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

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 .

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

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

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

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

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

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

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

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

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

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

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

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

http://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 polynomiàl that the concept of i was first noticed and exponded upon by Bernoulli . However despite Hamiltons Förderung it is the conception of Hermann Grassmann based on the works of all the Grassmans that is foundational.

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

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

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

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

In this light a twistor records in the complex part of the quaternion exponent, the frequencies ànd phases of any quaternion Fourier Transform and the real part records the amplitude. Thus expanding and dilating or condensing spheres combine in Rotation to make a static or dynàmic form appear in a region of space "

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