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 Author Topic: Twistor  (Read 8065 times) Description: some notes on Hermann Grassmann's Ausdehnungslehre 1844 0 Members and 1 Guest are viewing this topic.
jehovajah
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 « on: April 28, 2013, 09:31:15 AM »

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
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jehovajah
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May a trochoid in the void bring you peace

 « Reply #1 on: May 08, 2013, 04:34:00 AM »

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
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jehovajah
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May a trochoid in the void bring you peace

 « Reply #2 on: May 17, 2013, 08:50:38 AM »

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
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jehovajah
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May a trochoid in the void bring you peace

 « Reply #3 on: August 08, 2013, 05:13:08 AM »

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

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

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.
 « Last Edit: January 17, 2014, 07:59:47 AM by jehovajah » Logged

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jehovajah
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 « Reply #4 on: September 12, 2013, 12:30:43 PM »

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.

 « Last Edit: January 17, 2014, 08:01:59 AM by jehovajah » Logged

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jehovajah
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 « Reply #5 on: January 11, 2014, 12:16:06 AM »

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.
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jehovajah
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 « Reply #6 on: January 16, 2014, 11:25:17 AM »

Conservation of rotational momentum.
Compare with torque.

http://hyperphysics.phy-astr.gsu.edu/hbase/n2r.html#n2r
 « Last Edit: January 17, 2014, 10:57:43 AM by jehovajah » Logged

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jehovajah
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May a trochoid in the void bring you peace

 « Reply #7 on: January 16, 2014, 11:35:32 AM »

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.

///////////////////////////////////////////::::::::::::::::::::::::::::://///////////////////////////////

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
Ft = €r

Finally traditional torque is defined as
T =Ftr

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 Ftr 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)

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.
 « Last Edit: January 17, 2014, 09:10:35 AM by jehovajah » Logged

May a trochoid of ¥h¶h iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
jehovajah
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May a trochoid in the void bring you peace

 « Reply #8 on: January 16, 2014, 11:41:30 AM »

Defining the Twistor

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

@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

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 $\ sqrt{2}$ 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.
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 $\sqrt{-1}$? 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.

 « Last Edit: January 17, 2014, 10:20:42 AM by jehovajah » Logged

May a trochoid of ¥h¶h iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
jehovajah
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May a trochoid in the void bring you peace

 « Reply #9 on: January 16, 2014, 11:46:12 AM »

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.

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I seem able to characterise pressures by their internal source and external action. Some pressure self actuate by an internal source or potential which cannot be located no matter how narrowly we search: others are activated by an external system or medium directing these internal pressures by boundaries and passing between boundaries in the most curious ways.

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

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PAm = RAm + TAm

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Grassmann Point Algebra and Scakar Fields

this is a copy of a blog post, now moved to Wordpress.com
Grasmmann point algebra
http://jmanton.wordpress.com/2012/09/03/introduction-to-the-grassmann-algebra-and-exterior-products/
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Abstract

The dissemination of Grassmann’s ideas to the larger mathematical public in Germany intensified with the interest in this scholar’s achievements shown by Alfred Clebsch (1833–1872) in the early 1870s. The premature death of Clebsch prevented him from deepening his adaptation, but the friends and disciples in the Clebsch school continued the reception of Grassmann’s work. I intend to show the important role of Clebsch’s school, and in particular that of Felix Klein (1849–1925) in making Grassmann’s work ac

basis

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Abstract

Basis and dimension are two elementary notions in the theory of vector spaces. The origin of the term ‘basis’ comes from the possibility of expressing any element of a given set as a linear combination of the basis elements. Therefore, the origin lies in a question of generation; on the other hand the condition of unicity brings out the question of independence. The connection between generation and dependence is certainly one of the most interesting characteristics of the concept of basis: any maximal set of independent vectors or any minimal set of generators, is a set of independent generators and vice versa and such a set is a basis. Moreover, the dimension, beyond its “natural meaning”, is the merging point from which the question of invariance is to be drawn out. Indeed, the fact that all bases have the same number of elements entails two results: there cannot be more than a certain number of independent vectors, and fewer than the same number of generators. With a suitable starting point in the presentation of definitions and first properties on dependence and generation, these different aspects seem quite logically connected and easily explainable, but historically, the development of these two concepts was less straight-forward. For various reasons, in the approach to the concept of basis, the connections between dependence and generation were not always exhibited. Therefore the concept of dimension could only partially be drawn out, and some of its aspects were smothered, or even considered as obvious and assumed to be true without proof. On the other hand, the relation between the dimension of a subspace and the rank of any system of linear equations by which it can be represented, played a role in the history of the concepts of basis and dimension.

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Abstract

Michael Crowe has shown in his History of Vector Analysis (Crowe 1985) that Grassmann’s earliest mathematical work, the Theory of Tides, contains almost all of the key vectorial notions that appeared four years later in the Ausdehnungslehre. On the other hand, Grassmann never published the Theory of Tides—it first appeared in 1911 in Grassmann’s collected works (Grassmann 1840). Many of the physical applications, however, did appear in 1877, the last year of Grassmann’s life, as “Die Mechanik nach den Prinzipien der Ausdehnungslehre” in the Mathematische Annalen. The only essential difference between this later version and the 1840 appearance appears to me to be ostensibly minor changes of notation. I believe, however, that it is just this difference that points to the contribution that the Theory of Tides can make to our understanding of how the Ausdehnungslehre came to be.
http://mathoverflow.net/questions/22247/geometrical-meaning-of-grassmann-algebra
http://mathoverflow.net/questions/102917/urge-reason-for-inventing-interior-product-grassmann-algebra

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.
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 « Reply #11 on: January 16, 2014, 12:21:03 PM »

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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!

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The fundamental action behind Grassmanns analysis is synthesis or construction.mthus Strecken are construction lines, points are construction points, and the combinatorial actions of synthesis simply mean draw and construct using these TYPES of element.... The construction of a line read draw a straight line using points A,B. a computer would ask for specifics, a smart computer would access the types and use the default instances. An even smarter computer would ask for specific properties to be input to alter the default. This is how Grassmann synthesis works, but we must know the default construction to use it!

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.
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Without this superstructure most of the anti commutativity  notion collapses, and with it antisymmetry.
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jehovajah
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 « Reply #12 on: January 17, 2014, 11:50:24 AM »

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..
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 « Reply #13 on: January 18, 2014, 02:19:01 AM »

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.
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 « Reply #14 on: January 18, 2014, 10:17:34 AM »

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.
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