These so called infinite iterations or ad finitum iterations were merely iterations to exhaustion. Thus it was that Newton's method of Fluxions was born, the first principles of which he called his laws of motion.

Behind them lay an extensive geometric calculus and some inkling of Newton's vector algebra and vector calculus.

The general frame of reference that Newton, Huygens, and Wren used was in fact curvilinear. The use of the sphere as the defining law of motion necessitated it, and the study of Archimedes by these gentlemen enforced it. Archimedes and these other engineers and mechanics use a curvilinear or vorticular framework to describe the motions in space.

My only contribution to these noble principles is to add the caveat" of space!"

While I intuitively apprehended that the notion of space was non physical, it took a while to understand that it is entirly forml and in our heads as far as Spaciometry or Grometry is concerned,

Consequently I may set out any Spaciometry but only as a measure of my experiences of the dynmic motions I experience all around me. In this regard, the relation of relatively fixed magnitudes becomes indicative. For example if I fix ( or my proprioception unconsciously fixes) a curvilinear reference frame ( LRUDrotate) to the position of my torso, thrn I can move my head in it without alteration of the reference frame. But as soon as my torso moves then my reference frame rotates relative to my external surroundings!

In the same way , if one part of my external surroundings moves relative to mother I will experience that change as some kind of external force , even if it is an invisible element in my surroundings.

Returning now to the three headline forces, the tendency of modern physics teachers is to obscure the centrifugal force, because they are blind to it. Therefore they say that gravity is not balanced by levity! In studying circular motions Newton was aware of the balance between centrifugal and centripetal force. But the net effect in circular motion was so small as to make the actual implied acceleration very weak! The strength of the forces however are communicated through a tensioned rope! This is why Newton invented the centripetal and centrifugal forces in the first place!

Newton was able to detect and model the so called centripetal and centrifugal force, and to describe the tangential force, but the Coriolis force he did not emphasise, though he undoubtedly envisaged I. In his work on inverse cube root laws he was aware of the spiral effect of rotational motion.

However, he was also aware that he could never plumb the depths of all the laws hidden ith in the serene volume of the sphere in terms of motion! His categorisation of cubics was an initial attempt to begin this kind of analysis. He soon realised he had set himself a task beyond him!

This is why he became so excited by Roger Cotes intriguing logarithmic identity! 

ix = ln(Cisco + isinx)

This later was rediscovered by Euler in his anti logarithmic form
Exp( ix) = cost + isinx

Quite part from defining vectors in the plane in terms of polar coordinates, Cotes form provided Newton with a easy way to investigate all inverse power laws with regard to possible models of gravity! If only he could remember!

Unfortunately, by this time Newton's faculties were failing. He could see the fundamental importance to his laws of motion, and to his Principia, but he no longer had the powers to do the math! He relied and came to trust in Cotes ability to explore this as he would ! Unfortunately Cotes died, De Moivre did not fully grasp the ideas and significance, and Was busy on his own interest in probability theory, and Newton was incapable of communicating what he intuitively felt was lurking in this curious identity!
" Cotes, if he had lived, we might have learned something", was his eulogy of his bright hope now layer waste by the vicissitudes of life.

There is much that is written concerning gravity, but it is mostly mathematical. The great principles are encrusted in the dead weight of formulary. And yet the most remarkable achievement of he Voyager missions was not achieved by these detailed formulas, but by a determined application of the basic principles under iteration in a computer!

The iteration of the simpe laws provided a so called numerical solution to the problem of a space ship travelling in a many body system. At each point where the rotational force field, (now called the gravitational field! ) was most effective. Course adjustments were made to make use of the so called angular momentum conservation principle to slingshot the voyagers on. While it is nice to talk about the angular momentum principle it only works in this situation because the approximation to a point mass relative to a centre of rotation is so apt at this planetary scale.

In a sense, this is also where Newton got confused with his fluid dynamic model. He did not understand what factors to safely leave out of his calculations, nor indeed which ones to include! The Cotes identity offered a way to get a better fit for his general equation for inter body "gravitational" force. It did not however, offer any physical notion of what gravity was.

While I have defined twistors and Twistorque, and in he athematical way that is supposed to pass for insight, I sm conscious the math says nothing other than define your measures in this way!

However, the motivation for extolling these definitions is that vortices are evident physical phenomenon, and in particular, electro,Thermo and magneto complexes exhibit these vortex behaviours. The implication is that vorticular force field and vorticular kinetic energy will be unifying principles for all behaviours of motioning our experience.

The mathematical process of averaging, is based on the geometrical process of equating measures.

What is mathematical is not always physical.

Thus if I set a pendulum swinging and count how many swings it traverses while a mn on a bike rides from A to B. I can write a ratio AB:T
I now have to assume an unempirical notion: AB:T is a constant!
Once that is accepted I can use my geometrical methods to make all sorts of proportions.
Am I justified in doing this?
The answer is no in the long term, that is for T very large and no for long journeys. That is AB very large!. The reason is that these are all dynmic situations!  Thus for situations which are dynamic and ongoing we need a flexible or Fluxions method which looks backward into the past for extremely short time periods. It then iterates forward on these short surveys of past statistical behaviour.

So how did we come up with a universal law?

Newton assumed Absolute time, and uniformity of density in a specified region and continuity in change particularly of the surface of a sphere. Since all spheres are similar it was a no brainier that the heavens, being the motion of point stars in the spheres would all obey the same rules, modified by local conditions of course!

Newton believed that space consisted of variable densities, but the region in which the earth and planets moved was particularly rare! So rare in fact that he had difficulty understanding what the medium for force transmission could be! It is instructive to note that the idea of space being a vacuum is non Newtonian. Today physicist dodge around this early teaching of Nothing in the vacuum!

How could something so rare transmit the centripetal and centrifugal forces he felt in his tensile string experiments.? It is unlikely that Newton had an idea of exactly how small the force transmitted is. Also, his exploration of fluid dynamics was inconclusive, even though he tried all kinds of fluids. He did not make it seems the connection with the electro Thermo magneto complex of forces that we know of today, although, in keeping with his time he posited Gilbert's philosophy of magnetic attraction and repulsionas a model.

There are no records of Newton attempting to measure the Magnetic force.Hooke on the other hand is recorded as making such measurements and many others.

If Hooke had not fallen out with Newton, the 2 certainly would have come to some advanced conclusions! I have written a blog post on Hooke in which I explore how Newton took Hookes law of fierce and  extended it dynamically into his second law!

In addition, newtons first law reles heavily on Archimedian hydrostatics! Newton, apart from Hooke, did not fail to acknowledge his debt to other philosopher scientists and engineers.

Returning to the description of rotation
http://hyperphysics.phy-astr.gsu.edu/hbase/rotq.html#avel

We see the use of the statistical average based on the equating of geometrical measures.
The notions of velocit and indeed angular velocity are presented as closed systems, not dynamically changing ones. The systems have a start ad a final state, both of which are assumed to be constant!

Familiarity with these closed systems does not prepare you for the open dynamic systems in which we live. The wide scae availability of fractal apps helps to remedy this failing. In addition he trochoidal app by Laz Plath is the only one I know that gives you freedom of motion in an n dimensional vector or rather circle space of trochoids.

While these rotational formulae are interesting as initial models, they are inadequate as descriptions of our experience. For that you need a fractal generator app and a trochoid motion and surface generator.. With these tools you can do " rocket science" and even high energ particle physics!

Oh yes you can!

What is rotational kinetic energy?
I say we must equate it to the lineal keychain energy generated by the object rolling long a surface. If the object is not circular we can expect some variation in the formula process used, due to the object receiving varying impulse of tangential forge generation.
While this has the same formulary as torque I would apply the method of Fluxions or calculus to any descriptive determination.

For example the average angular velocity from stand still, contributes the 1/2 scalar to the kinetic energy formulas but this ison the assumption of a constant acceleration!

We have a lot to learn about "real physicl" rotational kinetic energy!
http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html#mi

Now, out of habit I guess, I have introduced the tangential force acting on a disc at a radius r .
F_t is the label for this tangential force

The problem starts with the radian angle ratio.
Ø = S/r

I know what S is, it is the arc length. It is also the distance travelled by the centre of the circle if it rolls on this arc perpendicularly to a plane surface.
The plane surface is a tangent surface and in it I can draw any tangent line. The surface and the line are not instantaneous! The point on  the disc which contacts them is only instantaneously in contact . In fact physically it is a line in a cylindrical surface that contacts the plane surface or line in the surface.

So, if I draw a tangent to a disc to represent the arc displacement it has to be of length S.

Now let's look at rotational velocity, at a constant velocity.
In time t that is S/t

Thus  ŵ = ø/t = S/rt

If we say S/r is the velocity , what velocity is it?
Well it certainly is not a tangential velocity! If that was the case then all velocities are tangential velocities!

It is the velocity of the disc as a whole as it moves over the surface!

I have discussed how that velocity, measured at the centre changes with surface geometry.

Well if the disc is fixed at the centre it has no tangential or bodily velocity, it has only rotational velocity. Tangential velocity is meaningless for a fixed rotating disc. All we can measure is rotational velocity and rotational acceleration.

So what is F_t ?

It is a constant frictional force applied tangentially as it must!

We can work out the moment of this frictional force but that is meaningless.
Now if I apply a force that decelerates a spinning wheel, I am not justified in using a constant frictional force to slow it down. The reason is the force is not slowing it down by deceleration but the kinetic energy is being drawn out of the system by conversion to heat nd kinetic energy of sparks etc!

The experiments have to be done more carefully than that. We have to stop a wheel spinning without friction and very quickly , within the distance of an 1/8 th turn, to avoid obvious frictional force contact. A negligible hook mass must be attached and a spring force measure applied through tht hook to stop the disc quickly..

However such an arrangement does not apply a tangential force over the whole stopping distance or arc length.

So the only course left is to see what force stops the wheel from starting to spin. This means we have to apply the moment to measure the stopping moment! The applied moment may be through some pulley system which transmits torque through a connecting axle..

Now we know the experimental set up one can reasonably ask, what has that got to do with planetary orbit!?

When we apply these concepts outside there empirical basis, we are asking an analogy. And when we derive certain formulae by analogy with geometrical form we may not actually have a physicl mandate to do so!

The early mechanistic philosophers really did think these wheelers and spheres existed in nature , maybe as spiritual echmical gears and levers. Newton showed his insight when he said I frame no hypothesis! , and elsewhere he allowed for forces to be impulsive instantaneously not as the models would suggest constant and mechanical.

The physical picture for rotation is therefore mre complex. We cannot presume ome tangential force or torque drives the orbits of planets or the motions of fluids., at least not without complicating the dynamics with energy transformations which will be electro Thermo magneto complexes.

No matter how rarefied a bubble of SpaceMatter contains and is this rotational kinetic energy for which moves restive to other such bubbles generating trochoidal motions nd transforming save into other bubbles of energy in rotation! It is ths transfer and turns flatiron of rotational energy that we experience as " force" or electro Thermo magneto complex behaviours.

When Newton looked at Hookes spring experiments he did not wait for the spring to reach equilibrium. In that case he would only have noted extension of the sping. Instead he noted the dynamics. He saw that the acceleration of the bouncing spring was proportional to the weight/ force on it. And he knew that the weight / force was proportional to the mass or bulk of the object. A little thought made him realise the force was proportional to a quadratic form. His own Fluxions method informed him that the finl length of the extension was a derivative of the acceleration , the second derivative. So he could see behind Hookes law to the dynamical world in which we live. But rotation was and is different. It is different because it is the only natural form of energy we can witness. It is a store of kinetic energy and a means of transfortion of energy into electro magneto and Thermo complexes in and of space.

If I set rotational kinetic energy to kinetic energy , by passing torque
I get 1/2 x m x (ŵr)² = 1/2 x m x v²

Which implies that kinetic rotational energy is scaled up when it is turned into lineal motion! The energy densit in a rotating form measures less than its equivalent dynamic form when in contact with a flat surface.. If the surface undulates that lineal kinetic energy will vary with it. It is as if the rotating form pulls linelenergy out of the rotating space or compresses it back in.

There is much to learn about Twistorque and rotational force fields!

Electro Thermo magneto complex
http://www.youtube.com/watch?v=5I4rxfnCtxY

Rotation is not as simple as the circle or sphere!

In my early definition of Spaciometry I defined spaciometric rotation as traversing the boundary of a form. Not all forms are closed, so open and closed loops are fundamental to describing rotations. I also defined a vorticular wrap, for a crazy idea I had of spiral reference frames! Later I realised that reference frames can be curvilinear, and in fact they have to reference points in space uniquely to avoid confusion and mental meltdown!

However, the fundamental aspect of motion around curved open or closed loops was established.

Since then I have got a better handle on affine and projective geometries, and twistors , allowing for a projective interpretation of rotation from a fixed point  or an axis in 3 dimensions.

In general we talk about centres of curvature to define arbitrary rotations and these can involve normal,tangents and oscillating circles and parabolas.

Normans series on differential geometry goes into mathematical or algebraic detail

However I want to start with Newton, because the elliptic error is best explained by his work in Principia on orbits.

The elliptic error is the common view that gravity acts centripetally to the centre of mass. Few know that gravity acts or behaves like an ellipse. Those that do ignore the second focus generally and do introduce an error into our thinking! The second focus is often in Empty Space! Either empty space has a mass or we are mislead in our thinking! In fact it is both!

There is no empty space and Newton assumed an elliptical vortex ( keplers discovery) for his derivation!

http://www.youtube.com/watch?v=abwoL-g8SWc

http://www.youtube.com/watch?v=4IYDb6K5UF8

The things we are asked to believe  become suspect as one becomes older and hopefully more experienced.

I do not recall doing any momentum experiments in my secondary level physic classes. I do remember doing a gravitational acceleration experiment. I am fairly sure we did mostly calculations of momentum after observing a demonstration.

So, what is momentum?

Well it certainly is not m x v!
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html

I can remember not believing that even as a child, just as I did not believe force was m x a. It took me a long time to actually stir myself to look at newtons work, nd the philosophical arguments of his era. That is when it started to make sense. I knew momentum was a kind of force that would not allow me to stop dead as a crashed through the finish tape! What kept me going forward was called momentum . What it was no one could tell me. I found Descartes called it a force, and said it was of god and thus conserved. Later Leibniz took issue with that. He knew from experiments that Descartes momentum, if measurable by mass and velocity was not conserved., or preserved, it changed tempestuously! However he knew Huygens had demonstrated that mv² was conserved by Pythagoras law! This was a living mathematical law given by god, as far as Leibniz was concerned and he argued that the measure of this living force was mv². It was living, and that is why it persisted. Every other kind of force he considered as a dead force, because it did not move anything!

Finally Newton defined Descartes measure as what it is a quantity of motion! MV was a vector sum concept, only really understandable by Grassmanns Ausdehnungs Groesse. It was precisely what Newton had in mind. The density in a volume sloshed about like a quantity of an accelerant he called motive. Mv  was the measurement of the resulting celerity. It was a vector sum of all the implied internal motions, and thus a quantity of motion.
Why did it keep moving with celerity? Because the body contained motive, and this motive was the cause of acceleration.

Then why did the body not accelerate? Because the motive was opposed by an equal and opposite motive, dissipating the acceleration leaving only constant celerity or velocity.

Now I am not sure if Newton commented on Leibniz vis viva, but his work formula based on his measur of force F = ma certainly confirms Leibniz notion of mv² being important and conserved, only the factor was different.

Well, as we know Newton's formulae won out even if his metaphysics did not. But , instead of giving us his heirs a better metaphysics we got fed on a diet of mathematical tripe! Only, we were not allowed to question the dry crumbs we were being given . We would have starved to death if we had not been given a hard dose of mathematical vitamins by Einstein! Most of us who have persevered in this dietary combination have ended up malnourished, gaunt and somewhat crazy. We can believe a few impossible things at the drop of a hat, now we are delirious on mathematics!
http://www.youtube.com/watch?v=W9EqU1_DXUw

Is momentum conserved? Lol!
I doubt it.

So let's look at the mathematics and the toy.
http://www.youtube.com/watch?v=0DR8emfSsF0

http://www.youtube.com/watch?v=YlkTBbFikU8

Ok ok ! Do now you have to believe momentum is conserved , right?

http://www.youtube.com/watch?v=ZH0fiaYGTYQ

Well not exactly. In all demonstrations nothing went exactly as we were told to believe or trained to look and ignore! The point is we want our Maths to be exact, which it can be, but the physics is not, because it can't be!.

http://www.youtube.com/watch?v=kNNLJwoEbUs

You see, if you question hard enough the ground shifts!

But , hey , we can not live without conservation of momentum, so despite everything just said, let's put it back right in there!
http://www.youtube.com/watch?v=T3m4UrJ3efQ

It is unlikely that there are any isolated systems in our universe, so these riles are approximations only ad that is why they do not match the observations. So all collisions do not conserve lineal momentum, we just apply the concept to all systems, and get an answer.

Can we believe the answer? Not always! NASA and rocket science knows that they have yo adjust their long range trajectories constantly, and then at the last minute! The 7 minutes of terror is a case in point!

Newton, when calculating centripetal acceleration derives a formula by successive approximations called Fluxions or calculus. He iterates down to the best approximation.

Newton developed a metaphysical concept called motive. It is based on ancient ideas but particularly Aristotles analysis of the Prime Mover or first Cause. Motive cuts to the chase, and says all things accelerate but they accelerate in opposition. This is the fundamental basis from which he begs us to start.
My argument is similar: if rotational motion is not the first cause, then there can be no motion. If you say lineal motion is the first cause then there can be no rotational motion. If you say variation in lineal motion is such that velocity is in all directions thrn you imply rotation as a prior cause of this , and if you say that parallel lineal motion has a variety of velocities you ignore the perpendicular motions that must exist to hod a body together in order for it to rotate.

Starting with relative rotational motion avoids all these issues because it allows all motions to derive naturally from the interactions of such forces.

The nearest concept to motive is energy. And it is this energy concept that therefore underlies all other notions , and is a better explanation than momentum for observed behaviours.. Underpinning that is the geometrical notion of Grassmann's Strecken!

One can explain all behaviours in terms of motive and it's effect in doing work on and in a body or region. This is why rotational,motive is so crucial. We know from the venturri effect that work done inside a body reduces the work done by the body externally. Thus the creation of bubbles will reduce the pressure exerted by a fluid element and will be seen as turbulence!

The work done in a collision changes the kinetic energy in the particles and they fly off or roll away or both accordingly.work is a vector or Strecken concept!

 r x F and r x p are vector cross products, properly associated with the moment of a force and the moment of a momentum.

While the formula is quite general it is most commonly applied to lineal force and lineal momentum. Moments are quite applicable in mechanical lever situations , but often they are applied to situations where they act as potential torque or potential angular momentum.. This is not taught or understood and so a lot of mathematical chicanery is done before your very eyes to get what they want or more easily can get by simple definition.

The potential angular momentum or the moment of momentum is not constant. It varies as r sinø usually. Similarly the potential torque is not constant.

The usual argument is to show that if you differentiate the potential angular momentum you get the potential torque.

Then the argument goes if the potential torque is 0 then the potential angular momentum must be constant
But we know the potential angular momentum is not constant so where do we go from there?

We actually have to go to instantaneous versions of these 2 vector forms, and the instance when the potential becomes actual angular momentum, and some thing or particle actually starts rotating. In that case p is no longer lineal momentum and its path actually has to be defined!

http://www.youtube.com/watch?v=BLdlfWT7UN0

It seems to be worth investigating James MacCullagh, and Green
http://en.wikipedia.org/wiki/James_MacCullagh

http://en.wikipedia.org/wiki/George_Green

http://www-history.mcs.st-andrews.ac.uk/Biographies/MacCullagh.html
https://archive.org/details/collectedworks00maccuoft
http://books.google.co.uk/books?id=QmzfJUShGYUC&pg=PA448&lpg=PA448&dq=james+macCullagh+irish+academy&source=bl&ots=AQ63EtFQQf&sig=Oeh-tKIke4wYmEP4gMIi4ia0WDo&hl=en&sa=X&ei=08jeUp3aNuWK7Ab2jIGABA&ved=0CDkQ6AEwBTgK#v=onepage&q=james%20macCullagh%20irish%20academy&f=false
 It would seem that Stokes, Kelvin and Boole represent irelands finest mathematicians of wold renown nd influence. Yet a deeper grass root strin of Irish mathematical and physical thought is traceable by its influence on Geometry and Electromagnetic theory.

Hamilton, MacCullogh, Samon and Fitgerald and Lamor buried themselves in forming and maintaining an Irish tradition in these fields.

We see the influence in the rapid adoption nd development of the vector mathematics inspired but Hamilton in the late 1820's and the wildfire spread of Quaternions . MacCullogh was able to use these ideas to positor postulate a potential equation which Hamilton described as the Curl of the vector displacement field.

We find Maxwell making use of this Curl in his theory of electromagnetism, as a straight development of Quaternions as proposed by Hamilton. Thus MacCullogh's ingenious idea transmits to Maxwell through Hamilon.

However it is not deemed of natural importance by others , who seemed to have opted for a settled science.

It is noted in Lamor's assessment that H A Lorentz a Dutchman, Helmholtz a Prussian who guided Hertz's research topics and Fitzgerald in Ireland who directed Lamor's research topics were the only 3 centres in the world seeking to advance the physical sciences and that through a more mathematical methodology.

To this we must also add Heavisides considerable efforts, based in England close to Maxwell.

Both Lorentz and Lamor conceptualised the Aether as containing a dynamic array of electrons. But electrons were non physical. In one case they were vortices of the aether described by the curl concept in Quaternion math, in another, they were a slip knot of strain in the aether, again an idea inspired by quaternion curl descriptions and due to Tait.
Helmholtz had developed a notion of the perfect vortex , along with Kelvin and influenced Hertz in that direction, looking for and expecting each vortex to be balanced by a contra vortex. Both Maxwell and Heaviside relied on the mathematical curl more than a physicl model, but saw it as a non physical vortex in the aether

The discovery of the electron enters into this background. Why it should be thought of as a physicl quantity is not clear unless the deep and abiding resentment of settled science and Victorian sensibility is taken into account.

To the rest of society, these theoretical postulations were preposterous. Lewis Carroll lampooned the new imaginary mathesis in mathematics in his influential books Alice in Wonderland! Lorentz was always seen as a crazy chancer who always seemed to pull it off! But he was always being corrected by one mathematician or senior sinister or another. It was his careless regard for the conservative opinion that gave him his reputation, but he more often than not got something right!

Lamor's theory was lampooned as being a non material explanation of matter! Only Hertz it seemed enjoyed some quiet respectability in his directed research. However he was reluctant to do the research, having other interests. It was the influence of Helmholtz that kept him on track.

Maxwell was derided for his theory and completely ignored by the establishment. Had he not had such an illustrious career he would have been openly attacked. Instead, his end is reserved that bitterness for his self styled acolyte Heaviside.

In the face of this social and professional and in some cases commercial hostility, these men kept blue skying it and making slow progress refuted by their contemporaries.

JJ Thompson, therefore provided the perfect patsy or fall guy to save face for the establishment and to bury these wild fantastical ideas. The relationship between mass and charge Thompson found , and that is all it ever was: a ratio. Now that is what it was and that is what it still is!

Using this ratio Thompson studied and rewrote the corpscuar theory that Newton and all settled science accepted. In a stroke of clever communication he transformed a ratio into a region containing a smaller region of opposite charge! The large region he did not name but postulated a positive charge. The smaller regions which sat inside the larger region to comply with Earnshaws instability calculation he styled as negative, because it was these that flew out LIKE corpuscles of light toward the anode in the Crookes ray tube set up.

All he knew was these corpuscles had a charge to mass ratio as per his calculation. He did not know, and no one knows if they were matter . But he could model them using a supposed unit mass for them giving a supposed unit charge, which by mathmatical manipulation made the mass of the region where they were initially stored relatively huge!

A bit more manipulation and you get a relatively huge mass with the same charge as a miniscule particle.

Lamor suggested he call this " particle" the electron, and eventually Thompson conceded, justifying Lamor and Lorentz at a stroke!

This particle was of dubious constitution. Was it matter or was it these strange vortices and slip knots?

It was only really decided on the back of Einsteins 1905 explanation of the photoelectric effect. By then all  opposition to corpuscles as particles of matter became meaningless it was thought. Matter interacts with matter!

Or does it?

Einstein posited a massless electron he called a photon. The only " proof" was the mathematics! In addition Rutherford was not happy with Thomson's atom. He dud not question the constituents, that was a minefield he did not want to enter! Instead he reeified the electron and named the proton. And together with Bohr's work came up with the planetary metaphor for the atom.

He had learned from Thompson to give the public a metaphor they could accept! Calling Thompson's model a sticky plum pudding immediately endeared their version to the students and later the public. Scientific communication had entered the sound bite age!

However it was done , the establishment moved toward spin doctoring and public relations management. The war and the propaganda effort made this move almost imperative. From now on science would be done behind closed doors in secret on a need to know basis. Scientific advances would be published by the " committee ".

In America powerful commercial and governmental forces were conspiring against the aether concept for many reasons, not least the distinction between a progressive American science and a stultified European one, in addition, those trained in America had a particular core of concepts that were identifiably American. The European trained scientist and potential industrial or governmental espionage agent would not have these concepts ingrained and so would give themselves away!

We end the 19 th century with questions and avenues for research. By the beginning of the mid 20 th century everything is virtually settled!

It has taken some of us a while to realise we have had the wool pulled over our eyes.

http://www.youtube.com/watch?v=PeVo4SCJa_0

http://www.youtube.com/watch?v=2uWps9LczH8
you should be able to understand what they are talking about , now and give a better explanation of the phenomenon/
Moyive is not "lost" it is just opposed!
Energy transforms, and one of the phenomenoa it transforms into is called Force!

James MacCullogh's collected woks

http://www.forgottenbooks.org/readbook_text/The_Collected_Works_of_James_Maccullagh_1000152110/61

The curl of a vector field has changed its meaning as Quaternions have been pushed out of the way by Gibbs vector concept..mhowever it is still the same calculation and it derives from the multiplication of imaginary or complex numbers.

Hamilton describes the curl as an elastic  rotation.mthis has not changed and if you have done any multiplication of complex numbers you soon have to get your head around the general rotation and stretching of the product. However in Hamilton's Quaternions the vectors are described by the Combination of complex products. The initial element or product is the scalar.

Producing 2 Quaternions gives a quaternion resultant whic is a scalar and a vector, but part of the vector has an additional term called a cross product. This is the origin of the curl.

Because uatenionic were designed to do rotations a special vector product is constructed o as to cancel the elastic part of the rotation and set the scalar to 1.

What MacCullagh proposed was to use the bare product in a differential quaternion . The differential would model the displacement vector field, the product would rotate and expand ot contract the field. This was essential for strain transmission, but equally Fresnel and Arago had argued convincingly that a transverse wave was the only wave that status fied the experimental results under polarization. Essentially this meant the waves were only oscillating in a plane orthogonal to the lights propagating I reaction.

Young argued for a small longitudinal oscillation, but at those speeds shock waves are more likely than compression waves, it was argued.

The curl or as it is now called the Nabla cross product seemed to be the answer to MacCullgh.

This was all new, nobody wanted to adopt Quaternions with its deeply lien imaginary mathesis, nor did they want to understand it. Stokes comment about a curl not complying with angular momentum conservation is one of those deliberate misunderstandings.!

If the potential field is rotating and expanding it must have a source! Conservation of angular momentum, such as it is deals only where there is no source.

The idea that a wave is generated and then it moves on its own is like a billiard ball, is so engrained thst they  could not envisage strain moving into each point to generate and further the wave,. The poorly understood wave mechanics was being trounced by the rigid engineering paradigms tht were. Felt to be understood.

Angular momentum is conserved as much as the system can be isolated . It is not an issue. The issue is the whole set of conservation laws and what they mean physically and how and when to apply them.

We also see the elastic rotation engenders sn expectation of waves in the aether. Thegrometry not the mathematics leads to that induction!

Having said all this, I see the curl now for what it is, a poor measure of rotational behaviours underpinning electro magnetic theory.

Oh my God!
The penny has dropped, the light bulb has switched on, I straighten out another wrinkle in my understanding.

When I learned mathematics I old not wait to do calculus. It was supposed to be hard and that made it cool! I could do the hard stuff!
Unfortunately, when I got to university level I found myself floundering in the deep nd, drowning literally in unfamiliar concepts!

What happened?

I had applied for Cambridge entrance, but chose Manchester instead. Half way through the first month Cambridge sent the entrance papers up  to Manchester to my surprise! One of my tutors suddenly appeared waving some papers. I quickly grasped he meant for me to sit the papers!
What?
I was not even prepared or forewarned. But I took them anyway. I had no intention of going to Cambridge so what did it matter!

I read the papers and was immedately aware that I had been taught a different mathematics to the examiners who set the papers! I had been taught traditional Math, and the papers were set in modern math style. In that instant I knew I was right to have chosen Manchester. The new Math was not my cup of tea!

In 1960's  the English education system went through a number of reforms with regard to Maths. Cambridge county schools and Scottish schools were in the pilot study project areas. I was not brought up in those areas.

So back to the deep nd. An,yhis threw me for 6 and I never recovered from it until I started blogging. The applied Maths were more familiar but the notation and lecture style was hard to digest. The algebra class bewildered me. Group theory? What in heavens name was that all about!
Number theory , where were the numbers?it was all algebraic set theory, or so it seemed.

I went underground to the computer studies department to learn algol 60. But what was FORTRAN?

After 6 months of the most disorienting presentations I was ready to quit, but then things started to turn around a bit. I actually started to understand computer language. My faith kept me going to social and religious gatherings that gave me a goal in life , and the library had a book by David Hilbertbcalled the foundations of mathematics that I never took out, but rather I sat in the library reading it with fascination!

That book helped me to make sense of one question I had repeatedly asked myself since age 12. What is Mathematics?

I learned there that Mathematics was not consistent or could not be proved to be. I read Polya, about heuristics in mathematics and hw solutions might be obtained or how research into a solution might progress. And I read a smll treatise by Popper who argued for falsifiability as a measure of truth.

One of my gols when I went up to Mnchester was to develop a theory of gravity based on magnetism and magnetic vortices. Of course I did not know how I was going to do it, but I started a notebook to sketch out ideas. One particular day I came up with the notion of a continuum of points! It was more of a visual miasma of shifting points of light which just kept spreading apart nd revealing more points the closer you get.!

I sat outside a professors office for hours hoping to discuss it with him. Instead I eventually got to speak to obe of his colleagues or researchers. He listened and was kind and that was that! Or so I thought.

Later I was introduced to Cantors set theory and the uncountable sets. I loved these sets obviously, because they were just like my continuum of points! But I could not accept the proof!

I was busy trying to learn bout polynomials on the unit interval, or rather to get though the class. But every time one of the tutors would come to my seat and show me another unconvincing proof of cntor's uncountable sets. Eventually they sent a visiting Irish "Reader" in mathematics to show me a functional proof of it, using mapping terminology I had only just learned.  You know what I lied the guy, and I could not see any fault in his explanation. I just thought: this is tautological! So I said yes! I said yes to get them off my back!

By then I was too tired to argue and heading for a meltdown.

I have copious notes, and I wrote copious notes during my 3 years at Manchester. I put them in a box and said one day I might understand what all this gobble de gook is all snout! What I learned was mathematics was in trouble and I did not know how or hy, and by the time I finished I did not care!

My mathmatical training however stood me in goid stead hen it cme to analysing problems nd situations. And I appreciated that. I found also that I had lacked social skills and conversational skills. I did not realise this until I was brought low by my experiences. Building those skills was my obvious priority. And eventually, many years later I achieved that and found that I was autistic!
I was now ready it seems. Suddenly I was gripped by an unstoppable low of mathematical nudity! I filled book after book ith notes. I found out about fractls, nd blogging and forums all at the same time. I had to write!

Fractal foundations using set terminology took me deeper into mathematics than I ever thought possible.. It reveled the human failings of the subject , the mythologies that it is shot through with. It opened my eyes to deep and long TEM goals and conspiracies. It also revealed implement failings in communication by pedagogues!

It revealed my own myopic bias and misunderstandings .

So, I was taught geometry fom primary age. In secondary I taught myself logs and elementary calculus. I read Lancelot Hogben Mathematics for the Millions. At A level I took Maths Ohysicsvand Chemistry.  The Maths and physics departments combined to teach Calculus. My O level trigonometry came in handy but thst was all the only geometry I ever used beyond O level. The rest was dynamics and kinematics in physics, backed up by mathematics treatment of the topic of calculus, Algebra etc.
Graphs and curves were employed for physical data.

I never heard of coordinate Geometry!

Transformational Gometry, a little bit. Graph work a lot. Equations even more, and algebra. Polar coordinates and complex numbers briefly. Mostly I learned trics, methods, techniques, and problem recognition and decoding. I was a trained automaton. No wonder I found it hard to think through my ubject!

Despite all the research and a course by nomn in universal hyperbolic geometry, I still did not get the connection!
Suddenly, after working on Grassmann last year and toward christnpmas, I fell out of love with algebra! The meaning of the Arabic hit home clearly. It was pure "mind fluff!" almost literally! It was screwing with my brain!

I then saw clearly that Arithmetic and symbolic arithmetic had been the ages long goal of Astrologers and the Pythagotreans. Then I heard it again and again in Normans expositions- arithmetic! Symbolic arithmetic all of it.

I then saw clearly how geometry was a precursor to the Arithmoi via the Logos Analogos method of Eudoxus, a master Pythgorean, a qualified Mathematikos or Astrologer of the first rank!

But only this morning have I recognised coordinate GEOMETRY as geometry! That then immediately makes sense of differential geometry. The calculus was not just about dynamics or even dynamic surfaces, it was also about describing curvilinear surfaces in geometrical terms!

The equations that Norman developed for 3 dimensional surfaces have differential nalogues for more curvilinear ones. Solving differential equations is also linked to finding an ration o describe the Gometry of a set of surfaces!

The distinction between geometry and Mechanics and dynamics is the concept of time.

In the Principia Newton revealed this methodically and in multiple ways by use of his Fluxions. Fluxions are dynamic generated surfaces, differential Grometry is a geometric description or equation for a surface!

Einstein by replacing time with a fourth displacement measure created spacetime, that is a 4 dimensional space for 4 dimensional surfaces. Few realise he dd this to preserve the aether concept mathematically.

In America the aether was dismissed completely after Michekon and Moreley. Bitter debates broke out and with war looming scientists were divided on national lines. Those who held to aether dynamics were at once suspected as Nazi sympathisers and industrial espionage agents!. Einstein fleeing Nazi domination in Europe was aware of these struct wartime assessment protocols. Most of the time he spoke in German to preserve his thought processes, but when he wrote in English his advisers were careful to translate his words in acceptable forms.

Despite his media association with the nuclear bomb effort, the reality was he was too much of a suspect to be allowed in on sensitive US secret operations! However, the administration relentlessly used him as a propaganda tool after worked war 2 and during the Japanese war, and the cold war.

Towards the end of his successful career in America Einstein came out strongly in favour of the aether. He judged the time was right to reassert the fundamental philosophical and mechanical necessity of it. This was partly to counteract the strange logic that was being promulgated as quantum mechanics. He felt there was a flaw in it somewhere, but of course the powers that be we're not going to allow tht to ever be found!

Today we believe just about anything because of quantum mechanics and the statistical probabilistic approach!

There I a backlash brewing so be aware! Also be aware that when it comes it is not likely to be free from its own errors!

At you, your topic so working hard and using heavy your system resources?
At me , when open this page fully movies from your topic, my CHROME browser can stop working, other browser surely crash, or worse my computer can shut down with the error "System has recovered a serious error" so my browser perish.
  :death: :jam:

Sorry hgjf2, I don't mean to blow up your computer!
How old is your machine?
I write my posts on IPad first generation, now not updateable, but it copes with this page ok.
Do you use a broadband connection to the Internet ?
How much ram do you have? How much is shared graphics?
What version of Chrome do you have? I use chrome on my iPad and it works ok
I f I think of any suggestions I will post, but for now try ramming your computer in safe mode if it is a windows machine.

Yes. My computer is enough old.
You right . I must purchase a newer computer, but yet I don't have money, so must wait a few time.
If my computer go slow, normally that any page hard on my browser on any sites can stuck, and my browser perish.
My computer is ESPRIMO at FUJITSU SIEMENS with 15GB at partition (C:) and 22GB at partition (D:) memory, and not recognise DVD's and seem be from second hand.

It is the ram that is more important for dealing with modern web browsers, although if your hardware is too old it won't swap memory efficiently even with more ram. ( RAM is random access memory).

If you have a word with your computer technician/ service person they can advise if a simple ram upgrade would do the trick. If it will it might be cheaper than buying a new computer, but I do not know what deals you can get in your area.

Sometimes it is better to archive your hard drive onto a terabyte drive and upgrade your machine, if you can afford it.

You may want to consider a script blocking tool so that you can have all of these things blocked and unlock the ones you wish to see individually.  They wont even be able to load until you allow it, and this will solve the issues completely.

For Firefox there is NoScript - which I swear by - and also AdBlock.  I can't say what to use for other browsers, but I would think similar tools exist for them.

I can tell you for sure that using Firefox with NoScript and blocking Everything google speeds up the entire internet by a third or more. If using Chrome, reconsider.  That browser reports so much of what you are doing to google-analytics that it can really slow down an older machine  (I honestly have no idea why people tolerate google and their intrusive invasions of our computers!  I block everything google with every tool I can find).

Hope it helps, and good luck resolving this.

Thanks Sockratease.
I think google must use something like this on its own google+ software for mobiles! I see cracked video symbols every time I load that app. But one click and they spring to life!

This video details cracked science and its commercial and financial overlords.

http://www.youtube.com/watch?v=1kb_T8sCWL8

The results can be explained in terms of twistors and Twistorque , which are abstract concepts I know, but they measure the chemico physical phenomena.

The concept of electro Thermo magneto complexes as vortices is relevant.

This really cool video only partly explains the processes.
Electro Thermo magneto complexes are involved in the so called shear plane or slip plane forces! In simple terms gyrodynamics( I love that word ! ) or twistors and Twistorque are involved.

http://www.youtube.com/watch?v=GX4_3cV_3Mw

I wonder if that is reproducible in a CFD simulation.

I do not know, but Claes Johnson is probably the man go ask.
http://www.youtube.com/watch?v=t7e_6bkUFzE

The research on barycentres has an impact on the notion of a twistor.

It is clear that a "vector" or Strecken notion is associated with a complement in a Barycentric system. The moments in the system are usually considered as turning moments, and they are arrange to produce the null or zero vector, however, the non zero vector for a Barycentric system is usually defined as Torque.

When developing the notion of twistor I based it on the circular disc. In this case the Barycentric system is "uniform".  This gives a symmetric system and a Barycentric point coincident with the geometric centre of the disk.  The Barycentric system is usually in dynamic equilibrium. This is confusingly called static equilibrium because the system in balance has no autonomous movement.

Because the system is in dynamic equilibrium we tend not to realise the barycentre is itself dynamic !

In a torque system, the dynamic equilibrium is non zero, but it is a vector quantity. This means it has a direction, and that is a so called bivector or planar  entity.

Can this bivector move freely in space?

The answer seems to be dependent on the barycentre!

A spinning object with a rotation pivoted not on it's barycentre seemsbtonoscillate wildly. This indicates that the barycentre is dynamic in such a system, and there is only one point where the barycentre will remain in equilibrium during rotation. This dynamic barycentre may move outside the body of the rotating system creating the wild oscillations. It may move within the plane of rotation effectively providing a sliding rotation centre responsible for elliptical motion in orbits.

It is a question that needs to be asked, because the dynamic barycentre is clearly an instantaneous phenomenon in a torqued system.

The combination of vector sums both action and reaction in free space seem to be best modelled by a network of springs rather than rods. It is the stiffness of the springs and the transmission of strain forces that may determine the locally observed motion behaviours even in a torque system. This kind of complex barycentre of a much wider system deserves investigation.

Precession in a near constant gravitational frce field with rotation at the Barycentre.

http://www.youtube.com/watch?v=EJWIl4MYMbw

The research on barycentres And tangential points of contact, means my concept of a twistor is changing.

When Maxwell created the distribution of particles in motion theory, based on Gauss and Boltzmann error distribution theory and associated probability theory he used the Barycentric ideas in a different way. The assumed point mass particles moved in a collision field. Depending on whether the collisions are elastic or inelastic the distribution of the particles will be wide or regionalised. The density of the particles in a given volume, that is how many in a given volume affects the number of collisions and thus the progress of any test particle. Consider Brownian motion.

Maxwell determined that this effect. Was best measured by the mean square averages of the Pythagorean description of a general translation, which would be directly proportional to the velocity of the test particle described as some combination of its orthogonal component velocities.

There was a lot of data collection required to establish the validity of this analysis, and I do not know how this was experimentally achieved, but the root mean square measure was used as a representation of the average collision free path!

The idea attracted Kelvin who developed his Kinetic theory of gases around these ideas and measurements.
Kelvin eventually made some breakthroughs in his theoretical model, which established his theory for gases and Maxwells reputation for analysis. The idea was also visually communicable, even if the math was not! The bell shaped curve symbolised the nature of some gas properties. . As the gas warmed, the prediction was a dhifting and flattening of the bell.shape. It was the fact that data when processed showed this flattening that justified the statistical analytical tool Maxwell developed.

Because as a rule it does your head in thinking about how these mathematical models and analogies are cobbled together it is easy to overlook things.

Number one: rotation and the barycentres of these dynamic particles were not physically considered. The mathmatical means provide a Barycentric tendency by default, but this is statistical not physical.

By this I mean, for every 2 or 3 particles( assumed as point masses) scattered in space a physical Barycentre is computable in the constant gravitational or hydrostatic pressure field. Thus over a volume a combination of these sub barycentres should give the Barycentre of the gas as a volume. The summation of these sub barycentres allows weights to be introduced to describe regional mass or density variations.

The motion and distribution of these point masses contribute to these dynamic sub barycentres moving as "particles" themselves, and contributing to a Barycentre of barycentres.

For each level of this fractal, the tangential contact points can be computed, and in this way a fractal measure for strain deformations put in place.

These barycentres and tangential points can now be treated as Maxwells original point masses and represented by components. The resultant probability distributions of the method can now be in a similar way related to physical properties of stress and strain within the body forces of the gas volume.

The mathematical means are now means of physical point masses or barycentres.

To these physical centres we can add the properties of rotation and rotating tangential points of contact.

It is assumed in the kinetic theory that point masses are not connected, and that they are free to ieave each other. But this is a non physical idea. Gases expand  before dissipating. During this process, hydrostatic pressure defines a boundary to the gas, by collisions according to the theory. However, I wonder if this is only partially the case? When one introduces gyroscopic or rotational dynamics, boundaries naturally form independently due to the gyroscopic dynamics.

Collisions in this gyroscopic model result in a boundary layer like a surface tension, with fragments tha break away following trochoidal paths, as well as fragments of the external medium dragged inside this surface tension boundary.

The variation in density associated with these poit masses, and the associated barycentres and tangential contact points , would lead to a distinct regional surface tension dynamic. As it stands, diffusion studies and hydrostatic studies indicate that denser regional masses would drag in lighter densities to a certain extent, bu overall the denser regional mass would move through the lighter density episodically, like a Brownian motion, but with the lighter density doing most of the observed jiggling!

Similar densities with the same rotational dynamics would merge by anti gyroscopic action. Thus the gyre of the regions is important in terms of merging behaviours. Those with the same observed gyre will tend to repel, while those with ant gyres or contra gyres will collect together. Merging would be a factor of external pressures on the gyre antigyre system. The emergence of a single larger gyroscopic rotation is also probable , but not necessarily inevitable.

Mind the flashing images!

I just feel in my gut this has a lot of relevance to the ideas I am expressing abstractly!

http://www.youtube.com/watch?v=QxaHNurYgSI

Short and long springs behave differently.

A short spring  applies rigidity to any network of springs joined to point mass. A long spring allows and models rotational behaviour about the point mass!

Thus, as observed, gases and fluids tend to allow vortex behaviours, with gases particularly forming rotating plumes of billows( local rotations) . The kinetic theory does not predict this behaviour.

However, by introducing lon range spring action between point masses, not only is surface tension a consequence, but also the long mushroom like billows.

The long springs model the electro Thermo magneto complex energies and behaviours in fluid dynamics, while short springs model them in crystalline structures.

Removing springs in body force models altogether is a modelling flaw I think.

However, the motion behaviour springs model I think are rotations and their dependent trochoidal locii. The rotations are not just circles but in fact conic sectional including trochoidal forms based on conic section curves and surfaces.

For example Lissajous curves are a complex rotational motion that body forces might generate under certain conditions.

The concept of rotation, limited to the circle restricts the conceivable motions. But once trochoids are allowed and also conic sectional curve and surface trochoids the more complex motions, undulations and vibrations and frequencies are an inherent part.

The Barycentric and tangential contact points allow us to build strain gauge tools from materials up to youngs Modulus collapse behaviours. Beyond that we will have to use theory or real planet sized dynamics as strain gauges!

Do you have reason to believe that springs are removed from the body force models ... ? I think they are in use since the life of Robert Hooke in 16... something. Just replaced the spring constant with the Young's modulus.

Yes you are right, but I am thinking in general. Thus most engineers see the value of springs, but mathematicians ignore them under certain conditions. In the kinetic theory, based on Maxwells analysis, springs do not figure. Generally in the kinetic theory springs do not figure. The gas, when it becomes cool enough is considered as vibrating molecules close together. It is chemists who put the springs back in, under the "electrostatic" theory they developed for electron proton interaction.

Body force interactions typically require several eclectic theoretical models to describe, and springs are perhaps the least favoured, although Susskind and others develop quantum manifolds on such spring models.

You would not believe how insular many subjects are, preferring their own language and models over ones they classify as " mechanical". Mechanics has always been disdained through intellectual snobbery and religious rationalism. Mechanics is man aping god! ( meaning man is the ape that copies the superior handiwork of god).

I think it is a flaw to leave springs out of any model of fluid body forces, even high energy plasmas! My underlying reason is that motion is not rectilinear, it is trochoidal, I believe.

Twistors and Twistorque  rotational kinetic energy are new concept terms I am exploring to draw together our knowledge of rotation.

Rotation has always been associated with the sphere, but this is really the arbitrary paradigm model. Within it many forms may be specified, and their behaviours determined by empirical observations and hypothesis.

The main principle of rotation is that it is about a point. This point is special because it can have a direction line segment( vector) associated with the Barycentre of an object. This line segment is variously named a vector, a point vector an axis of rotation etc, but it is a point around which any spherical motion can occur at tHe same moment.

Thus combinations of rotations are normal and tHese would be labelled as combinations of point rotations . These are what I have labelled as twistors.

Rotation in space I have come to realise is not circular but trochoidal, especially space trochoids. Constraining rotation to a circle introduces forces that we have not fully taken into account generally. Trochoids or roullettes as they are now called can be visualised as 3d Lissajous curves of various frequency ratios.
http://youtube.com/watch?v=SePFwkbhxdw

Bearing in mind Grassmanns dictum " Länge mit der Richtung Festgehalten"

His development of the arc may have started in radian measure where arc length is given by rø and the arc segment described by ABC say giving a combined product

RøABC or røAC if we drop the middle point for convenience, although of course this was what constrained Möbius Barycentric method.

The retention of the 3 point description of an arc ABC immediately links it to the product of 3 points which is a triangle in Grassmanns method, and immediately the geometric fact of the exterior circle of a triangle!

Thus r becomes the radius of this exterior circle and ø the radian measure of the unit circle arc based at this circles centre.

røABC is a complete Grassmann inspired description of a circular arc segment!

By analogy, this work should develop on concentric circles as the analogue of parallel lines!

In the mean time

http://youtube.com/watch?v=bIr66quAKkU

The light shows it's spherical fluid nature, with absorption, scattering, re radiation and diffraction, refraction and polarisation.

Actually a little more thought makes me think Norman's universal hyperbolic geometry would be a better geometrical set up for twistors using Grasmmann notation.mthe reason , parallel as a concept works differently in the circle. Scaling occurs in the ratio r:R or 1 : R/r. This last form she's that rectilinear relationships ar the limit of circular ones for very large radii!

1:1 as r,R —> infinity, where all curved line become locally straight. Universal Hyperbolic Geometry has a corrective term that copes with these differences

I have reached a point in the Vorrede to the Ausdehnungslehre1844 where it is clear that Grassmann had already isolated the fundamental basis of a twistor in terms of his concept of primitive products, including products of points as straight lines. While he uses the radian measure in this conceptualisation he does not refer to the arc as a product of 3 points, but his diagram easily shows that this connection can be made.

Studying the Vorrede has revealed to me how very odd and whimsical the Grassmann method actually is!. Those like Bill Clifford and Hestenes who have extensively developed this style of approach have attempted to finesse the fabulous nature of this method, by relating it to traditional math concepts, methods and procedures. In fact, Robert, hermanns own brother substantially redacted Hermanns work!

There is something to note in Hermanns obstinate refusal to revoke his earlier work(1844), but instead to promote it as a heuristic paradigm. The meditative processes that give rise to knowledge and expertise are rarely written about.

Very interesting application of rotational ( pivoting et al) deformation.

http://youtube.com/watch?v=3I1Hl1gl3Uk

This treatment of Euler's thinking I hope is accessible. The interpretation of it is readily expressed by Grassmanns analysis, and we use that and refinements by others to apprehend it today?

http://mathsforeurope.digibel.be/Euler.html
http://www.songho.ca/math/euler/euler.html

For me it is only the first of an infinite number of trochoidal identities, which taken as a group fully define the term totation!.

In fact, Grassmann shows we have 2 ways of describing rotation based on the trig and hyperbolic trig analysis of Euler. Both spiral, but one is more spirally than the other! Lol!

The concept of a twistor is identical to the screw in a Clifford Algrbra. But of course that is more fully developed. However Grassmans concept is more general than both .

Based on 's reworking of the quotient algebra of the inner product in Vorrede to the Ausdehnungslehre 1844

I propose
  E^ix= coshx+ isinhx

This would be a twistor for hyperbolic curves and RE^ix would be hyperbolic curves of different " curvature".

These kinds of twistors would thus not generate circular rotation , but allied with the circular functions could create trochoidal or hyperbolic swings or pendula.

What makes you think "It's just FRACTALFORUMS maintanance for work properly?"

Sounds more like your browser needs maintenance, or your Operating System, or something else at your end.

There is NO WAY an embedded video merely being there can crash a browser unless you are either using an outdated version of the player or it is somehow set to start playing immediately instead of the default mode of waiting for you to hit the play button.

What OS are you using?

How much RAM?

Does it crash before or after you try to play the video?

Let's finish this in the help forum where you asked rather than here in somebody's thread   :police: :siren: :police:

And if anybody else experiences this, please post here:  http://www.fractalforums.com/help-and-support/trouble-with-fractalforums-videos/ (http://www.fractalforums.com/help-and-support/trouble-with-fractalforums-videos/)

I don't like to report errors, bui I want to this site to work properly.
Probabily that my computer is guilty, that is old and have only 15GB at C:\ partition, and going too slow, becuase the red led "capacity" sometimes stay on, cause for whick the windows can freeze a few minutes. Maybe my operating system have trouble.
Next times, I will checking browser and programs.
On certain phone tabletes, the movies on FRACTALFORUMS have play trouble, as example. I believe that FRACTALFORUMS going properly but need a good computer and good OS.
 :sad1: :sadness:

Hgjf2 I am aware of the problems caused by these video links. I experience some issues as well, but fractalforums does not stop working, instead i get the link visible to go to the site on YouTube etc.

Also i am trying to post less videos and perhaps use a link tag instead. However if your machine needs upgrading it might be an idea to see of some software might help until you can afford an upgrade.

Thanks for supporting the thread though, and if you want to contribute please do. :dink:

This is by my new Twistor formula with quaternion coefficients for each of the imaginary axes in the trig /circular functions

A twistor can be viewed as a dynamic line segment: it both swivels and dilates. Thus it can be summed in lineal combinations giving unprecedented " nodal control over path modelling, as well as form combining.

One relatively eay application is in tracing 3 d trochoidal paths nd surfaces, but using the notion of differentials wave like or curvier lines or surfaces can be modelled. A kind of Fourier sum can be used to model a signal, surface or dynamic structural combination. The possibilities are endless.

With regard to modelling the space time manifold they offer great spacetime path flexibility. In Eucliden geometrical terms they low space to dynamicalynonorm to actual data sets.

What they do not do is replace our need to gather mpirically data and design models that conform to the data.

It is common to believe science predicts behaviours . It is better to understand the cyclical nature of all behaviours ad our ability to model these cycles. Thus when we have a good McLennan of a cycle we I'll understand when each phase is due. It took maybe 10 000 years for us to process the cycles of the planets as data sets. We are still new in this game of understanding the cycles, despite our technological advances.

Chaos or the decline of the mechanical Philoophy of the universe has placed us in a brand new ball park! Computing all the likely probabilities is now possible for defined cyclical events. What we still have to do is choose wisely which scenario to act upon. That is and always has been the Kairos, the responsibility of the Astrologer to tell the rest of us the opportune time!

We start in the continuous world and then we get discrete!

Rather let us always remain in the contiguous world!

http://www.cs.cmu.edu/afs/andrew/scs/cs/15-463/2001/pub/www/notes/fourier/fourier.pdf
http://cs.calstatela.edu/wiki/images/5/5b/Understanding_the_Fast_Fourier_Transform.pdf

http://youtube.com/watch?v=DdlqgNV7LkM

Wave is a rotation phenomenon. Waves are phenomena associated with rotational motion, and without understanding rotation we will have poor insight into what we might define a wave to be!

Suppose I define rotation as a motion that visits every point on a sphere, and is approximated by visiting N evenly distributed points on a spherical surface, or the radials to those points on that surface, then it is plain to see that rotation is a complex notion which we only understand approximately.

Any other motion that does not fulfill these definition points of a rotation can be called a partial rotation, and in the vernacular it can be called a Wave.

Thus the so called sine wave is not representational of the general wave! The general wave has to be described by both sine and cosine functions in space and generally within a quaternion exponential form in at least a sum of 2 such forms for a trochoidal wave.

Complex or what? But at least you avoid the fantasy that you or I understand waves?

In my research into Wave Mechanics, it is John Strutt or Lord Rayleigh who stands out as asking the most insightful contributions. And yet they are necessarily very mathematical as models. Using Grassmanns analyss I can attempt to give an intuitive exposition.

Firstly the bulk properties of any medium are used to describe deformation of that medium. Such properties are stress longitudinally, stress shear and stress compression. Some may distinguish transverse or cross sectional sheer as the medium dictates..

Having obtained measurements for thes bulk properties , then a time dependent differential is sort for the inertial effect of deformation in any or all these stress modes. It turns ou that all these stress modes have time dependent action on mass  proportional to a square root of the compression property divided by an inertial measure constant appropriate to the material.

Compression is not just longitudinal . It is in all orthogonal directions, and since we resolve forces and velocities orthogonally usually to understand what is going on, it makes sense.

However I claim that rotation is ultimately involved in wave progression.. Longitudinally this would seem not to be the case. However, the reason why all faces are compressed in the compression test is to prevent, buckling in the free faces. Buckling is just another term for surface rippling which is a rotational phenomenon.

The viscous restorative forces also act either side of a rounded vertex , however sharp. Where this is not the case the surface fracture into a crack, which releases the restorative energy!

Now an Analysis of a travelling pulse reveals that for it to have a velocity it must have an impinging impulse force as well as a stress compress force! The bulk motion properties give us the speed characteristic but not the speed omparting mechanism! In many cases this mechanism is similar to a blow or an explosion, thatis a sudden impulse is given which releases before the material has time to counteract! This in effect enhances the effect of the restorative force which having nothing to push back against achieves a remarkable velocity!

However the forward moving compressive force which is a reaction to the initial compression causing buckling now combines to stress the forward material by stretching. This then creates dimpling which now attempts restoration as the rear medium begins to return to its natural position. These  two forces combine to create new buckling and the cycle repeats at the bulk dependent speed, if the forces do not dissipate.
But the initial velocity which is being transmitted was given by the first impulse.

This first impulse is not a sine wave despite what the oscilloscope shows. It is a whip lash, that is it follows a sine curve up to a certsinly point and then a cosine curve down very quickly . This creates a leading crest curve which helps to maintain the forward netted velocity of the dress compress wave..

Our example of this in a skipping rope shows that the stress compress wave using tension in the rope is powerful enough to lift sections of the rope against gravity and to ling it back down with gravity to keep the stress compress wave functioning.

Do we have these same rotational and stress compressive forces in the fabric of space?

Yes if you understand that rotation is the fundamental motive.

When we measure electric tension and magnetic tension we are measuring the same thing, a rotation in space that superposed not sine wave on sine wave bur sine wave on cosine wave! Due to rotational super positioning not longitudinal .

http://www.usna.edu/Users/physics/ejtuchol/documents/SP411/Chapter2.pdf

This provides a fuller derivation of the bulk properties of a medium determining wave speed.

The essential point is conservation of momentum considerations as a fluid element moves. However , despite stating that a system has a pressure wave moving at a given relative speed, it is noticeable that the block is given its own unexplained background speed. The wave moves within the block but is not the cause of the block moving. Thus this derivation , setting block speed to 0 means the pressure wave speed is a speed of deformation, that is how quickly the block deforms from one shape to another. How quickly, not in time but in velocity.

The question of propagation is not directly addressed mathematically.  We have to assume that this deformation continues to move st this speed in the unbounded situation of the open medium. Empirically the behaviour of leading and trailing fronts is not discussed.

At all scales we have to accept that if there is a net force motion will occur. This motion has to be characterised by the observer. In this case the observer describes it as a travelling deformation.because it can be seen in liquids. However in gases and electromagnetic theory, these deformations cannot be seen!.mwe thus trust to the calculation and utilise an analogy., by the analogy an electromagnetic ' wave" does not need to convert to magnetic and then electric forms to propagate! It propagates because of a difference in pressure in the medium. However, that difference in pressure is associated with a rotational deformation

We have developed techniques of seeing the invisible. While the expositions above do not require sine wave motion for propagation, in fact they simply state that wave propagation is due to conservation of an initial impulse derived " momentum" or quantity of motion , yet it is still observed that spatial deformation accompanies all these conservation of momentum progressions.
http://www.ivorcatt.co.uk/x18j184.pdf
This explains it well, but makes no bones about the irrelevance of any sine wave to the derivation. The wedge constitutes a change in space at a point resting on the given velocity of the blocks. With a constant change in density/ pressure


The so called plane wave solution refers to an impulse derived solution. An impulse is so narrow that it looks like a plane in 3 dimensions for the constraints given. However it is more accurate to call it a surface solution in 3d in fact a spherical surface solution.  This solution also does not take into account surface variation, ie transverse variations. When this is done, the spherical surface is no longer undisturbed. In fact it becomes highly complex in patterning and able to develop non tangential solutions in addition to the simple surface.

These non tangential solutions appear as filaments or fibres vortices that extend beHind the advancing surface like plumes?
message#7 (http://www.fractalforums.com/let's-collaborate-on-something!/and2350and2339and2381and2337and2354and2348and2375and2341-(maand7751and-t4935/msg27443/)

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

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.

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.

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

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.

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.

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

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.

Penrose spacetime Twistors

http://youtube.com/watch?v=hAWyex1GKRU

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.

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

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.

http://youtube.com/watch?v=Mv9NEXX1VHc

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,

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.

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
 

http://youtube.com/watch?v=ULDj_LkynIo

Rigid rotation

http://youtube.com/watch?v=lGaMKrtiTc8

No matter how they explain it away the Grassmann twistor is a rotation of space. In this view there is no justification for discarding the imaginary part.

http://youtube.com/watch?v=kUft1l6Xrjw

http://youtube.com/watc?v=wGVue66jVBs

I post this to illustrate a great misconception in the teaching of rotation using vectors. The misconception derives from the unwillingness of physicists to accept Newtonian analysis. At the same time it gives me a chance to make clear what Newton meant by centrifugal force.

Today many eminent physicisypts go so far as to claim newtons centrifugal force was erroneous. This is because they confuse a model with reality! Newton made it plain in his introduction that this was a mistake.

Newton recognised that to decompose circular motion requires 3 not 2 vectors. Two vectors are collinear but contra( centripetal and centrifugal) plus one that is orthogonal to both. This is always the tangent vector.

Introducing this reference frame you can better understand the dynamics of rigid rotation . Necessarily Newton states that the forces involved in rotation must include impulse forces. While it is possible to posit some continuously varying force or system of forces it is also physically possible for a similar path to be " force" by an impulsive , or vibrating or " undulatory" system!

Given rigid rotation is not as you have been lead to believe it is easier to understand fluid or vorticular rotations in fluids as impulsive or continuously varying pressure systems.

The Laplace transform is also a Grassmann form.

http://youtu.be/MRy8xxvsZA4

Clearly Hermann was aware of LaPlace as well as Larange.

I have spent some time apprehending the Fourier story, as well as the Fast Fourier Transform. The historical perspective is helpful.

One source linked the idea to Laplace as Fouriers teacher, but this soorce explains the interactions in more detail.

http://todayinsci.com/F/Fourier_JBJ/FourierPoliticianScientistBio.htm#series

We find that Fourier worked on various forms: bar,annulus, sphere and prism . When he tried to use the Fourier transform to describe the temperature gradient Lagrange objected. So his full ideas were written in an unpublished memoir. Of the forms the annulus and the sphere are to me the most fundamental, but the method itself clearly derives from rotationally dynamic systems . The important thing was the notion of period.

I think this term is misleading because the derived  ratios are trigonometric and relate to a static form. Thus the superposition is always defined over a spatial region and should be called the bound form. Because of the nature of the description several parameters are used which means that the transform is inherently multidimensional

The introduction of t as a 4 th dimension  in 3d case is misleading, if it is considered as time. It is another dimension that is used to coordinate data sets that describe a bound form. But in the 1 dimensional case it is precisely a parametric dimension used to "represent" time.

From Fouriers conception this method was a way of modelling or describing a form.

While it is correct to cite Euler, the true source of these ratios are the ancient Greeks, who derived trigonometry from Pythagorean philosophy and more ancient astrological practices. The works of Ptolomey in his Almagest particularly inspired the Indian astrologers to develop the sine ratio from the chord ratio.. The Arabic and Islamic scholars then initiated a centuries long calculation of the sine ratios for the right triangle in the semicircle. In the course of this the binomial expansion was developed and eventually the Multinomials expansion was studied in the formulation of spherical trigonometry and spherical geometry.

In Europe these things were hardly known, but in the Islamic and Indian traditions Multinomials forms were discussed in "algebra" , that is symbolic arithmetic, as a result of developing methods and algorithms to calculate the sine ratios by difference formulae. This is what is referred to as interpolation. However Newton and his acolytes De Moivre and Cotes, excelled in this area. In fact DeMoivre was accepted into the Royal Academy due to his astonishing papers on Multinomials! Both he and Newton shared a private joke. Newton had instructed De Moivre in this published but generally unread area of mathematics!

Further, in his mathematical papers and Waste book it is clear that Newton developed a deep apprehension of spherical trigonometry of the unit sphere and unit circle, and the imaginary magnitude . It is also clear that Cotes together with DeMoivre took this to the furthest extent in the theorems on the roots of unity.

That Euler, the Bernoulli's and Lagrange were aware of this is undoubted, but it was not in their nature to give credit where it was due! This is something Fourier decried. Thus with this background we can see that Fourier made great use of available mathematical methods to support his own methodical investigation of heat transfer. Clearly inspired by Newtons observations on temperature he developed his geometrical differential forms in an attempt to describe the " form" of heat in certain bound forms. The logical objection to his initial concepts derive specifically from his spatial description. Heat transfer clearly has to be dynamic. He corrected one differential from a spatial into a time varying one, but few noticed or even understood the difference.

In effect he had created the first formulation of " wave" mechanics, but it was not until Helmholtz, Lord Kelvin and Lord Rayleigh that a corpus of able physicists were able to draw out the incredible applicability of what he had done. We might also include Navier and Stokes who struggled to describe fluid motion by differentIal equations which had simplistic solutions in this form.

It is an important aside that Kelvin was a great advocate of Fouriers analysis and his series. He apparently mastered it within a short period and felt it was of fundamental significance. He developed his own kinetic theories on the back of it. Thus his bitter opposition to Hamiltonian mechanics, based as it was on" nonsensical" imaginaries was in part fueled by his abreaction to the general ignorance of the Fourier transform.

When Gibbs, an American acolyte of Kelvin developed the fledgling vector analysis of mechanics which relied not on imaginaries but on good old trig  ratios he fully supported Gibbs baudlerised version of Grassmann algebras!

Further he lead a private campaign against the use of Quaternions, forcing Maxwell to recant his erstwhile proclamation of them. The debacle ends when the American academies decline the introduction of Quaternions into the curriculum in favour of Gibbs vector and statistical mechanics, supported by Kelvin.

This is one of many shameful incidents in Academia. However, in Europe the value of Grassmanns analytical and synthetical method was being fully appreciated. Bill Clifford in England was among a number of English academics who were heavily influenced by the Prussian adoption of Grassmanns style. Consequently the imaginaries were never divorced from the original Grassmann lineal Algebra of Strecken.

This had a profound effect on the study of the Barycentric calculus and how it was presented, and the study and development of the Fourier transform, as well as the Laplace transform. These innovations did not pass Lord Rayleigh by and helped him considerably in his notes on Wave Mechanics.

Today the Fourier transform is hard to explain without the imaginaries, although it was and still is a fully trigonometric analysis. Its meaning however is more clearly expressed in Grassmann Ausdehnungs Größe, or what I now call Grassmann Twistors.

Grassmann twistors, in fact the whole Grassmann methods derive from a bound form: the line segment. This line segment is of 2 Types: ordinary and trigonometric. It turns out that circle arc segments can be represented by these trig line segment in a modified product sum AB + BC= AFC  where AFC is an arc segment described by 3 points and AB , BC and BF are radii of the arc.

It is to be noted that this product sum encompasses the notion of a rotation being representable as 2 reflections or a single reflection in intermediary line segments or radii. But i am more convinced as time goes on that Grassmann thought of rotation as a projection of a line segment onto another or into another 'position". The trig line segments as vertical projections thus record this rotation. In this sense the cosine laws are projections onto other line segments but clearly elliptical or even hyperbolic.

Of these trig line segments the simplest are those in the unit circ le or sphere, and they represent circular arc dynamics..

In the context of this representation of rotation the "sign" takes on a differemt meaning. In a single line segment the "sign" represents a half rotation about some arbitrary and unspecified point. The principal orientation is preserved but the principal direction is contra in that orientation.

Howeer in a product of 2 line segments the sign" represents a "quarter " turn . To be precise it represents a cyclical rotation of the points by 1 out of the 4 possibilities, that is the next in the cycle. So far from meaning negative in the contra sense it actually means a rotation! Thus when we annihilate AB with BA w are in fact not making physical sense, because the form exists but not in its original orientation.

For example, if i turn a picture from portrait into landscape position the object still exists! However negation is really only about a discounting process, it annihilation! This is why our math often gives the wrong result in physical reality, because objects or entities do not appear and disappear as we imagine from relying on the math!   These sculptures represent a Fourier / Laplace transform method with the transform and its inverse. The form is aome kind of vorticular spiral the inverse just demonstrates that the Fourier/ Laplace consists of small products that do not exceed the bailout condition, doing the inverse recovers the original form.

I have to say i had no idea what the original form was, as i simply made up a geometric series of Fourier coefficients!
f=1*exp(z-c)+0.5*exp(2*z-c)+0.25*exp(3*z-c)
z=f*exp(-1*z)+c

f=1*exp(z-c)+0.5*exp(2*z-c)+0.25*exp(3*z-c)
z=f*(exp(-1*z+c)+1*exp(-2*z+c)+0*exp(-3*z+c))+c

This is a full version.

It turns out that in the mandelbrot type that final c outside the exponentials is important for visualisation.
By inspection you will note that i can remove the contributing exponentials by zeroing them.

This video about gimbal lock is interesting in terms of Quaternions.

Gimbal lock results from poor notation
 If we name a rotating axis x then we confuse it with a lineal axis x! As you see the " axis" is really a 2 dimensional polygon or circle. This Grassmann identifies as a fixed plane of swivelling, analogous to the fixed direction of displacement.

The key then is not to label the rotation axes as axes, but as planes .
Thus labelling the plane xy indicates clearly the rotation track..

The three planes  of interest( ie,  we will include but not identify the negative directions) are xy,xz and yz.

Without the confusion caused by using a normal to these planes as an axis of rotation it is clear how the Rotation proceeds.

Now gimbal lock occurs when the planes are not independent but nested. Normally axes are independent and the function or relation depicts the dependency of the parameters. Of course the question is what are the parameters in the plane?

The simplest parameters are arc magnitudes or polygonal perimeters. We can then relate a point on the arc magnitude in plane xy to a point on an arc magnitude in plane xz and finally to a point on the arc magnitude in yz.

To plot such a combined point we have to drop out of the rectangular frame and the polar coordinate frame  in our minds, where we have been taught to use parallel and concentric displacements to mimic or model dependency.

In a circular arc magnitude frame within fixed planes we have to generalise the dependency process.

We can consider the other arc segments as moving rotating by the third. However in the Euler system this coupling is " unlocked " in our minds, and so in our computers. Thus we get gimbal lock by uncoupling one of the planes from the other 2!

Hamilton's Quaternions have 6 planes  of rotation. Because the 4 th axis is actually not fixable as orthogonal to the other 3 orthogonal axes this axes can actually be fixed in any relative orientation? Choosing such an orientation always gives us a maximum of 6 planes of rotation. Usually this is too much for mathematicians to idle with so they do not uncouple the structure and hence we do not get gimbal lock!

I have to say that Grassmanns analysis has proved invaluable in clarifying this in my mind, though I have been working towards this for some time,
https://www.youtube.com/watch?v=zc8b2Jo7mno

As you see the nesting means gimbal lock cannot be avoided. But always rotating 2 by the third releases the lock. That does  mean rotating 2 axes not 2 rings .

In most physical or viscous situations there is a difference between rotation and twist. Twist involves at least 2 counter rotations . Thus I feel Twistor as a general name ought really to refer to that intubation. The single exponential I will call a gyre. Thus a Twstor for me is at least 2 counter gyres.

How are they combined?

I have 2 ideas . A lineal combination seems the mst logical, in that the two gyres are distinct. However the combination devolves down to a line segment sum. Thus it must be a three dimensional line segment sum .

The question is: should it be a  sum of opposite gyres or a subtraction of opposite gyres?

The other alternative is a gyre product. This means the exponents will directly cancel or subtract from one another. However if the twist is is two contra gyres in one plane only the other planes are free to move independently and can gyre as they please. This is of interest but somehow does not " feel " like an effective or physical twist, but rather an opposition of Newtonan motives.of rotation, Newtonian fluid motives.

Trying out the definition of twistor as 2 contra gyres gives the following sculptures.

One is the mandy procedure the other a julia. The mandy is also a product of quotient operators or Grassmann gyres while the julia is a sum of gyres/

It is important to note that Newton did not originate the notion that Descartes Vortex explanation of the motion of planets was wrong. Newton merely ailed  to be able to demonstrate that it was a foundational idea. We see from this quote that " students of Newton twisted his work to their own ends.http://www.ampere.cnrs.fr/parcourspedagogique/zoom/courant/electrodynamique/index-en.php

It is thus clear why the lineal algebra heavily relies on the straight line segment and the trig line segment .

Despite Eulers work on the arc , and that of DeMoivre and Cotes there was little enthusiasm for a spherical or circular arc analysis of the physical phenomena.

Understand this one simple point. Mathematics is a human creation that follows the predilection of its creators.

Despite the fancy notation and extreme regard mathematics is as philosophical as any philosophy and as experimental as any experiment. It derives from our mechanical interaction with space and our tendency to be lazy. Thus we call the product of laziness perfection, elegance and some other such epithets.

This is not a bad thing perse for we need to keep things simple, but as soon as we ascribe laziness to natural events we are in trouble! Nature is extremely busy, intricate and detailed. It is a complex fractal and no lazy man will fathom it.

However that is not to say that the repeated and diligent application of simplicity  at all scales cannot reveal some of the complexity of and beauty in our space.

Ampères electrodynamic theory is a fractal theory of loops, and it underpins modern electromagnetic theory and the concept of matter as a system of corpuscles, atoms molecules energies etc.

The trig line segments are a historical device used by mathematicians of great ingenuity : Laplace, Lagrange, D'Alembert , who believed there was no circular or spherical force in nature.

You must examine yourself, as I have to see how infected you are with that belief system. It is very difficult to avoid years of training without some effort.

The significance of Örstrds findings , and his philosophy dd not penetrate my mind until I did the research on Ampère. What I have believed and promoted as fundamental, namely relative rotational motion , and which I make this reasonable argument for:- if there is only straight line translation, then all things can move in only 2 directions., for to move in any other is an implicit rotation. This implicitly contradicts the notion of only straight line translation; this is demonstrable around a wire and indeed in any tornadic system, but simply in eddys and vortices in running water.

However to start with relative rotational motion presents no implicit difficulty with translation in straight lines.in any orientation or indeed in any curve.

What is the fundamental natural rotation? It is not the spheroid! The mushroom headed smoke ring like torus is the fundamental natural rotation.

Mechanically speaking a rotation cannot occur without a radial quantity and an arc quantity. Thus a radius thdt swings in an arc is a potent symbol of a natural dynamic.

Examples: an explosive event sees a plume Nader pressure push out in all directions, formin thr top of a mushroom or a bell shaped curve. The radial pressure is pretty constant, but then the bell orm becomed buoyant in a higher density environment. As it lifts the plume pressure in the direction of levity is enhanced by higher density pressures. This accelerates the bell shape and radial pressures in the belly are counteracted external denser pressured. The resulting gradient reveals the rotational pressure in the bell which curls the bell inwards into a vortex ring structure.

Torque is the usual but inadequate model for this behaviour. It requires one to posit a memory for parts of space which have to remember where to move in a dynamic system!

We cannot have rotation without a radial that acts to define the Spaciometry of many arcs. We may have many varying radil and thus several arcs and several inherent rotating pressures which typically feel like mushroom surfaced plumes and bubbles,

Plumes and vortex rings are the fundamental rotational structure.

The plumes are equivalwnt to radii and the vortex ring in the mushroom head equivalent to the arc. However this is where we get real, and recognise our perfected geometry only provides labelling terms.

the reality of rotation is a physical phenomena not a geometrical one , thus we who are accustomed to representing force by a geometrical form to whit a line segment are in trouble. The mechanical apparatus we use to draw our circles ought to have taught us a lesson. Circles are realised by mechanical principles. the chief mechanical principle nowadays is energy, but back in Newtons time it was Pressure. Pressure was reduced to force or vis in latin without loss of implication, and could equally be used to describe emergy work and many other such notions

However, over time the force measure came to be stated As force, a thing Newton warned against, but few took heed, However, before that happened it was common to understand that pressur was reciprocally proportional to surface area in its dynamic effect or vis.

Many issues due to their symmetry could be "solved" without recourse to the surface area of a body, but in fluid dynamics this was not possible/ The use of area was as mathematical as possible, little consideration was given to the physical reality. Thus a pressure measure was derived simply by dividing a force measure. It was known that the pressure in a fluid was the same on all the sides of the container under pressure, but this gave no preferred direction and so sn line segment could represent pressure.

Bernoulli found that in a moving fluid the pressure was not equal on all sides ofthe container, and hydrostatically this was also not true, Few therefore paid any attention to the orthogonal differential in the "forces" that constitute a general fluid pressure profile. fluid pressure in an uncompressed fluid is not equal in every direction, but rather equal in planar surfaces. this means that fluids in motion never have a uniform pressure profile , and therfore will not support a uniform pressure while in motion.

Bernoullis findings are obscured under conservation laws, but the pressure differential in fluids is real but poorly described at the elementary level.

If a plume of water say rises up from the depths, the pressure in that plume is not equal. The lower part of the plume will tend to drive the higher part to spread out due to its greater pressure. similarly the core of the plume will move under greater acceleration than the outside of the plume where pressure equilibrium dynamics with the environment will e occurring. The top of the plume will thus fan out from the centre forming a familiar mushroom head initially, but then external pressure dynamics will shape the head into the familiar vortex ring providing the internal and external pressure dynamics and the material viscosity pressures allow. Often in water the head forms into a spherical ball that pinches off as equilibrium is reached. The lower column drops away and the spherical ball reveals its higher energy or pressure content by continuing on before it too achieves dynamic equilibrium and falls back,

The plume thus actas as a mechanical system that transmits pressure to where a circular or spherical arc can be generated by dynamical interactions. Consider now that the compass mechanism does precisely this by transmitting human pressure, manual. to the point of the pencil or marker which now opposes frictional forces to deposit the shale on the paper, or shear the surface.

We cannot understand  rotation without understanding pressure. 

In modelling pressure the concepts of absolute and differential or gauge pressures become important.

It is important to establish a pressure gradient as this is often an empirical data set which underpins further model building and interpretation. Pressure gradients , like temperature measurement gradients are often calle scalar data sets. While scalar data sets are often also called potential data sets  it is important to realise that they are empirical measurements or the formulae to determine an empirical measure.

Because they are empirical measurements it is often or gotten or not explained that the Barycentric methods are most applicable to any summation or aggregation. Thus the atmospheric pressure at a height is a hydrostatic measurement. The mass of the air at the surface is a Barycentric sum of ll the particles in a test volume from the top of the atmosphere down.  Each point as it descends in a vertical track " gains " mass, that is it is weighted with a greater factor than one above it, and corresponds to a Grassmann weightpoint or weighted point.. Because of this one can sum the atmosphere by evaluating the weight points in a volume close to the surface.

Traditionally the mass of the column is given by a density factorisation of the volume.( or density volume product if you prefer), but the density is then a mysterious ratio, which in fact is defined by use of Archimedes principle. In fact it is better to consider each point as a Barycentric sum of a line , tht is as a Schwerpunkt. We can then see immediately how a directed line segment derives from such a scalar data set at each line segment initial point. The irected line segment pointing own indicates a mass gradient that is increasing . The same line segment is used to represent increasing " gravitational" acceleration . The same line segment in meteorology is used to represent the pressure gradient while the line segment pointing upwards is used to represent the pressure force gradient..

The multiple uses of the same line segment should be related to the Barycentric method. The exception is the gravitational acceleration, which is defined to vary by the inverse square. This means the empirical data set in a Barycentric line in fact is a sum of inverse square weighting factors.  We do not notice that the weight of an object gets less as we lift it, but surprisingly lifting a pendulum clock can illustrate this difference empirically in the difference in the time of swing.

Barycentric modelling is thus fundamental to understanding how a mathematical model captures physical data sets.  This then impacts on more derived models which may involve directed line segment quantities . If those line segments are derived as trig line segments, the hidden Barycentric nature must not be discounted, especially in fluid dynamics and pressure heuristics.

Consider a point A at a lower height in the atmospheric line. If it is moved higher it carries with it it's greater Barycentric factor.  In its new environment it can add aitional mass to the line, creating a line which has a higher potential than its neighbours. The other alternative is it gives up its additional factor to the new height producing a disturbance or perturbation at that point centre! In either case the Barycentric data informs physical expectations ..

Because of this Barycentric interaction with physical phenomena it is crucial not to exclude the rotation of point A. This too has an impact and has a Barycentric description. Grassmanns quotient operators can be summed as Barycentric points, and in that guise they encode rotation of the point. However this summation is akin to the Fourier transform of a spatial form , and this to has implications for empirical data sets.

Currently vorticity is the only measure of rotating material points, but it is not ended as such. Mathematicians have used streamlines and variation between streamlines to define a curl operator, ompletely ignoring the direct spin of a material point! This direct spin is modelled by a Grasdmann twistor.

Placing Grassmann twistors at every point in a data set oes add a complexity, but it should be noted that Fourier transforms do this quite easily nowadays, and the streamline data can be encoded with the same Fourier transform using directed line segments at each point.

The Fourier transform thus represents the needed Barycentric calculus for rotating points and rotational pressure gradients.

These forms  when processed should account for the spiral behaviours in fluids.

Norman Eildberger is doing some fabulous work in exploring the dot product and his notion of Quadrance and spread.

In so doing he has adopted a stance against the use of angle measure, which has softened over the years. However I have adopted the view that arc measure and sector area are missing fundamentals in our understanding of space.

Ŵithout adopting the arch restrictions on mechanical solutions to fundamental issues I feel that there is some fundamental comparison I am missing.

For example rolling the circumference of a disc on a flat surface only gives us a right triangle whose area can be formulated in plane geometric terms. Or a differential argument can be used to split a circle into fine sectors which interleaved to give a rectangle of the area of a disc again in geometrical terms related to the plane.

It thus makes sense to set the area of a unit circle as a standard unit. But some insist on making the radius a standard unit of length which then forces aea to be a transcendental value 2.

It makes pragmatic sense to use the diameter as the unit which still makes the area /4.

What are we tring to preserve by doing this? It seems to be the tessalators of the square. Whereas the circle does not tessellate the square does. However this makes no difference to the count of a unit  factoring an area.  Thus a shape that is precisely 4 unit squares is also for unit circles whose diameters are the unit standard.

What does tessellation give us that is valuable?

What does the space between contiguous circles or spheres give us that is valuable?

The conversion factor tells us we are covering up something that is iteratively endless, ad infinitum! That alone is of great value to know. The square obscures all that rich reality, or rather it confines it to its diagonal!. The circle places it on its perimeter. With such a boundary we have the excitement that anything is possible if we cross ove it into fractal space!

I have just reread the Frontespiece to Ausdehnunglehre, and Hermann claims an application to The doctrine of Magnetism. I suspect it will involve twistors so when I come across it I will post it here? :dink:

String Theory

http://youtu.be/cCkGD76OEwA

http://www.youtube.com/watch?v=cCkGD76OEwA

That old bugbear of dimensions! Dimensions mean cut measures. We cut measures in any of an infinite variety of directions and orientations. Plus we have locii that move in all these directions in sequence. Spirals and circular locii are ways of cutting measures.

We can measure any way we like!

Grassmann freed us from 3 dimnsional thinking in 1844.

This shows how systematically working through Grassmanns idea of projections and products  can cope with varying frames of reference grounded on a real point in space.

http://youtu.be/3gagkzfcWHc

http://www.youtube.com/watch?v=3gagkzfcWHc

Worldview is Grassmnns real geometrical basis.

Puzzling things out by these labels is more insightful than trying to imagine it all in your head?

What is a period?
It is a thought pattern evoked by an every time realised form that comes to be and passes away in a sequence. Thus a dynamic form realises and flows away in a repeted pattern. It is the identifiable form which is the element of this repeatng pattern that is called a period. Thus the pattern is now renamed as periodic ,

This relation between a dynamic thought pattern and a larger repeating pattern in which it is a distinguished element is the formal structure of our construction of time as a periodic, episodic and duration, so a marker or metron.  However more importantly it reveals a fundamental structure to the concept of frequency. Thus all these concepts are tautologies of one another based  on spatial extension and dynamic variation, or intensity and its dynamic variation.

Consequently the Fourier transform seeks to model this period as a dynamic form by dynamic rotations, the geometrical essence of a repeating pattern, the conic sectinss. In making this transformation a fundamental view is realised by ancestral linking It has far reaching conclusions and supports what the  ancients  already knew  of the attributes of space

I remember reading briefly about Newtons water bucket experiment with interest when much younger.

http://www.ifi.unicamp.br/~assis/Mach-Principle-PPP-p37-44(2003).pdf

Of course then I was fascinated by water creeping up the side of a bucket and not spilling out when swung round your head.

Later in some more investigative research I read a bit of a Latin original to check the translation. And remember trying to understand what absolute time was in the face of relativistic time. At the time I equated that TEM as a synonym for Gods time. Everything was relative to gods perception. Thus I was fascinated by the idea that circular motion somehow locked you into gods frame of reference, and could reveal the true force as god knew it!

Later, philosophically  I realised my own relativistic double bind: I could not determine absolute truth because everything I know is relative. Thus I intrinsically believed there was an absolute space  out there. .

Later again I approached this through Einstein   relativity with no reference frame being preferred except by the observer, and still rotation was the mystery behaviour

I Carried that notion of rotation as the key to absolute space to the limit of my imagination and investigative powers and slowly it dawned on me that rotation is fundamental to every motion. I had to generalise the concept of rotation to break free from the simple minded circle idea of it.

I gradually came upon a geometry that is prior to plane geometry , that is more ancient than plane Geometry and indeed is ancestral to it: spherical geometry and trigonometry. I began to realise my concepts of angle were restricted by convention, and finally by studying Euclids first book in the Stoikeia I realised that point and line were derived concepts from a previous circle geometry.

It took time to reshape my views and to try out ideas and logical constructions to test for consistency, but over none existen time, rather observing many cycles of a given frequency, I statistically verified the soundness of my notion.

My studies of Grassmann have reoriented my view of how philosophy relates to specifically Mathematics so called in these modern times, and in fact the source of much of what was called traditional Maths in my secondary education. But the secret of the unit sphere and circle was kept from me

Knowing what I do now has made sense of what otherwise was confused and muddled thinking by Mathematicians, supposedly the clearest and cleverest thinkers of all! I found out that this is all a tissue of lies or halftruths .

Then this year 2014 I was moved to understand Galileo and Newton as a piece.. The fractal thinking of Galileo passes on to Newton who elaborates more fully on it free from catholic persecution., and for the first time I looked up the meaning of Absolute! It just means independent of all else. Even of God!

Galileo was the real source of Newtons ideas on absolute space, time and force, not the water in the bucket!

So Galileo imposed a rule on systems in space only mentioned briefly in the Dialogo, and illustrated to explain it! It was a picture of a circular fractal. Each circular system as an element within a greater circular system was independent because of its rotational centre! Thus any rotation in any system is independent  if it's centre differs from the larger system.,

Newtons Astronomicsl principles are not based on the straight line but on the circular arc segment. This is why Cotes was so sure the logarithm of the imaginaries was the key to a better version of the gravitational formula!

But there is something more fundamental and ancient going on: the descriptive models laid out by Galileo are man constructed ones. If they were divine then the catholic church would not have imprisoned him. And how men come up with these constructions is never taught. The study of heuristics is rarely a school curriculum topic. Hermanns work puts that lack to rights. However he hoped to teach it as a professor and thus pitched his material at too high an academic level.

The more I study Hegel, the more I see his intended target Audience: Studium or Cassical Studies students, not mathematicians!  Hegel only wrote 2 books, the rest are lecture guides, notes and aids. There are over 28 volumes of this kind of material. Needless to say Hegel did not get a chance to edit his material into further books. Other students and supporters did this. Thus we get a view of Hegel that is warts and all! Contradicting himself and his principles, making invidious value judgements , spin doctoring historical accounts to illustrate a point etc, all of which would have been edited to consistency had he had the chance before dying of Cholera .

Hermann thus has the Hegelian principles to work from, and boundless material to sift through if in fact he required it. But he comments in the Vorrede that the contemporary editors were reducing Hegels work to Nothing! In this regard his Ausdehnungslehre is a work attempting to illustrate the power within the Hegelian system .

This system underpins the thought patterns of Galileo and Newton and helps to explain the system within systems that they constructed to model 3d spherical space.

Newton's Absolutes are Grassmanns RechtSystem, the accelerative force system is the ruled local system and finally the motive system is the on the surface centre seeking behaviour. These 3 systems represent a far, a local and an immediate view of an absolute "system within a system"

I have to correct my transforming of Hermnns Richtsystem or orientation system, into Rechtsystem by which I meant Cartesian coordinate system after Wallis, that is an orthogonal axes system. Hermann of course includes the Cartesian system, but in fact thinks more widely and generally than that particular example!

Here  now, this video is of interest because not only is it very helpful, but it in fact shows how rotationl waves are represented by sinusoidal ones. Despite this being declared it is not encouraged to think further about this misleading transformation and its implications on just about every phenomena we observe as deformation of space!

http://www.youtube.com/watch?v=zxrPm7gxID8
//www.youtube.com/watch?v=zxrPm7gxID8

This is a very long video which I have posted in the light thread.

http://m.youtube.com/GObB67ETvRQ

What Eric does is unite Grassmann, Norman Wildberger, Steinmetz  and A good few others into one consistent system . As he presents it means that I can now model a magneto Thermo sono electro complex by a quaternion fractal equation , introducing the time varying and phase varying into a linear combination of twistors or a Fourier series or Laplace series of twistors.

I have done this many times over the years without understanding the fractals I got, but now I can see that they are fractals of the magneto Thermo sono electro complex particularly as they realise in 3d space, or any 3dimensional  reference frame where the dimensions are not just length and time, but other measurable magnitudes used as a dimenßional axis.

One of the more important realisations I have had with regard to twistors is that these trochoids represent not jus locii, but also force and acceleration Actions! Thus when I look at a sinusoid I can see it as an actuall acceleration locii, not just a third party representation of some variation in some " other" parameters!

For example we say sound is a longitudinal compression wave( first demonstrated ingeniously by Newton), and imagine little material molecules springing between one another, but now we can represent it as a standing wave that moves the air  by a ccelerative forces and decelerative force action in this sinusoidal motion in say 3d, giving an actual structure of rotating beads on a string!

If we look at a vibrating string we can see such a structure in action as an example. And of course sound does have a transversal component we have now " discovered". In other words, the theoretical model, though a good approximation, has had to give way to actual measurement, not theory dominating reality!

So Grassmanns style is slowly percolating through my thinking.

The Grassmann -Euler twistor as I have defined it is a very astute product design or function design. But the role of the cyclus group is hidden in this design . The cyclus group 1,2,3 is approximated by , with the constraints

This lays out the cyclic interchange group along radiativey opposing axes. Because of this we lose sight of the "running into against set " nature of the cyclus group. 1,2,3 is one status 3,2,1 is its opposite status, but dynamically, clearly it runs through the numerals" backwards". The axes do not run through the numerals backwards per se, we have to consider 4-1,4-2,4-3 to get the same effect. In short we have to use modulo arithmetic notation to accurately capture Hermanns "running into against set "concept.

Consequently we have to use


to capture Hermanns notion. In addition we have to set to as runs , with this switch occurring every time achieves a value that is not a multiple of 2, switching back when it fulfills tht condition.

This allows the cyclic interaction to fulfill the condition on , and makes any plotting of the arc segment run smoothly around the circle coordinate system imposed.

With some modifications we can now model superimposing rotations as running into against set cyclic interchanges.

We will use 2 Grassman-Euler  twistors set up as discussed. If we add them they represent a constructive standing wave, in which the reflection occurs at a free end . If we subtract them they represent a destructive standing wave at which the free end is clamped . To see what happens when the phase of the cycling group changes relative to the other twistor, we add or subtract a small amount to the in one of the twistors.

What results is a standing pattern with a phase shift.

We may then interpret the phase shift as a cycle start position in a generator, or a time difference in a wave reflection at a boundary, or a free end( in which case it relates to the tensile relaxation time in the deforming medium) or a speed of wave deformation travel.

We can alter the frequency by multiplying by cycles, and the amplitude by multiplying the twistor by a radial scalar.

If a twistor is supposedly translating through a medium then we have to relate the to time and distance. This modification is in fact different to the twistor itself, but uses the embedded time and cycle period to define a length in space. Such a length is then allowed to extend with the embedded time , and the phase positions are marked off to imply wave progression or reflection.

Steinmetz use of these Grassmann Euler twistors to calculate and describe the rotational behaviour of the alternating magnetic current in and around  a transmission line is legendary. The more complex the Twistors( that is if quaternionic twistors are used) the more complex the described results.

Importantly, however, one must not fall into the trap that mathematics is describing nature accurately! Steinmetz was renowned for his expertise, his ability to intuit and comprehend the product design of the Tesla systems, but he refused to claim he knew the secrets of the aether! A small man he lived and survived by bigging himself up in these early days of "electric current" generation and transmission design, but like Tesla he knew he was tapping into something bigger than he could imagine.

The thread V9 now discusses the combinatorial doctrine of Justus Grassmann that underpins all of Hermanns work and ideas.

In particular what an Ausdehnung really is and how the combinatorial label form represents whatever it is.

The Grassmann twistor has taken a leap into an extended form in the thread here, where grassmannnderives it in its quaternion form!
http://www.fractalforums.com/complex-numbers/der-ort-der-hamilton'schen-quaternionen-in-der-ausdehnungslehre/345/
How it applies firstly Combinatorially, then topologically and only then Geometrically is avdiscussionnshowing how far we have come in applying fractal geometrical principles expertly and wisely , given the computing power now available to us!

http://m.youtube.com/BLloW3zzTrc
http://www.youtube.com/watch?v=BLloW3zzTrc

This is ridiculousness in the extreme!

Simply he is constructing combinatorial structures using common or garden processes and number spaces in Mathrmatics. But why use Hieroglyphics?

There is a disease that humans fall prey to. It is called ivory towerism! In ancient universities no mathematics student would become qualified until they got an A+ in rhetoric!

Rhetoric was that class of communication skills which enabled the learned man to communicate accurately and appropriately with all levels of society educated or uneducated.

This style of presentation from the extreme abstract hieroglyphics to specific instances was definitely utilised by Justus and Hermann, but whether it was pioneered by them I do not know. Maybe Lagrange et al were doing this kind of anodine presentation.
Normans latest video suggests that set theory and Aristotelian logic have lead to this style of hieroglyphics. However studying Justus combinatorial method gives us a key into how it may best be communicated.

Normans assessment

http://m.youtube.com/5LsdsnjXT_Y
http://www.youtube.com/watch?v=5LsdsnjXT_Y

A period is a very tricky idea. Its expression in time derives from its expression in space.
So forget about time( Lagrange did) by thinking of a single moment. In that moment a magnitudinal structure of space always exists: different forms and landscapes. To describevthisvtopology we consider timeess dynamical processes that sketch this distribution in space. So when we identify the processes that draw out a spatial structure we can then make these timewise( moment wise) dynamic.

The drawn structure in space was called by Fourier a period! This period is usually a finite spatial structure. By fourier analysis we can draw this by a Fourier transform. Thus we describe a spatial structure entirely by frequencies snd amplitudes of the combined trig functions cosine and sine!

Thus a semi circular disc may be the spatial structure, but the time wise dynamic reveals only a portion of this disc semi circle in each moment. Consequently we may see a swinging pendulum in a specific region, but ln superposition it looks like a semi circular disc.

The quantum question is:is the semi circular disc extant in space or is it really just a swinging pendulum? Or : is what my eyes are seeing what my fingers will feel if I put it into that space?( not recommended!)

So a wave deformation in space as a timewise progression can be modelled as a momental deformation that progresses as a whole, in which case we would see only a step change, or it progresses by superposition, in which case we would see constructive and destructive interaction regionally, and standing wave forms.
Or a deformation could actually progress through a specified space resulting in timewise elastic transverse deformations.

Stress stressors topology and magnetic bonds.
One of the plane facts is that matter is variable in its intensity and that it has continuous and discrete forms .

Further we recognise energy -phase changes that is to say simply : solid liquid gas plasma. States of matter with different vibrational signatures.

It was determined by Newton et al that heat was best described or pictured mentally as a vibration in a corpuscular region corpuscles were thus conceived as flexible plasmas from the nascent biological discoveries of the time.their flexibility was merely a way of accommodating the vibrational modes conceived as heat .
We have since developed the heat model into the Energy ,Odell.nsuch a model really has its foundation in Gilbert's magnetic philosophy which recognised 2 forms of magnetism: crystalline lattice or mineral magnetism and biological magnetism, that is organic lattice type. The one was called the iron or ferro magnetism, the other Electra magnetism or amber magnetism.

These specs were later intensively studied and distinguished into models that were fundamentally mysteriously related. However the various guilds of magnet makers and electric charge makers  jealously hung on tontheirbtrade secrets and further distinguished the common link, that is the fundamental material experience and phases of matter at different energy or vibrational states.

Matter and space were philosophically divided in the most natural way , but space was called empty for religious reasons. It was desired that spirit be distinct from matter, and invisible able to manifest to conscious souls, themselves spiritual. 

Such a model kept the material world that technology was exploiting as a distinct environment that mankind could "play" with like a child, but spirit was the playground of the gods and those ascended humans who struggled to grasp it.

Of course the material philosophers were brought up in that Matrix of ideas ad models, but younger more ambitious material scientists saw no valid reason to limit material science in this way and technology indeed advanced to miraculous inventions gadgets and contraptions which supported that viewpoint. 

Unfortunately the mathematical clan or guild retained enough obscuranting mystics to cover the material  viewpoint with a pseudo mystical or religious analogue of religious theology . BishopmBerkely wrote a powerful attack on mathematical religious views pointing out that material scientist of the mechanical philosophy were in fact no more clear of superstitious nonsense or beliefs and practices than religious theologians and clergy

Accepting that point we may progress toward a common view proposed in detail by Hegel et sl and eastern mystics since way back, that spce is in fact something that is connected to a singular concept that contains all: if you like spiritual and material are two aspects of an entity with many aspects.

The Aether or Ether philosophies attempted to place that view as foundational to the material sciences, but all clans and guilds fell apart in disarray: a modern Babel of jargon and gobble de gook .

We do not have to be so inclined.  A simple and useful combinatorial model underpins the technological achievements of mankind. It is rather humble in its scope and often  symbolically and ritually obscure. Attempts to make it clearer usually do the opposite, academically. But those who actually make things using their whole organism develop inexpressible expertises that often seem to those not involved in the physical actions of making , well they seem like magic, even magycke of the old kind.

Regularly organic and rock crystals cry out to us ! We call them TV's and Radios!! 

So we can take material science or specifically the science of material and develop an expertise that Tribology or tribologists express. This level of expertise is an art form found among the few material scientists with a broad enough range of experiences and understanding. It is eclectic and very much in its infancy .
For example, topology would help to provide a structural overlay akin to corpuscular philosophy of Newtons time. Topology fundamentally is a combinatorial art that gives forms or formats for space in its most general Aetheric model . These formats may then be used to count and measure regions and distinguish behavioural changes in material over time, circumstance and internal and external conditions. 
Topology allows us to place atomic or molecular models into regions of space so we can characterise certain regions by their phase and nearby status and their visual characteristics . The stresscandvstressors in such a space become characteristics of the phase of the element or molecular material as well as contact behaviours between such characterised materials.

Clearly a chemistry and organic/ biochemistry based on these characteristic measurements and observations has proved very fruitful in organising our technological advances. But we have found some who want to isolate these behavioural and technical expertises away from common knowledge to support some particular world view or religious ideology. 

While the knowledge is presented as iconic, eclectic, difficult to understand, even iconoclastic, it is in fact not. The simplicity of the energy/ vibration - phase model of matter is how it has been hidden in plain sight.

It is a simple robust and powerful combinatorial system that was advanced by Navier Stokes and yes even Newton into the fluid dynamic model. Helmholtz and Kelvin added the Vottex model rules (inaccurately: see Claes Johnson work) but all of this was obscured by the Mathrmaticians. Helmholtz following perhaps Euler and Lagrange felt that the differential equation format was the safest way to express natural laws of motion in space-like material especially if they were fluid.

In a sense , given the prevailing attitude that aether was basically solid billiard balls at the femto metre scale( an assumption since demonstrated to be misleading especially by quantum Mechanics) his reliance on the continuous nature of differential equations by design does preserve the fluidity of the foundational corpuscular theory of Newton et al. 

The use of hard billiard ballasts representing corpuscles is a mistake we can trace back to Leibniz et sl . But even then he was not insisting that matter was thus. The mathematical topology was simpler if this was assumed , and produced useful models. It also produced strange aberrational models which rightly were ignored! Now they are mystically clumped together in the quantum mechanical world viee.

The fluid mechanical expertise is very difficult to establish except in the simplest cases, but in fact computationally it has provided us with some excellent models within the boundary conditions appropriate to the study. The advent of more powerful computers and computational schemes has made it possible to model complex data increasingly accurately, by applying a whole mish mash of equations topologically to a spatial region.

The one fundamental breakthrough for physical topology is ascribed to Benoit Mandelbrot. His Fractal Geometry  inspired many computer geeks to model the particular types of inductive or iterative( recursive) equations that hitherto were avoided by Mathematicians as monsters. Ad yet we know Focault and many others were studying them quietly prior to the wars with little academic support . 

We can laugh, but it was mathematicians that were the Luddites when computers were first introduced!! 

Today I watch a lovely flowing scene of actual waves lapping onto a shore with all the sound effect on a flat, crystalline organic matrix presenting the display of raw data as processed by billions of logic gates in some silicon(rock) chip processor with a certain vibration/ energy which has to have the heat removed by invisible gases passing over it as winds! 

The fluid dynamic models expressed as differential Equations may be nonsense. But the expertise that organises itself in this way around the energy phase model really has a powerful set of models with which to develop further models and technological advances .

Why have I not mentioned Maxwell, and Einstein? Because they represent a deliberate mystification of the simple model. Faraday was not happy with how Maxwell fundamentally distorted his ideas. In fact Örsted Faraday, Ampère Volta and others held that a circular force o rather a rotational force was fundamental to our understanding of all models built on observation. Newton, unlike his successors took a circular or rotational force for granted. 

This simple topological element distinguishes models that account for quantum behaviours from those that do not. 

And yes the mathematics becomes confusing and complicated, but only when the circle and the trochoids based on circles are excluded.

Include trochoids in space in your models and you can model magnetic behaviours topologically in a consistent way.

The gravity of this point should not be missed, as Gravity to this day is still touted as a mysterious force,  all forces are fundamentally mysterious characteristics of topologically described space. Their iscavsimple reason why conic curves describe both gravity and magnetic attraction and repulsion. The topology of space that best describes them is rotational or circular. Add to that fractal geometrical topologies and you can model fluid dynamics in appropriate small boundary conditions.
Add phase changes at different energy and thus vibrational states( frequency amplitude and trochoidal topologies) and you arrive at discrete regional distributions with phase changes marking the boundaries of regions fractally.
What we know about bonds is dependent on these fractal regional surface phase change behaviours at differing energy/ vibrational statues.

Hopefully you will bear with the many typos in the above post and be kind enough to point them out to me!!

I am continuing my fundamental research of thecGrassmann ecology of ideas. As you may know they were on the edge of the intellectual revolution in Europe but by no means totally isolated. They particularly had access to many good books on the Philosophical implications of events and discoveries in their time.

For me they provide an accessible entry point into some of the grass root fundamental issues of the day which more importantly resonate today.

While the post is clearly based on fluid dynamical terminology, the deeper connection is the non subject based philosophical enterprise of modelling the space around us. I believe the AusdehnungscLehre gives all philosophers a combinatorial framework to develop appropriate models and counting and measuring formats to empirically test ideas of extension and contraction of space-like objects.

I believe the twistors or rather Fourier decompositions of appropriate regions are fundamental characterising descriptions of material and spiritual space .

As far as the metrical aspect of the analysis goes it can characterise fractal regional behaviours sufficiently for technological manufacturing projects. Nevertheless, despite the film flam of exact sciences this is still an artisanal enterprise. Skilled expertise is demanded in the manufacture of entities at such small scaled.
 A fluid dynmic corpuscular dynamic approach seems to be essential.

Particle Physics is of course the biggest scam on our finance, but it has a basis in corpuscular dynamic theory traceable back to Gilbert and Newton et al.

Quantum and classical mechanics are a smoke screen. two schools of thought fundamentally dis agreeing on approach are duking it out to get or keep their hands on the money!!

Twistors effortlessly combine the two.

http://m.youtube.com/watch?v=6dW6VYXp9HM
I want you to have a physical model of a twistor!
This Fourier Analydis machine shows how engineers know how the "wheel works " of nature structure in space and time.
Yes these are ideals and require an inductive fractal mind set , and yes fractals and vortices are way more complex, but this is the einfachste model, which we iterate by scales to describe more complex  structures.

http://m.youtube.com/watch?v=TAHczLeIUTc
This technique was well known from the time of the Akkadians and Sumerians! Thus gears cogs cams were all engineering structures that could be built and designed by Arithmoi , that is spaciometric forms that used topological combinatorics to design desired motions( kinematics)

http://m.youtube.com/watch?v=jiejNAUwcQ8
Notice the role of the vorticular topology

Thanks,  I found it very informative. :)

http://m.youtube.com/watch?v=r18Gi8lSkfM
When considering trochoidal rotations be aware of the dynamic nature of rotation. Hence recognise the need for space-time .
Traditionally we are taught axes as orthogonal triplets. We do not question this presentation as we have no idea what axes are?  We are then hoodwinked into some faux reverence of Rene Des Cartes "Coordinate" system. The truth is DesCartes did not establish such a system!
The Greeks already used and established several flexible reference systems for locating proportions in space . Apollonius in particular refined a system that was " shape" or fom concordant!
In all greek thinking the circle shape was fundamental. Similarly both in India And Egypt the Sphere was a fundamental form. Not neglecting the Sumerian/Akkadian contribution in which many wisdoms of the ancient world were collected by the Empire and cuneiformed.

So the line segment became a fantastic tool for recording ratios / proportions ( Logos Analogos ).
The hardest won ratios were the chord and later Sine ratios. These ratios were not tabulated until much later, instead an Egyptian practice of using the circle to freely construct any right angled Triangle was later ascribed to Thales .

Wallis after studying greek thought combined many ideas into the measuring line concept, and many circle properties into the orthogonal coordinate Ordinate system we attribute to DesCartes! Really DesCartes ideas survived as the Lagrangian Generalised Coordinate system.

So what is an axis? It is a Spindle around which a wheel spins . It is derived from the concept of an Axle. The same notion underlies the word "axiom".

Wallis is responsible for the Law of Cosines, through which he was able to write down a general expression for the conic section curves. Until his breakthrough revision, Conics were carefully carved out of blocks of wood shaped into cones!

These cones themselves could form a reference frame, but we are so indoctrinated into the measuring line concept that we do not find that a natural thought!

We have many coordinate systems all based on a Fractal measuring schemes, but we typically obscure their inhomogeneity  . We replace it by some pseudo reverence of Pi ! Because of pi many have no understanding of i that is sqrt(-1). And yet t most profound transformation scheme was set down by Cotes, and Decades later by Euler in greater detail that expressed how to use i as an axial lable for circular arc "lengths " or more accurately magnitudes as Euler and all classically trained philosophers described extensive and intensive experience.

Thus Fourier reintroduced this freedom of coordinate reference frame at a time when many of the intelligent elite were adamantly stuck in the straight line mode of measurement, believing their own assumption of Newtonian principles for Astologers and ignoring his general rotational , vortex exploring dynamics.

The Fourier transform enable us to reference a point in space using curved planes not merely straight edged planes. Typically you will require at least 3 intersecting surfaces to identify a point in der Raum!

This video introduces the method in the most accessible way for Fourier Analysis along one axis of rotation. Using 3 or more axes of rotation means any spatial form can be represented by a data set of Fourier transform coefficients.

We can view space-time as this rich description of rotating axes for which we can measure a data set of coefficients unique to any particular form.

More than that: we can characterise not only form, but also haptic force, colour intensities , physical signal response characteristics especially magnetic resonance responses for  specific space - time topology. That is, any object is a space-time object not a classical " geometric" on.

For space-time we may substitute a Fourier aether .

http://m.youtube.com/watch?v=PNHSIEO-KOQ
Angular momentum

http://m.youtube.com/watch?v=leZX0GpV5W0
Rotation here is represented by a bivector or a parallelogram . This is a result of a stepping out product of two line segments.
Otive however how the dynamic is not explained for the circular rotation! The momenta in this case are instantaneous . Thus we observe a changing momentum for  a point moving in a circle and thus a Circular Force!
A circular force is one of those observed "realities" that we are taught to ignore! It includes centripetal and centrifugal forces as orthogonal resolutions and thus orbs a complex system of forces that can beadptec to any convenient measurement scheme
 The issue of action at a distance as pondered by Newton is a philosophical one, particularly in the light of th magnetic behaviour / vortex bout a wire . Newtonnalready observed that a tether or a series of impulses directed centripetally could produce circular motion, but he was less clear about vortices in lids as he regarded these as resistive media . In that light he posited differing densities in outer space resisting planetary motion , thus providing reason for planetary bodies to lose there motion over time . But he could not posit magnetic behaviour as causative because what little he knew about magnetic behaviour both ferro and Electra as Gilbert distinguished them were too occult in his day to apply to planetary motions in a believable way.

However he was well aware of Robert Boyles Theoretical exploration of magnetic planetary attraction as an occult explanation of observed behaviours. These writings were very carefully written and circulated secretly to avoid persecution by the church and social rejection, something Newton could not abide. Newon therefore only uses magnetic behaviour metaphorically In the Principles for Astrologers.

However we should know better today how magnetic behaviour is a universal vorticular behaviour making action at a distance understandable as a aether/ spacetime phenomenon due to fluid motions. The notion of a fluid as a resistive medium only has long since become observationaly obsolete!

Within a luid vortex we may at last understnd pressure as th primary manifestation of energy density and thus force as a Lineal symbol of that pressure where the area of action is theoretically a Point .
Spaciometrically if we apply the same density of an object in all it's volume to a smaller area the effect observed is an increased pressure .  In contrast if we divide the density of an object into a larger volume the pressure on any test area decreases. Density therefore is a measure of stored energy in an object. That energy is relative to a given region throughout which a Centripetal accelerative force holds .

For those who have read Newtons principles they will recognise this as a description of Newtons 3 " vis" system, derived from Galileos observations of Jupiter and Venus.

What is this region that gives relative density to parts within it? It is principally a  Magnetic region, often called a gravitational region! Around any lodestone magnetic behaviour is characterised by Newtons 3 vis system with an additional observational factor: vortex behaviour!

We are traditionally focused on magnetic attraction and repulsion rather than on magnetic vortex behaviour!
In space a pair of magnetic bars will repel rotate and attract in a single dynamic . The magnetic accelerations and dampings are simply ignored ! In addition the highly coherent structure of a bar magnet is ignored .

Magnetic vortices are a necessary condition for any proper description of astronomical behaviours.
Setting that as a sound basis as Gilbert and Boyle proposed would be a sound direction or modern astronomy and indeed modern Science as a whole .

The combination of 2 line segments is a third line segment in the plane of the combining pair, but the combining pair are also factors of a Parallrlogram in that plane.
It is the notion of combination and the notion of factorisation that we use to construct any symbolic arithmetic. This symbolic arithmetic is therefore based on spaciometric forms . It isbthereforeva spaciometric Algebra
The equivalent spaciometric Algebra for circular arc segmnts is far more interesting .

The combination of 2 circular arc segments is a third curved line segment in the same plane called a trochoid.
This is a simple introduction to arc segmnt combination Inthe same plane.
http://m.youtube.com/watch?v=rz8A5l_yn34
It is far more complex.
In addition 2 combining arc segmnts do not necessarily define a unique plane. So we have to allow a unique   surface that contains the two combining arc segment must also contain a trochoid linking the arc segments . In this case the arc segments are factors of a form that lies within this surface containing the two. The form is equivalent to a Parallelogram in that surface.
The symbolic algebra therefore describes trochoids and trochoidal surfaces volumes etc.

The forms especially in 3 or more combining arc segments are very complex vorticular forms.

Some are clearly bubble like in form as qqazxxsw demonstrates, but generally they are energetic dynamic forms capable of describing any dynamic behaviour : especially charge separation, attraction and repulsion and fundmental magnetic behaviours.
http://m.youtube.com/watch?v=KCJ3d_CUb-Y

http://m.youtube.com/watch?v=LpqwdIoWgw0
The quaternion Fourier transform is what I define as a twistor magnitude .
This particular application uses the vortex decomposition of a colour sphere with a light to dark tonality axial centre . These are not physical twistors but as the decompositionis plotted the great twistors show a twisted shape while the higher frequency ones show definite regional sharpness until the full image is now perceived and the twistors ignored

http://www.ijcsit.com/docs/Volume%205/vol5issue03/ijcsit20140503378.pdf
A reference for the method based on the original work by J J Sangeine .(1996 ?)

https://www.researchgate.net/profile/Jian-Jiun_Ding/publication/3318160_Efficient_implementation_of_quaternion_Fourier_transform_convolution_and_correlation_by_2-D_complex_FFT/links/0046351f971f308e17000000.pdf

The methodology utilised here to improve efficiency obscures the basic combinatorics involved in achieving this complex result!

The reason to point this out is simply that Justus Grassmann made a special study of combinatorial process and. Methodology which informed Hermann Grassmanns work on the Susdehnungslehre 1844. Now modern combinatorics is a sprowling field of study, expertise and technique that underlies and motivates all mathematical, and spaciometric disciplines. In fact it underlies every system or methodology of thought and theory whether linguistic or spatial/visual , extensive or intenive in magnitude.

The simplicity of putting blocks together is obscured by the complexity of the blocks!

Twistors may be a complex quaternion building block requiring expertise, but how we combine them or anything else to do pragmatic things is essentially and heartwarmingly simple!

http://slideplayer.com/slide/9379777/
One of the open secrets of radio waves is the frequency bands. These frequencies immediately are obscured behind a sinusoidal representation. In fact all electromagnetism is. Nature is observable and die waves are not! In fact ancient societies observed the trochoidal surfaces of large bodies of water in a continuous rolling motion. Undulation means rolling motion.

The wave description of electromagnetism has been overshadowed by a misleading notion of a wave. Led Rayleigh made ome important observations regarding wave Mehanics but chiefly the rotational dynamic of surface waves. This of course is obscured behind mathematical expression.

What rotates in space!? The only observed rotational force on a local scale is magnetism around a inducting/ conducting wire. It is that same wire as an antennae that transmits and receives radio frequncy signals( and higher) .
The magnetic rotation ( polarity rotation ) spirals outward and also inward as a pulsating system. The spheroidal nature of these point rotational emissions are described by twistors . That these rotational systems can be used for image processing , radio and TV signalling is just part of te intensive/ extensive magnitude utility of these Ausdehnungs Größe!

http://m.youtube.com/watch?v=z_6B2M12H9w
This is a very important video to begin to grasp why theoretical physicists back away from physical or topological spin.
Now Susskind explains the observed results with the theoretical thinking associated with it.
http://m.youtube.com/watch?v=ZCymP87zFwc

Firstly note the fundamental requirement for a magnetic field to measure " spin" or prepared orientation against output in this Case called a photon.

The description is probabilistic. That means the topological behaviour is not physically observable !
Normally for small or invisible to our senses behaviours we can induce a topological behaviour from the consistency in many experimental set ups : invariance in the results .  When you get a probabilistic result you can not induce a damn thing!

However we can use a different type of thought pattern, not induction based on invariance but deduction based on a model behaviour. The model used is a complex vector space. .
Using this model we can perform a calculation called complex multiplication. This allows us to find the magnitude of a complex vector. It turns out that the magnitude squared is related to the probability distribution in terms of its standard deviation .

So we have a tenuous link between measured data set, capturing a probability distribution and a theoretical complex vector space. It has no topological counterpart in physicl space! We therefore deal with the vector space topology!!
 The vector space topology can be modelled in real,space but it describes the vector space topological behaviour.

Hamiltons quatermios are such a space, found to be useful for describing out puts for otational in real opologicl space.
Careful Mathematicians do warn against conceiving quaternions as describing actual rotations in physicl space . The calculation does not determine the physical rotation only the output for 2 specified inputs!

Ok, so twistors deal in this same kind of mathematical jiggery pokery, but instead of using a pivoting arm as fundamental I use a arc segment of a specified radius.

This spatial coherence to trochoidal motion means that I am more intuitively linked to physically observable trochoidal behaviour.

So does the set up describe anything physical? The only thing observed is magnetic flux behaviours.
Magnetic flux has to be theoretically determined/ defined before we can make statements of interpretation!

Change that fundamental definition and we change the fundamental interpretation .

I make no secret thst I do not think the standard model is the only or best model!

I prefer topological rotation / vortices as fundamental fluid dynamic material points .
http://m.youtube.com/watch?v=5UqDb2BcxZk

http://m.youtube.com/watch?v=uRKZnFAR7yw
If you are struggling with Susskind Njwildberger makes it easier.
Now understand Eulers treatment of arcs of rotation and why you have to go round twice to get back to the start.
http://m.youtube.com/watch?v=0_XoZc-A1HU
Notice how the usual orthogonal vector description for axial rotation is in fact misleading. Njwildberger shows how to properly sum two arc rotations about a point!!
again as Norman is careful to point out, these rotation arcs represent rotations twice there arc segment magnitude.  
so for a quarter turn we can expect the rotation to be passing through the initial point on its way into the opposite orientation! We expect a figurre of 8 or some other Lissajou precession to occur depending on frequencies involved.

So called electron spin is far more complex than some would allow, and the probability distribution for the results reflects the fourier analysis of the frequencies involved.

Spin up and spin down reflect the transition between Lissajou patterns which are very stable dynamical ones.  
https://m.youtube.com/watch?v=vNuDxc9tZMk

The emision of a "photon" represents the precise frequency differences between the stable Lissajou patterns,
https://m.youtube.com/watch?v=wvJAgrUBF4w
cymatic Lissajou or Chladini patterns

http://slideplayer.com/slide/6231443/
The versatility of quaternions lies in their rotational metaphor. However the belief that quaternions are numbers but higher dimensional structures are not is a chain around the neck of Natural philosophers . The weak link in this chain is the concept number . As we may garner from Grassmanns efforts the Combinatoril and compositional nature of our interaction with spatial forms is not bound by numerical Aritmetic. Instead Arithmetic is freed into a greater spatio entry of symbolic algebra that serves to describe both xtnsive and intensive magnitudinal experiences and processes.

http://m.youtube.com/watch?v=oW4jM0smS_E
http://youtu.be/oW4jM0smS_E
In this video Norman demonstrates a method ascribed to LaGrange and Euler but in fact developed by Newton in his scrapbook.
The binomial expansion of a polynomial/polynumber or any function expressible in the form of a power series ' in the manner LaGrange advocates  gives the binomial coefficients as precise Derivatives/ sub derivatives for a given argument of the function or symbolic polynomial.

These Lagrangian concepts underly the iterative application of quaternions!

The quaternion is a form of infinite series spatially determined by circular planes orthogonal to one another. Becausebof notation we miss the lineal combinatorial nature of a polynomial. But it is from a polynomiàl that the concept of i was first noticed and exponded upon by Bernoulli . However despite Hamiltons Förderung it is the conception of Hermann Grassmann based on the works of all the Grassmans that is foundational.

So the Fourier series at first expressed as a lineal combination of sine and cosine functions eventually becomes expressed as a lineal combination of exponential functions with complex thên quaternion logarithms!
Powerfully we have relabelled a combinatorial sum of polynomials as a lineal combnation of exponentially expressed quaternions. And we can go further, vut that is beyond what I need at the moment.

In this format it becomes clear that amplitude, frequency and phase are vital and that these are interpretable not only as standing or static forms, but also as dynamic forms with reflection , refraction and diffraction .

The complexity of these kinds of formulations mentally requires a thought pattern associated with an expertise. The correct expertise actually dïd not arrive until the advent of electronic computation using binary arithmetics stored in magnetic dipole cores with a facile hysteresis response!
But even then it has taken until now to bend our minds to the modelling possibilities perceived by Hamilton, yet glimpsed with ardour by the Grassmanns . That is not to say Euler and LaGrange and Newton and Cotes did not also glimpse these things but the notation of notions , the Begriff' was not standardised or even perspicacious until Hermann made it so! And then he was accused of being obscure, introdcing too many new terms ' not academic enough like for example Moebius!

Well I foregive him for trying to make what he was doing so clear that any child not educated can see what he is referring to !  It's a trip for those academically trained, but for the layman he uses clear metaphors and references. Who would not understand decorating the interior of an array with enduring condition markers , or how one array grips into another final array , or colliding vertical line segments or outwardly stepping linè segments ?

In this light a twistor records in the complex part of the quaternion exponent, the frequencies ànd phases of any quaternion Fourier Transform and the real part records the amplitude. Thus expanding and dilating or condensing spheres combine in Rotation to make a static or dynàmic form appear in a region of space "

http://m.youtube.com/watch?v=8SiuTk2ZiyU
More uses of trochoids and vortices  and how rotation underpins magnetic behaviour.
The diffraction grating effect is a quaternion Fourier Transform structure.
Coherence is what is induced in MASING .
As Facinaying as Prof Lauthwaites demonstrations are and as helpful as they are, they are inconsistent with one another. The Archimedian screw is a much better analogy explaining how the copper barrel rotates in the alternating "magnetic field".
And why does not a reflected screw stop the rotation? Because the copper and the aluminium are both induced with their own Archimedian screws. Net resultant is a motion determined by initial conditions the second Archimedian screws usually ignored by modern theoretical explanations. The 2 screws opposing each other still pass through each other and this superposition creates the " gear tooh" that bites into the material.  Inertial forces (Lenz law effect) give the resultant from initial conditions.
Finally, phase trains of higher rotational harmonics (often ignored)  preponderance the initial impetus but vary in expression depending on material inductibility / conuctibility.

http://m.youtube.com/watch?v=QfDoQwIAaX
There are 2 types of Quaternion Fourier Trnsfotm. The traditional one is based on Eulers exponential nittion , the less familiar one is based on Sir Roger Cotes Notation.
The first type models forms explosively.  The second models forms expansively but more in keeping with human visual.expeience.
Often distant objects do not appear to be moving very fast because the scales are compressed by the visual field effect. Objects far away present a low angle of incidence on the retina while those nearer present a large effect, the behaviour of the logarithmic function mimics that.

The logarithmic quaternion presents a large modulus quaternion within accessible scales while an exponential quaternion soon blows large modulus uatenionic out of sensibility.

The video is a demonstration of how a 3d fractalmgeneratorbapplication like Quasz By Terry Gintz handles both types very naturally but very differently in terms of output sculpted form.

The projectiles are the Fourier forms as classically expressed, the impacts represent the effective pplication of Fractl iteration on the Fourier transforms. The explosive decomposition of the form is either exponential in nature or logarithmic. Or some combination of both.

http://portail-video.univ-lr.fr/Quaternion-Geometric-Averaging-in
The video here shows the thought process that designs the product required for filtering an image presented as signals for each colour map in RGB , plus an additional signal for depth say or intensity/ tonality/contrast..

These signals come in as voltage variations but essentially they are phased. Thus the complex part of the quaternions represent phase arcs controlling a pointer. This picks out the colour point and the local technology implements that colour reference on screen pixel by pixel.
Because of this pixel by pixel nature we in effect have a quaternion Fourier transform being transmitted term by term . The sum is the image on screen.

This topic is about manipulating the transform to give the best image output so the signal is not sent direct to screen but through filters that do the desired job.

While rotation seems not to be actually involved this is far from the case ! Rotation and cycling are everywhere involved in signlling we obscure it by calling it a wave!

https://en.m.wikipedia.org/wiki/Absolute_rotation
In the system of reference framescNewton constructed absolute rotation was key to his interpretation of thebGalilen Fractal Solar universe.
It is to be observed that abolute means simply independent , thus it is  a sine qua non that a circular rotation is independent of any other centre than its own . This means that  a fractal systm of bsolutes can be constructed on circular orbits conistently.

However the implications of the absolute nature of circular dynamics is tht Fourier Analyis of an object can be comprised of the um of rotations round a single centre making each uniqued centre of an object not only the place to sum all translations but also all rotations of the whole or parts of that object.
The further implication of Fourier analysis is that a rotationl description of dynamics is achievable and potentially more in keeping with observed behaviours.

http://m.youtube.com/watch?v=LpqwdIoWgw0
Grassman Twistor used to do digital image processing.

http://m.youtube.com/watch?v=4Rx35q-zJRk
How twistors encode biological information

Most of us as students of Newton have a problem with force.
We are taught the proportion mass x acceleration . It is in factor form and it is as Newton explained, a proportion of observable quantities with an experiential but non visible phenomenon. It is actually a kinaesthetic experience called pressure and is clearly in proportion to surface area of contact.

However the mass of an object is not proportional to its intrinsic volume and thus it's surface area, which may in fact increase without change of volume. So the inverse or reciprocal proportion is used in any definition of the proportion for pressure.

We have 2 related proportions for the concept of force : mass x acceleration and pressure x area of contact.

The difficulty comes when we wish to describe a contractive force as in contracting materiality. And also, but less obviously to most in describing attractive forces!

Most of us have been taught to use the negative of a force to describe an attractive force. We do not even realise how misconceived that is.

So when Dirac arrives at the idea of negative energy we are totally unprepared to understand what that might be!
Like imaginary numbers negative integers were a very taboo subject for most of our top western philosophers especially DesCartes who preferred to work in the positive quadrant of the so called Cartesian plane.

It was only when Euler established the exponential function in generality that most became aware that Sir Robert Cotes and Isaac Newton had been discussing the use of the proportion we call e, the evaluation of the binomial compound interest growth factor in some detail as it related to Gravity.

It was later understood that Cotes had formulated the logarithmic form of the Euler expression for the evaluation of the imaginary exponent .

What few realise is that Newton was on the verge of reworking his by now famous gravitational equation, based on Cotes ideas.

Newtons system of forces based on Galileos observations of the Jovian system and the behaviour of Venus would have then been based not on the inverse square law, but the more general logarithmic law as expressed by Cotes.

In newtons system the absolute behaviour in an orbiting planetary system is proportioned down to the planet surface by a spatial influence that acts centripetally at every point  in the system, and objects of mass are a measure of that orce by the process of the balance omparisons. Thus mass object acquire their motive from this centripetal force behaviour which is accelerative.

Philosophers are arguing whether this is the first expression of the modern Field conception, but of course none of them really understand what the field idea really is.

The consequence of this change would have been to make force an exponential proportion, and thus to model expensive and contractive forces by exponential ombinations.
Dirac was forced to make sense of his ideas only in the accepted format, o negative energy could not be understood as the contractive energy because by common acceptance energy was positive only. And negative forces are not contractive forces in everyday physics!

The exponential versions of physics were carefully separated into real physics and imaginary or heretical constructs in physics.

Complex functions in physics are still routinely explained in this old mistaken way of understanding negative and co plex values, or they are smothered in quantum mumbo jumbo !

Few want to do physical calculations as exponents to he base e because they misguidedly think mathematics is the language of the God of Nature and nature is direct, simple and real!

Twistors are in fact the basis for doin physical calculations because they are a better model of general behaviours than the direct measurement schemes of biased Natural Philosophers from DesCartes onwards.

I am sufficiently happy now to regard the Grassmann Twistor as a circular arc extension, or circular vector.

The notation, which derives from Euler, I would modify or simplify in today's computer rich world .
Exp( iø) is a function form which obscures the direct circular vector .
Some use for this circular vector but I am not familiar with that notation yet and it does not appear to accommodate the necessary radial or diametral line vector that must be associated with a circular vector. The radial vector has to record the phase of the circular arc.
The exp(z) format certainly allows all of that which is why I like it, but it just does not look like a vector , but then again neither does the vector cross product!

That is the nub of the difficulty . I want to be able to write a cross product of circular arcs of differing phase and amplitude without confusing it with ordinary power form multiplication .

In any case the general cross product is only a vector in vector algebra, not in general Clifford or Grassmann algebra, where a rotation about a point in a plane is directly referenced

Some applications of Grassmann Twistors

http://youtu.be/qm5I_D9NN3g

http://m.youtube.com/watch?v=qm5I_D9NN3g

The explanation is very sketchy but it is saying : take geometrical forms as magnitudes that are extensive/ intensive, do nt ignore the quarter turn/ rotational magnitude, ke the dynamic relation between all magnitudes clear and then use this analysis to synthesise a useful product.
Recent work on oriented magnitudes highlights that orientation is subjugate to rotation, and the minus sign does more than imply spatial orientation, it implies relative quantity , that is >=<. So the quarter turn magnitude may be used to sore process until it becomes synthetically relevant , and impacts on the Final product. the key point is not to give power to the mathematical process but to the person who designed the process to produce a useful end.
The use of complex forms in electrical nine ering is a good example. : the neatness of the process made writing down the circumstances more systematic and symmetrical, and the combined product made it clearer how to do the calculations which gave the expected results.
It was quicker and clearer and thus more practical. It was not " math magic" it was a better designed product.

When Grassmann explained Quaternions, demarcated by Hamilton, it was not to denigrate Hamiltons achievement, but to show clearly the system of synthesis to which they belonged: the Ausdehnungslehre
. And Clifford was a respectful Acolyte of Grassmann. His Algebras continue the geometric doctrine laid ot initially by Justus Grassmann, in summary of the then current state of Science, nature and Mathematics. In so doing he drew on the great minds of his times and presented the essential intersection of all their idea, in a accessible geometric firm rather than a pure symbolism of logic which frankly failed to capture the basis of multiplication! The language of logic is thus subservient to the apprehension and expression by drawing of spatial relationships and magnitudes.

In time we learned how to use extensive magnitude to depict and node inteni e magnitudes like colour and touch and smell!