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 Author Topic: Twistor  (Read 8581 times) Description: some notes on Hermann Grassmann's Ausdehnungslehre 1844 0 Members and 1 Guest are viewing this topic.
jehovajah
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May a trochoid in the void bring you peace

 « Reply #45 on: February 22, 2014, 05:31:57 PM »

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
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jehovajah
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 « Reply #46 on: March 17, 2014, 11:36:53 AM »

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.
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jehovajah
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May a trochoid in the void bring you peace

 « Reply #47 on: March 19, 2014, 07:22:00 PM »

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

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jehovajah
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May a trochoid in the void bring you peace

 « Reply #48 on: March 28, 2014, 09:32:09 AM »

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!
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jehovajah
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May a trochoid in the void bring you peace

 « Reply #49 on: April 22, 2014, 09:40:44 PM »

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 .
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jehovajah
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May a trochoid in the void bring you peace

 « Reply #50 on: April 26, 2014, 05:08:12 AM »

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

I propose
Eix= coshx+ isinhx

This would be a twistor for hyperbolic curves and REix 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.
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Sockratease
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 « Reply #51 on: April 26, 2014, 01:02:43 PM »

Oops! your movie can make CHROME to become unresponsive and occur the error "next plugin is inactive:unknown, do you want to stop plugin (Yes/No)?"
Please Jehovajah, try to not posting too many movies and movies to long
I'm sorry, but this message isn't complaint, but I need to close CHROME with WINDOWS TASK MANAGER.
It's just FRACTALFORUMS maintanance for work properly.

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

And if anybody else experiences this, please post here:  http://www.fractalforums.com/help-and-support/trouble-with-fractalforums-videos/
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hgjf2
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 « Reply #52 on: April 26, 2014, 03:34:15 PM »

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.

 « Last Edit: April 26, 2014, 03:36:36 PM by hgjf2 » Logged
jehovajah
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May a trochoid in the void bring you peace

 « Reply #53 on: April 26, 2014, 08:40:15 PM »

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.

This is by my new Twistor formula with quaternion coefficients for each of the imaginary axes in the trig /circular functions
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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 #54 on: April 30, 2014, 08:46:46 PM »

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!
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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 #55 on: April 30, 2014, 08:52:48 PM »

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
 « Last Edit: April 30, 2014, 09:15:01 PM 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 #56 on: May 02, 2014, 01:23:55 PM »

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!
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jehovajah
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May a trochoid in the void bring you peace

 « Reply #57 on: May 02, 2014, 02:55:38 PM »

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 .
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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 #58 on: May 02, 2014, 10:52:45 PM »

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
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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 #59 on: May 02, 2014, 11:28:13 PM »

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?
by terry gintz (http://mysticfractal.com/)
we expect to release "μυστική मण्डलबेथ" (mystic maṇḍalabeth) 3D very soon! and also the 4D version later

if you would like to include the मण्डलबेथ (maṇḍalabeth) 3D & 4D fractals in your fractal program,
please email me, and ill be happy to send you the source code!

bethchen@umich.edu
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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!
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