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What is a period?
It is a thought pattern evoked by an every time realised form that comes to be and passes away in a sequence. Thus a dynamic form realises and flows away in a repeted pattern. It is the identifiable form which is the element of this repeatng pattern that is called a period. Thus the pattern is now renamed as periodic ,

This relation between a dynamic thought pattern and a larger repeating pattern in which it is a distinguished element is the formal structure of our construction of time as a periodic, episodic and duration, so a marker or metron.  However more importantly it reveals a fundamental structure to the concept of frequency. Thus all these concepts are tautologies of one another based  on spatial extension and dynamic variation, or intensity and its dynamic variation.

Consequently the Fourier transform seeks to model this period as a dynamic form by dynamic rotations, the geometrical essence of a repeating pattern, the conic sectinss. In making this transformation a fundamental view is realised by ancestral linking It has far reaching conclusions and supports what the  ancients  already knew  of the attributes of space

I Carried that notion of rotation as the key to absolute space to the limit of my imagination and investigative powers and slowly it dawned on me that rotation is fundamental to every motion. I had to generalise the concept of rotation to break free from the simple minded circle idea of it.

I gradually came upon a geometry that is prior to plane geometry , that is more ancient than plane Geometry and indeed is ancestral to it: spherical geometry and trigonometry. I began to realise my concepts of angle were restricted by convention, and finally by studying Euclids first book in the Stoikeia I realised that point and line were derived concepts from a previous circle geometry.

It took time to reshape my views and to try out ideas and logical constructions to test for consistency, but over none existen time, rather observing many cycles of a given frequency, I statistically verified the soundness of my notion.

My studies of Grassmann have reoriented my view of how philosophy relates to specifically Mathematics so called in these modern times, and in fact the source of much of what was called traditional Maths in my secondary education. But the secret of the unit sphere and circle was kept from me

Knowing what I do now has made sense of what otherwise was confused and muddled thinking by Mathematicians, supposedly the clearest and cleverest thinkers of all! I found out that this is all a tissue of lies or halftruths .

Then this year 2014 I was moved to understand Galileo and Newton as a piece.. The fractal thinking of Galileo passes on to Newton who elaborates more fully on it free from catholic persecution., and for the first time I looked up the meaning of Absolute! It just means independent of all else. Even of God!

Galileo was the real source of Newtons ideas on absolute space, time and force, not the water in the bucket!

So Galileo imposed a rule on systems in space only mentioned briefly in the Dialogo, and illustrated to explain it! It was a picture of a circular fractal. Each circular system as an element within a greater circular system was independent because of its rotational centre! Thus any rotation in any system is independent  if it's centre differs from the larger system.,

Newtons Astronomicsl principles are not based on the straight line but on the circular arc segment. This is why Cotes was so sure the logarithm of the imaginaries was the key to a better version of the gravitational formula!

But there is something more fundamental and ancient going on: the descriptive models laid out by Galileo are man constructed ones. If they were divine then the catholic church would not have imprisoned him. And how men come up with these constructions is never taught. The study of heuristics is rarely a school curriculum topic. Hermanns work puts that lack to rights. However he hoped to teach it as a professor and thus pitched his material at too high an academic level.

The more I study Hegel, the more I see his intended target Audience: Studium or Cassical Studies students, not mathematicians!  Hegel only wrote 2 books, the rest are lecture guides, notes and aids. There are over 28 volumes of this kind of material. Needless to say Hegel did not get a chance to edit his material into further books. Other students and supporters did this. Thus we get a view of Hegel that is warts and all! Contradicting himself and his principles, making invidious value judgements , spin doctoring historical accounts to illustrate a point etc, all of which would have been edited to consistency had he had the chance before dying of Cholera .

Hermann thus has the Hegelian principles to work from, and boundless material to sift through if in fact he required it. But he comments in the Vorrede that the contemporary editors were reducing Hegels work to Nothing! In this regard his Ausdehnungslehre is a work attempting to illustrate the power within the Hegelian system .

This system underpins the thought patterns of Galileo and Newton and helps to explain the system within systems that they constructed to model 3d spherical space.

Newton's Absolutes are Grassmanns RechtSystem, the accelerative force system is the ruled local system and finally the motive system is the on the surface centre seeking behaviour. These 3 systems represent a far, a local and an immediate view of an absolute "system within a system"

This is a very long video which I have posted in the light thread.

http://m.youtube.com/GObB67ETvRQ

What Eric does is unite Grassmann, Norman Wildberger, Steinmetz  and A good few others into one consistent system . As he presents it means that I can now model a magneto Thermo sono electro complex by a quaternion fractal equation , introducing the time varying and phase varying into a linear combination of twistors or a Fourier series or Laplace series of twistors.

I have done this many times over the years without understanding the fractals I got, but now I can see that they are fractals of the magneto Thermo sono electro complex particularly as they realise in 3d space, or any 3dimensional  reference frame where the dimensions are not just length and time, but other measurable magnitudes used as a dimenßional axis.

So Grassmanns style is slowly percolating through my thinking.

The Grassmann -Euler twistor as I have defined it is a very astute product design or function design. But the role of the cyclus group is hidden in this design . The cyclus group 1,2,3 is approximated by , with the constraints

This lays out the cyclic interchange group along radiativey opposing axes. Because of this we lose sight of the "running into against set " nature of the cyclus group. 1,2,3 is one status 3,2,1 is its opposite status, but dynamically, clearly it runs through the numerals" backwards". The axes do not run through the numerals backwards per se, we have to consider 4-1,4-2,4-3 to get the same effect. In short we have to use modulo arithmetic notation to accurately capture Hermanns "running into against set "concept.

Consequently we have to use


to capture Hermanns notion. In addition we have to set to as runs , with this switch occurring every time achieves a value that is not a multiple of 2, switching back when it fulfills tht condition.

This allows the cyclic interaction to fulfill the condition on , and makes any plotting of the arc segment run smoothly around the circle coordinate system imposed.

With some modifications we can now model superimposing rotations as running into against set cyclic interchanges.

We will use 2 Grassman-Euler  twistors set up as discussed. If we add them they represent a constructive standing wave, in which the reflection occurs at a free end . If we subtract them they represent a destructive standing wave at which the free end is clamped . To see what happens when the phase of the cycling group changes relative to the other twistor, we add or subtract a small amount to the in one of the twistors.

What results is a standing pattern with a phase shift.

We may then interpret the phase shift as a cycle start position in a generator, or a time difference in a wave reflection at a boundary, or a free end( in which case it relates to the tensile relaxation time in the deforming medium) or a speed of wave deformation travel.

We can alter the frequency by multiplying by cycles, and the amplitude by multiplying the twistor by a radial scalar.

If a twistor is supposedly translating through a medium then we have to relate the to time and distance. This modification is in fact different to the twistor itself, but uses the embedded time and cycle period to define a length in space. Such a length is then allowed to extend with the embedded time , and the phase positions are marked off to imply wave progression or reflection.

Steinmetz use of these Grassmann Euler twistors to calculate and describe the rotational behaviour of the alternating magnetic current in and around  a transmission line is legendary. The more complex the Twistors( that is if quaternionic twistors are used) the more complex the described results.

Importantly, however, one must not fall into the trap that mathematics is describing nature accurately! Steinmetz was renowned for his expertise, his ability to intuit and comprehend the product design of the Tesla systems, but he refused to claim he knew the secrets of the aether! A small man he lived and survived by bigging himself up in these early days of "electric current" generation and transmission design, but like Tesla he knew he was tapping into something bigger than he could imagine.

http://m.youtube.com/BLloW3zzTrc
<a href="http://www.youtube.com/v/BLloW3zzTrc&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/BLloW3zzTrc&rel=1&fs=1&hd=1</a>

This is ridiculousness in the extreme!

Simply he is constructing combinatorial structures using common or garden processes and number spaces in Mathrmatics. But why use Hieroglyphics?

There is a disease that humans fall prey to. It is called ivory towerism! In ancient universities no mathematics student would become qualified until they got an A+ in rhetoric!

Rhetoric was that class of communication skills which enabled the learned man to communicate accurately and appropriately with all levels of society educated or uneducated.

This style of presentation from the extreme abstract hieroglyphics to specific instances was definitely utilised by Justus and Hermann, but whether it was pioneered by them I do not know. Maybe Lagrange et al were doing this kind of anodine presentation.
Normans latest video suggests that set theory and Aristotelian logic have lead to this style of hieroglyphics. However studying Justus combinatorial method gives us a key into how it may best be communicated.

Stress stressors topology and magnetic bonds.
One of the plane facts is that matter is variable in its intensity and that it has continuous and discrete forms .

Further we recognise energy -phase changes that is to say simply : solid liquid gas plasma. States of matter with different vibrational signatures.

It was determined by Newton et al that heat was best described or pictured mentally as a vibration in a corpuscular region corpuscles were thus conceived as flexible plasmas from the nascent biological discoveries of the time.their flexibility was merely a way of accommodating the vibrational modes conceived as heat .
We have since developed the heat model into the Energy ,Odell.nsuch a model really has its foundation in Gilbert's magnetic philosophy which recognised 2 forms of magnetism: crystalline lattice or mineral magnetism and biological magnetism, that is organic lattice type. The one was called the iron or ferro magnetism, the other Electra magnetism or amber magnetism.

These specs were later intensively studied and distinguished into models that were fundamentally mysteriously related. However the various guilds of magnet makers and electric charge makers  jealously hung on tontheirbtrade secrets and further distinguished the common link, that is the fundamental material experience and phases of matter at different energy or vibrational states.

Matter and space were philosophically divided in the most natural way , but space was called empty for religious reasons. It was desired that spirit be distinct from matter, and invisible able to manifest to conscious souls, themselves spiritual. 

Such a model kept the material world that technology was exploiting as a distinct environment that mankind could "play" with like a child, but spirit was the playground of the gods and those ascended humans who struggled to grasp it.

Of course the material philosophers were brought up in that Matrix of ideas ad models, but younger more ambitious material scientists saw no valid reason to limit material science in this way and technology indeed advanced to miraculous inventions gadgets and contraptions which supported that viewpoint. 

Unfortunately the mathematical clan or guild retained enough obscuranting mystics to cover the material  viewpoint with a pseudo mystical or religious analogue of religious theology . BishopmBerkely wrote a powerful attack on mathematical religious views pointing out that material scientist of the mechanical philosophy were in fact no more clear of superstitious nonsense or beliefs and practices than religious theologians and clergy

Accepting that point we may progress toward a common view proposed in detail by Hegel et sl and eastern mystics since way back, that spce is in fact something that is connected to a singular concept that contains all: if you like spiritual and material are two aspects of an entity with many aspects.

The Aether or Ether philosophies attempted to place that view as foundational to the material sciences, but all clans and guilds fell apart in disarray: a modern Babel of jargon and gobble de gook .

We do not have to be so inclined.  A simple and useful combinatorial model underpins the technological achievements of mankind. It is rather humble in its scope and often  symbolically and ritually obscure. Attempts to make it clearer usually do the opposite, academically. But those who actually make things using their whole organism develop inexpressible expertises that often seem to those not involved in the physical actions of making , well they seem like magic, even magycke of the old kind.

Regularly organic and rock crystals cry out to us ! We call them TV's and Radios!! 

So we can take material science or specifically the science of material and develop an expertise that Tribology or tribologists express. This level of expertise is an art form found among the few material scientists with a broad enough range of experiences and understanding. It is eclectic and very much in its infancy .
For example, topology would help to provide a structural overlay akin to corpuscular philosophy of Newtons time. Topology fundamentally is a combinatorial art that gives forms or formats for space in its most general Aetheric model . These formats may then be used to count and measure regions and distinguish behavioural changes in material over time, circumstance and internal and external conditions. 
Topology allows us to place atomic or molecular models into regions of space so we can characterise certain regions by their phase and nearby status and their visual characteristics . The stresscandvstressors in such a space become characteristics of the phase of the element or molecular material as well as contact behaviours between such characterised materials.

Clearly a chemistry and organic/ biochemistry based on these characteristic measurements and observations has proved very fruitful in organising our technological advances. But we have found some who want to isolate these behavioural and technical expertises away from common knowledge to support some particular world view or religious ideology. 

While the knowledge is presented as iconic, eclectic, difficult to understand, even iconoclastic, it is in fact not. The simplicity of the energy/ vibration - phase model of matter is how it has been hidden in plain sight.

It is a simple robust and powerful combinatorial system that was advanced by Navier Stokes and yes even Newton into the fluid dynamic model. Helmholtz and Kelvin added the Vottex model rules (inaccurately: see Claes Johnson work) but all of this was obscured by the Mathrmaticians. Helmholtz following perhaps Euler and Lagrange felt that the differential equation format was the safest way to express natural laws of motion in space-like material especially if they were fluid.

In a sense , given the prevailing attitude that aether was basically solid billiard balls at the femto metre scale( an assumption since demonstrated to be misleading especially by quantum Mechanics) his reliance on the continuous nature of differential equations by design does preserve the fluidity of the foundational corpuscular theory of Newton et al. 

The use of hard billiard ballasts representing corpuscles is a mistake we can trace back to Leibniz et sl . But even then he was not insisting that matter was thus. The mathematical topology was simpler if this was assumed , and produced useful models. It also produced strange aberrational models which rightly were ignored! Now they are mystically clumped together in the quantum mechanical world viee.

The fluid mechanical expertise is very difficult to establish except in the simplest cases, but in fact computationally it has provided us with some excellent models within the boundary conditions appropriate to the study. The advent of more powerful computers and computational schemes has made it possible to model complex data increasingly accurately, by applying a whole mish mash of equations topologically to a spatial region.

The one fundamental breakthrough for physical topology is ascribed to Benoit Mandelbrot. His Fractal Geometry  inspired many computer geeks to model the particular types of inductive or iterative( recursive) equations that hitherto were avoided by Mathematicians as monsters. Ad yet we know Focault and many others were studying them quietly prior to the wars with little academic support . 

We can laugh, but it was mathematicians that were the Luddites when computers were first introduced!! 

Today I watch a lovely flowing scene of actual waves lapping onto a shore with all the sound effect on a flat, crystalline organic matrix presenting the display of raw data as processed by billions of logic gates in some silicon(rock) chip processor with a certain vibration/ energy which has to have the heat removed by invisible gases passing over it as winds! 

The fluid dynamic models expressed as differential Equations may be nonsense. But the expertise that organises itself in this way around the energy phase model really has a powerful set of models with which to develop further models and technological advances .

Why have I not mentioned Maxwell, and Einstein? Because they represent a deliberate mystification of the simple model. Faraday was not happy with how Maxwell fundamentally distorted his ideas. In fact Örsted Faraday, Ampère Volta and others held that a circular force o rather a rotational force was fundamental to our understanding of all models built on observation. Newton, unlike his successors took a circular or rotational force for granted. 

This simple topological element distinguishes models that account for quantum behaviours from those that do not. 

And yes the mathematics becomes confusing and complicated, but only when the circle and the trochoids based on circles are excluded.

Include trochoids in space in your models and you can model magnetic behaviours topologically in a consistent way.

The gravity of this point should not be missed, as Gravity to this day is still touted as a mysterious force,  all forces are fundamentally mysterious characteristics of topologically described space. Their iscavsimple reason why conic curves describe both gravity and magnetic attraction and repulsion. The topology of space that best describes them is rotational or circular. Add to that fractal geometrical topologies and you can model fluid dynamics in appropriate small boundary conditions.
Add phase changes at different energy and thus vibrational states( frequency amplitude and trochoidal topologies) and you arrive at discrete regional distributions with phase changes marking the boundaries of regions fractally.
What we know about bonds is dependent on these fractal regional surface phase change behaviours at differing energy/ vibrational statues.

http://m.youtube.com/watch?v=6dW6VYXp9HM
I want you to have a physical model of a twistor!
This Fourier Analydis machine shows how engineers know how the "wheel works " of nature structure in space and time.
Yes these are ideals and require an inductive fractal mind set , and yes fractals and vortices are way more complex, but this is the einfachste model, which we iterate by scales to describe more complex  structures.