Twistor

[jehovajah]

http://portail-video.univ-lr.fr/Quaternion-Geometric-Averaging-in
The video here shows the thought process that designs the product required for filtering an image presented as signals for each colour map in RGB , plus an additional signal for depth say or intensity/ tonality/contrast..

These signals come in as voltage variations but essentially they are phased. Thus the complex part of the quaternions represent phase arcs controlling a pointer. This picks out the colour point and the local technology implements that colour reference on screen pixel by pixel.
Because of this pixel by pixel nature we in effect have a quaternion Fourier transform being transmitted term by term . The sum is the image on screen.

This topic is about manipulating the transform to give the best image output so the signal is not sent direct to screen but through filters that do the desired job.

While rotation seems not to be actually involved this is far from the case ! Rotation and cycling are everywhere involved in signlling we obscure it by calling it a wave!

[jehovajah]

https://en.m.wikipedia.org/wiki/Absolute_rotation
In the system of reference framescNewton constructed absolute rotation was key to his interpretation of thebGalilen Fractal Solar universe.
It is to be observed that abolute means simply independent , thus it is  a sine qua non that a circular rotation is independent of any other centre than its own . This means that  a fractal systm of bsolutes can be constructed on circular orbits conistently.

However the implications of the absolute nature of circular dynamics is tht Fourier Analyis of an object can be comprised of the um of rotations round a single centre making each uniqued centre of an object not only the place to sum all translations but also all rotations of the whole or parts of that object.
The further implication of Fourier analysis is that a rotationl description of dynamics is achievable and potentially more in keeping with observed behaviours.

[jehovajah]

http://m.youtube.com/watch?v=LpqwdIoWgw0
Grassman Twistor used to do digital image processing.

[jehovajah]

http://m.youtube.com/watch?v=4Rx35q-zJRk
How twistors encode biological information

[jehovajah]

Most of us as students of Newton have a problem with force.
We are taught the proportion mass x acceleration . It is in factor form and it is as Newton explained, a proportion of observable quantities with an experiential but non visible phenomenon. It is actually a kinaesthetic experience called pressure and is clearly in proportion to surface area of contact.

However the mass of an object is not proportional to its intrinsic volume and thus it's surface area, which may in fact increase without change of volume. So the inverse or reciprocal proportion is used in any definition of the proportion for pressure.

We have 2 related proportions for the concept of force : mass x acceleration and pressure x area of contact.

The difficulty comes when we wish to describe a contractive force as in contracting materiality. And also, but less obviously to most in describing attractive forces!

Most of us have been taught to use the negative of a force to describe an attractive force. We do not even realise how misconceived that is.

So when Dirac arrives at the idea of negative energy we are totally unprepared to understand what that might be!
Like imaginary numbers negative integers were a very taboo subject for most of our top western philosophers especially DesCartes who preferred to work in the positive quadrant of the so called Cartesian plane.

It was only when Euler established the exponential function in generality that most became aware that Sir Robert Cotes and Isaac Newton had been discussing the use of the proportion we call e, the evaluation of the binomial compound interest growth factor in some detail as it related to Gravity.

It was later understood that Cotes had formulated the logarithmic form of the Euler expression for the evaluation of the imaginary exponent .

What few realise is that Newton was on the verge of reworking his by now famous gravitational equation, based on Cotes ideas.

Newtons system of forces based on Galileos observations of the Jovian system and the behaviour of Venus would have then been based not on the inverse square law, but the more general logarithmic law as expressed by Cotes.

In newtons system the absolute behaviour in an orbiting planetary system is proportioned down to the planet surface by a spatial influence that acts centripetally at every point  in the system, and objects of mass are a measure of that orce by the process of the balance omparisons. Thus mass object acquire their motive from this centripetal force behaviour which is accelerative.

Philosophers are arguing whether this is the first expression of the modern Field conception, but of course none of them really understand what the field idea really is.

The consequence of this change would have been to make force an exponential proportion, and thus to model expensive and contractive forces by exponential ombinations.
Dirac was forced to make sense of his ideas only in the accepted format, o negative energy could not be understood as the contractive energy because by common acceptance energy was positive only. And negative forces are not contractive forces in everyday physics!

The exponential versions of physics were carefully separated into real physics and imaginary or heretical constructs in physics.

Complex functions in physics are still routinely explained in this old mistaken way of understanding negative and co plex values, or they are smothered in quantum mumbo jumbo !

Few want to do physical calculations as exponents to he base e because they misguidedly think mathematics is the language of the God of Nature and nature is direct, simple and real!

Twistors are in fact the basis for doin physical calculations because they are a better model of general behaviours than the direct measurement schemes of biased Natural Philosophers from DesCartes onwards.

[jehovajah]

I am sufficiently happy now to regard the Grassmann Twistor as a circular arc extension, or circular vector.

The notation, which derives from Euler, I would modify or simplify in today's computer rich world .
Exp( iø) is a function form which obscures the direct circular vector .
Some use for this circular vector but I am not familiar with that notation yet and it does not appear to accommodate the necessary radial or diametral line vector that must be associated with a circular vector. The radial vector has to record the phase of the circular arc.
The exp(z) format certainly allows all of that which is why I like it, but it just does not look like a vector , but then again neither does the vector cross product!

That is the nub of the difficulty . I want to be able to write a cross product of circular arcs of differing phase and amplitude without confusing it with ordinary power form multiplication .

In any case the general cross product is only a vector in vector algebra, not in general Clifford or Grassmann algebra, where a rotation about a point in a plane is directly referenced

[jehovajah]

Some applications of Grassmann Twistors

<a href="http://www.youtube.com/v/qm5I&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/qm5I&rel=1&fs=1&hd=1</a>

http://m.youtube.com/watch?v=qm5I_D9NN3g

The explanation is very sketchy but it is saying : take geometrical forms as magnitudes that are extensive/ intensive, do nt ignore the quarter turn/ rotational magnitude, ke the dynamic relation between all magnitudes clear and then use this analysis to synthesise a useful product.
Recent work on oriented magnitudes highlights that orientation is subjugate to rotation, and the minus sign does more than imply spatial orientation, it implies relative quantity , that is >=<. So the quarter turn magnitude may be used to sore process until it becomes synthetically relevant , and impacts on the Final product. the key point is not to give power to the mathematical process but to the person who designed the process to produce a useful end.
The use of complex forms in electrical nine ering is a good example. : the neatness of the process made writing down the circumstances more systematic and symmetrical, and the combined product made it clearer how to do the calculations which gave the expected results.
It was quicker and clearer and thus more practical. It was not " math magic" it was a better designed product.

When Grassmann explained Quaternions, demarcated by Hamilton, it was not to denigrate Hamiltons achievement, but to show clearly the system of synthesis to which they belonged: the Ausdehnungslehre
. And Clifford was a respectful Acolyte of Grassmann. His Algebras continue the geometric doctrine laid ot initially by Justus Grassmann, in summary of the then current state of Science, nature and Mathematics. In so doing he drew on the great minds of his times and presented the essential intersection of all their idea, in a accessible geometric firm rather than a pure symbolism of logic which frankly failed to capture the basis of multiplication! The language of logic is thus subservient to the apprehension and expression by drawing of spatial relationships and magnitudes.

In time we learned how to use extensive magnitude to depict and node inteni e magnitudes like colour and touch and smell!