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Author Topic: Twistor  (Read 8570 times)
Description: some notes on Hermann Grassmann's Ausdehnungslehre 1844
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« Reply #105 on: February 27, 2016, 12:08:20 PM »

http://m.youtube.com/watch?v=jiejNAUwcQ8
Notice the role of the vorticular topology
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« Reply #106 on: February 28, 2016, 09:05:52 AM »

Thanks,  I found it very informative. smiley
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« Reply #107 on: March 18, 2016, 07:09:39 AM »

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

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

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

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

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

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

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

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

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

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

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

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

For space-time we may substitute a Fourier aether .
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« Reply #108 on: April 02, 2016, 07:19:22 AM »

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

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

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

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

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

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

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

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

Magnetic vortices are a necessary condition for any proper description of astronomical behaviours.
Setting that as a sound basis as Gilbert and Boyle proposed would be a sound direction or modern astronomy and indeed modern Science as a whole .
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« Reply #109 on: April 05, 2016, 07:00:05 AM »

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

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

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

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


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« Reply #110 on: April 06, 2016, 09:06:05 PM »

http://m.youtube.com/watch?v=LpqwdIoWgw0
The quaternion Fourier transform is what I define as a twistor magnitude .
This particular application uses the vortex decomposition of a colour sphere with a light to dark tonality axial centre . These are not physical twistors but as the decompositionis plotted the great twistors show a twisted shape while the higher frequency ones show definite regional sharpness until the full image is now perceived and the twistors ignored
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« Reply #111 on: April 07, 2016, 12:09:40 PM »

http://www.ijcsit.com/docs/Volume%205/vol5issue03/ijcsit20140503378.pdf
A reference for the method based on the original work by J J Sangeine .(1996 ?)
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« Reply #112 on: April 07, 2016, 12:39:15 PM »

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

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

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

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

Twistors may be a complex quaternion building block requiring expertise, but how we combine them or anything else to do pragmatic things is essentially and heartwarmingly simple!
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« Reply #113 on: April 08, 2016, 07:27:19 AM »

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

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

What rotates in space!? The only observed rotational force on a local scale is magnetism around a inducting/ conducting wire. It is that same wire as an antennae that transmits and receives radio frequncy signals( and higher) .
The magnetic rotation ( polarity rotation ) spirals outward and also inward as a pulsating system. The spheroidal nature of these point rotational emissions are described by twistors . That these rotational systems can be used for image processing , radio and TV signalling is just part of te intensive/ extensive magnitude utility of these Ausdehnungs Gr廲e!
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« Reply #114 on: April 13, 2016, 06:57:56 AM »

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

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

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

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

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

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

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

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

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

Change that fundamental definition and we change the fundamental interpretation .

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

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

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« Reply #115 on: April 13, 2016, 12:50:47 PM »

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

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

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

The emision of a "photon" represents the precise frequency differences between the stable Lissajou patterns,
https://m.youtube.com/watch?v=wvJAgrUBF4w
cymatic Lissajou or Chladini patterns
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« Reply #116 on: April 28, 2016, 01:42:24 AM »

http://slideplayer.com/slide/6231443/
The versatility of quaternions lies in their rotational metaphor. However the belief that quaternions are numbers but higher dimensional structures are not is a chain around the neck of Natural philosophers . The weak link in this chain is the concept number . As we may garner from Grassmanns efforts the Combinatoril and compositional nature of our interaction with spatial forms is not bound by numerical Aritmetic. Instead Arithmetic is freed into a greater spatio entry of symbolic algebra that serves to describe both xtnsive and intensive magnitudinal experiences and processes.
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« Reply #117 on: April 28, 2016, 12:17:47 PM »

http://m.youtube.com/watch?v=oW4jM0smS_E
<a href="http://www.youtube.com/v/oW4jM0smS&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/oW4jM0smS&rel=1&fs=1&hd=1</a>
In this video Norman demonstrates a method ascribed to LaGrange and Euler but in fact developed by Newton in his scrapbook.
The binomial expansion of a polynomial/polynumber or any function expressible in the form of a power series ' in the manner LaGrange advocates  gives the binomial coefficients as precise Derivatives/ sub derivatives for a given argument of the function or symbolic polynomial.

These Lagrangian concepts underly the iterative application of quaternions!

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

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

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

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

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

In this light a twistor records in the complex part of the quaternion exponent, the frequencies 跣d phases of any quaternion Fourier Transform and the real part records the amplitude. Thus expanding and dilating or condensing spheres combine in Rotation to make a static or dyn跩ic form appear in a region of space "
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« Reply #118 on: April 28, 2016, 01:37:24 PM »

http://m.youtube.com/watch?v=8SiuTk2ZiyU
More uses of trochoids and vortices  and how rotation underpins magnetic behaviour.
The diffraction grating effect is a quaternion Fourier Transform structure.
Coherence is what is induced in MASING .
As Facinaying as Prof Lauthwaites demonstrations are and as helpful as they are, they are inconsistent with one another. The Archimedian screw is a much better analogy explaining how the copper barrel rotates in the alternating "magnetic field".
And why does not a reflected screw stop the rotation? Because the copper and the aluminium are both induced with their own Archimedian screws. Net resultant is a motion determined by initial conditions the second Archimedian screws usually ignored by modern theoretical explanations. The 2 screws opposing each other still pass through each other and this superposition creates the " gear tooh" that bites into the material.  Inertial forces (Lenz law effect) give the resultant from initial conditions.
Finally, phase trains of higher rotational harmonics (often ignored)  preponderance the initial impetus but vary in expression depending on material inductibility / conuctibility.
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« Reply #119 on: April 30, 2016, 01:53:49 AM »

http://m.youtube.com/watch?v=QfDoQwIAaX
There are 2 types of Quaternion Fourier Trnsfotm. The traditional one is based on Eulers exponential nittion , the less familiar one is based on Sir Roger Cotes Notation.
The first type models forms explosively.  The second models forms expansively but more in keeping with human visual.expeience.
Often distant objects do not appear to be moving very fast because the scales are compressed by the visual field effect. Objects far away present a low angle of incidence on the retina while those nearer present a large effect, the behaviour of the logarithmic function mimics that.

The logarithmic quaternion presents a large modulus quaternion within accessible scales while an exponential quaternion soon blows large modulus uatenionic out of sensibility.

The video is a demonstration of how a 3d fractalmgeneratorbapplication like Quasz By Terry Gintz handles both types very naturally but very differently in terms of output sculpted form.

The projectiles are the Fourier forms as classically expressed, the impacts represent the effective pplication of Fractl iteration on the Fourier transforms. The explosive decomposition of the form is either exponential in nature or logarithmic. Or some combination of both.
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