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 Author Topic: The space time manifold tensor candidates  (Read 8358 times) Description: Iteration statements that capture the strain tensor of Einsteins space time 0 Members and 1 Guest are viewing this topic.
jehovajah
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May a trochoid in the void bring you peace

 « Reply #30 on: March 15, 2016, 02:38:34 AM »

Many images show the fractal patterning achieved by fractal apps . It suggests that iterative processes have fractal outcomes because rotation is involved at all levels
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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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 « Reply #31 on: March 25, 2016, 07:38:47 PM »

Firstly Tensors are exactly what Hermnn Grassmann describes as,
Ausdehnungs Größe. It is rather technical but basically they are descriptions of extensive or intenive magnitudes in space . Relative to a specified space/ spatial object they describe in a lineal combinatorial format any or all relevant extensive magnitudes associated to that region. Thus a tensor provides a full description of a region and an opportunity to model the full dynamic behaviour of said region under dynamic circumstances.

It is a very general view which only has purpose if the specific aspect of interest falls out of the more general procedural manipulations.

So now we take a tensor as specifying a region. A manifold therefore is a varied collection of such Tensors. Together they describe a much bigger region or more regions of space. The many "folded ness " of these tensor collections represent the vorticular/rotational dynamic of space. Consequently the best representation of a manifold will be a complex lineal combinatorial arrangment of twistors, or if you prefer a complex 3d analysis of space in the Fourier transform format.

There are 2 Grassmannn representations of this type of high level(Stüfe) or grade Ausdehnungs Größe: the line segment version and the arc segment version. The arc segment version correspond to what I named a twistor ( not claiming priority) and which correponds to a very general Fourier Transform or a system/ collection of such. As a system they may be considered as a reference frame or a basis.

So now we come to spacetime. In fact the notion goes back at least to Galileo  and probably back to the Greek mechanics. You are not able to describe a dynamic without some reference to positions( space) and some regularly changing dynmic! Yes it is circular!
Our system of measurement is Fractal and ased on the topological and Dynamical distinctions we adopt as Metron. We then proceed by proportioning these Metrons . The process is idntical for all measures, thus we hardly can in general proceed differently with spatial magnitudes and dynamical ones.

Distance and time thus fall out as a particular interpretation of a more general n-level process.
The concept of space-time in its general sense is a mathmatical abstraction( that is a contentless label / topology) which holds the general direction we wish to go in in our development of a dynamic descriptive system.

So we describe an aether philosophically according to Grassmann in this way specifying the content in general form and developing specific interpretations of the general tensor formats.
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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 #32 on: March 26, 2016, 09:06:04 AM »

https://see.stanford.edu/materials/lsoftaee261/chap8.pdf
In this particular presentation, the Fourier is notated in vector form. This is a simplification that avoids a more compact quaternionic format, but one which is too unfamiliar for most scientists and engineers.
The quaternionic form is topologically describing arbitrary rotations in space about a point. Even Grassmann found this difficult to apprehend in 1844, but by the time 1877 came around he had written a paper called " w
Where Halmiltonian quaternions can be found in the Extensive Magnitude Doctrine". I am part way through translating this paper but have elected to do the background work on Justus seminal combinatorics which underpins all Hermann and Roberts thinking or thought forms.
Suffice it to say that Bill Clifford appreciated this and laid the foundation for the Grassmann Clifford Algbras elicitly for describing "undulatory" systems, that is to say Manifolds!
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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 #33 on: March 26, 2016, 09:38:31 AM »

http://glow-wacky.com/2013/03/21/2d-and-3d-fourier-transforms/
Here you will see a simple manifold on the left and the " array" of amplitudes of the Fourier transform on the right.

We can represent the manifold by arrays of amplitude data in the traditional
Complex format . The quaternionic format or higher will thus be represent able as arrays of data.

The process of linear transformations therefore encapsulates at a general level how we process arrays of data sets. MRI scanning is perhaps the most heard of , but least understood commonly , application of this kind of processing. Without computational machines this representation would be completely useless to us even though conceived in principle by Grassmann in 1844.
Despite the greater acceptance of quaternions over the Lineal Algebra of Grassmann, Hamilton was generous enough to regard Hermann Grassmann as his Master! Of all the people who could appreciate the Ausdehnungslehre 1844 Hamilton was the best placed to appreciate its philosophical significance as well as its scientific one. He strove to generalise his work beyond quaternions both for personal and academic reasons but mainly because he believed he had the. Methid of describing the electrical aether!
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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 #34 on: April 21, 2016, 01:44:01 PM »

My recent meditations lead me to concentrate on Sir Roger Coaes natural logarithmic form for arc segment identity, especially as it is expressed in quaternionic format?

If the twistor is a Fourier transform in general, the "function" the transform is modelling is a form in space.
Thus the twistor or quaternionic Fourier transform models a spatial region or trochoidal bubble form. This form is adequate as a space-time manifold but is generally too complex to visualise. The Quasz sculptures of twistors are more the results of QFTs of these manifolds.

Using that line of analysis, the natural logarithm of the twistor should return exponentialed manifold form . That is, the natural logarithm of a general twistor should sculpt a space-time manifold directly, consisting of frequency and amplitude data.

Quite apart from suggesting that the data array  of a Fourier transform Is  a logarithmic transformation, it also makes possible the concept that a logarithmic transformation of a twistor/ QFT will give back the appropriate space- time manifolds analysis( into frequencies and amplitudes)  as plotable sculptures.

I have experimented blindly with this in a couple of Gallery posts called Coatesgravity etc.

The reason for that name is my belief, as yet unfounded on documentation, that Cotes in his Harmonium Mensuraram was about to suggest this idea to Newton as a formulation of Gravity equations , but far more as a description of any absolute space with the Newtonian 3 force characteristics. Thus a dynamic space encompassing magnetic, sonic, thermic and electric behaviours.

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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 #35 on: April 23, 2016, 12:41:31 PM »

It occurs that QFT might automatically by the process provide a Finite Element Grid for this method and thus provide a speedier analysis of complex dynamical situations
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jehovajah
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May a trochoid in the void bring you peace

 « Reply #36 on: April 23, 2016, 01:41:28 PM »

The theoretical method which Hermann Grassmann was conceiving to present by lineal Algebra. The classical problem was laid out by the French Mathematicians in a way difficult Tom"see" on the page . It was a clear as mud! Hermann claimed to offer shining clarity!
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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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 « Reply #37 on: April 28, 2016, 02:27:12 AM »

http://math.stackexchange.com/questions/2552/the-logarithm-of-quaternion
One of the major difficulties I faced was understanding logarithms. But my research into the history dispelled the myths I had been taught. Later I became aware of the role of identities, over equalities and the role of labels and product design, and it's attendant rigorous responsibilities.

Euler in expressing the exponential in an infinite  series , that is a power series that evaluated to the exponential of a rational power , was able to show that the structure could bear an identity. Thus the same structure allowed any symbol to be replaced by any other symbol throughout. Thus he was able to found the Cotes Euler formula on an algebraic structure which encoded a process of calculation. That process resulted in a remarkable identy
Exp(ix) = cosx + isinx

This was later developed to express the quaternion and then to express Banach and Clifford algebras.
ix= ln( cosx + isinx)
From which we can easily expect the logarithm of a quaternion to be the complex combinatorial part .

Exponents and logarithms are just another use of an Arithmos that is a mosaical structure which is countable as well as measurable and spatial. As the symbols are identities we do nothing really new by implementing them in our computer applications. They are all quaternions in this discussion and as such the same structure produces the same results regardless of the underlying elemental behaviour.

Therefore the fact that we can develop a Fourier quaternion Transform means that regardless of whether it is expressed in exponential combinatorial forms or logarithmic combinatorial forms the same out put should result for the same structural format. That is up to the variation in the way the exponential and the logarithmic functions increase toward infinity and decrease toward 0
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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 #38 on: September 03, 2016, 10:25:34 AM »

This is Jupiter imaged by the Juno satellite.
The fractal images and sounds have been naively explored in this forum by some of our most innovative explorers.
I will try to post links to where , but if you know post them here p,ease xxx
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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 #39 on: September 11, 2016, 06:43:08 AM »

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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 #40 on: October 06, 2016, 10:18:12 AM »

One of the difficulties of understanding a fractal sculpture is the so called iterated function format.
The quaternion format is misrepresented at a fundamental level among physicists and astronomers and mathematicians. But amongst Fractalers we have an intuitive grasp of its dynamical applications.

Hermann Grassmann claimed that his Ausdehnungslehre could depict many physical dynamical and static situations as well as crystallography., Phorometry and fluid dynamics. Amongst those claims he mentioned magnetism.

Here Bill Gaede attempts to explain magnetism in a dynamical system. The ropes or threads are representative. His mechanical modelnraisescas many questions as it answers but at least does not fob the engineer off with nothing!
Of course the aether is back in vogue as the old guard die off and new technology substantiates plasma and a ethereal substance. The dark matter dark energy response is a breathtaking Mathemythical nonsense parading as physics. It is much simpler to replace it by aether, and to correct the irrational fallacies.

In this thread I have sculpted a quaternion output for Einsteins gravitational theory. The outcome is interesting but hard to interpret. Moreover it applies equally to magnetic as well as gravitational behaviours.

Black hole cosmology and dark matter and dark energy are farcical attempts to hold on to funding streams. At the same time the trochoidal dynamic of the current gravitational theory does create a surface distribution which highlights curvature and rotation in an evolving system at any scale.
Gravity fails at a radius depending on the mass distribution , and that is why Newton was only concerned with our solar system and bodies moving within it. Outside that he had no idea, data or even comprehension of the vast known universe!

But for sure he would not rely on occult theories to develop astrological principles. The new Newton, that is Sir William Rowan Hamilton was keen to move to an electromagnetic description of the known universe even perhaps ahead of Grassmann who was limited by his circumstances and yet cleared the ground for the modern astrological revolution.
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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 #41 on: October 06, 2016, 10:42:26 AM »

I love digging around in the dark, and then blue-skying with what i find, just as i feel it!

However Doug really knows what he is talking about.

Doug's metric i suddenly realised is a kind of Trochoidal matrix.

doug Explains it actually was discovered by Rosen.

Here is the metric in tensor form

<Quoted Image Removed>

in quaternion form

$q=\left\{ \exp \left( -\frac{2GM}{c^{2}R} \right)\hat{e}-\exp \left( \frac{2GM}{c^{2}R} \right)\hat{i}-\exp \left( \frac{2GM}{c^{2}R} \right)\hat{j}-\exp \left( \frac{2GM}{c^{2}R} \right)\hat{k} \right\}$.

I have to figure out how to Quasz this, but it does remind me of a whole load of trochoid experiments i did late last year.

Here's Doug explaining what it means.

And the sculpture I figured out.
The two versions show the output from the elements varying directly( as you move through the quaternion block from - through o to +, and secondarily as the inverse of the elements do the same.

The inverse elements give a positive sculpture to the "negative" sculpture  of the basic elements .

What we see is a snapshot of how the force dynamics move quaternion points( as representing material points) in the quaternion aether. The spiral form near the origin is hidden in the simple elemental sculpture but revealed in the inverse.
This spiral is a characteristic of the inverse square law and reveals the so called black hole , in the first sculpture to be the entrance to a magnetic curling funnel dynamic.
This is true in the sculpture however the centre is approached so the form is only apparent to this snapshot view, it is a dynamic curvature in space however the centre is approached.

While this is claimed to be gravitational, the result is similar for magnetic dipole forces of rotation

I am exploring some formulae around gravity and electromagnetism using Quasz. Certain vectors and vector/tensor combinations are becoming familiar to me, and complex and quaternion vector algebras are providing models of space-time . Because the fractal generator is effectively a cutting tool the models produced are in "negative" in a sense, with surfaces and bound regions representing less volatile regions.
<Quoted Image Removed>
The models so far exhibit a big rotational vortex precedes a calmer steady state-like phase. The rotational region continues out at the edge of the universe, clearly weaker.
<Quoted Image Removed>
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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 #42 on: October 10, 2016, 09:30:05 AM »

This derivation follows the vector Algebra of Gibbs and Heaviside supported by Kelvin ..
The products designed ( dot and cross products) have to be explained . Sir William Rowan Hamilton took the imaginary magnitude as founded by Newton DeMoivre and Cotes and Euler and argued over by the Benoullis.
The work of Bombelli created a fundamental basis to the mathesis of the imaginaries, named imaginary by DesCartes, but it is Newton who set out the mathesis in its practical use. It was the basis of Newtons " vector" algebra .
The Fluxion Theory of Hermann Grassmann(  Ausdehnungs Größe) explicated the work of Newton to the nth degree, based on the induction process of Newtons fundamental principles.

Hermanns work was takn forward by many but particularly by Bill Clifford. .he made it clear that which Hamilton Acknowledged  : The Fluxion theory of Grassmann is more generl, but the same corpus of ideas as Hamiltons. The product design allows many useful models to be constructed for complex lineal algebras, both arced and straight.

This Calculus derivation obscures the direct formulaic processes. The exponential formulaic presentation gives a direct set of applicable measurement schemes that can encode Fluxion and fluent changes . These changes enable engineers to design robust mechanisms or calculate complex trochoidal surface forces in fluid dynamic of rigid dynamical( high viscosity dynamics) consistently within the limits of agreed measurements of constant ratios.

The twistors, therefore are the general olution to all physical situations that are dynamic and measurable in terms of surface interactions.

The B field surfaces indicating dipole orientation in magnetism are therefore expressible in exponents form. The Equipotential electric surfaces have the same form  the phase angle difference is indicative of the gyroscopic principles in arc rotation about a generated centre
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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 #43 on: October 10, 2016, 09:05:25 PM »

The design of all Arithmetics is founded on topological combinations as identified in Stoikeia book 7
Starting with book 7 we may analyse topology  by proceeding toward book one  and synthesise the topology of regular forms by proceeding through the later books .

The fundamental process of Arithmetic is factorising larger by smaller topology or more by less!!

This gives not only the basis for comparison or ratio but also the impetus to count, order arrange and sequence .

The misdirection to Number obscured this general approach. Hermanns Grassmann was the first to rediscover this general approach in the 1830's  and to make sense of a Pythagorean system or method of expertise development or even an ancient Indian subcontinent system of rhythm patterning ..

Within the process of topological factoring addition as a sequential arrangement along with subtraction as
the reverse or analytical sequence arrangement informs the counting response , the ratio response  and the proportion response,

These manipulations of topological forms inhere translation and rotation . The mirror reflection informed thinkers of the topological duality in  reflection and its use in analysis and synthesis.

Thus factorisation is a complex process by which we measure topology .
Appended to this topology is direction, orientation and intensity and extensivity.

When we experience weight for example we
Say weights are equal by a visual measure!

This our subjective proprioception is laid against a visual   Identifier
This visually cued measurement system fails to capture intensity and density directly but nevertheless we can interpret these kinaesthetic and olfactory responses by the standard topological pattern to which we can assent as analogising these sensations .
The matching of intensity to a topological pattern no matter how precise does not inform our senses of the actual process merely of the approximate outcome

The mosaics which form the basics of any factorising method  allow calculus to be constructed in lieu of actually performing the modelled process , one can only have confidence in such calculations if they prove by rigorous comparison to represent actual outcpmes.
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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 #44 on: October 12, 2016, 04:26:28 AM »

In the thread " Twistors" I attempted to deal directly with rotation from first principles. I had no real clue how to go behind the A level Physics I had studied quickly , or how confused I was by the edifice of mathmatical Physics.  I knew that Mathematics was fundamentally incoherent but not how, or where, or when.
Luckily my years at university were not entirely clueless. David Hilberts book on the foundations of mathematics was my prize find in the library there. I spent my time devouring that book, amongs other things.
I should have flunked Mathematics, but the university decided to give me a pass degree.
Why? Because I had been a thorn in the flesh of much of the faculty. I had managed to upset most of their lecturers and readers by questioning what they were teaching. I had suggested to my professor that space was a continuum of points, I had impeached Cantor and I had mastered computer programming in Algol60.

So I had potential, even though I was thoroughly disenchanted with the whole subject as taught, as promoted and as revered.

In the thread "Fractal Foundations" I set out like Norman Wildberger to set things straight. But I did not know what I did not know! So that thread records how I learned the truth about mathematics and mathematical physics in the context of the forum.
In the meantime I was encouraged by a misunderstanding to start blogging on Opera .  There I could really go to town on my growing understanding.
Opera blogs no longer exist, but Wordpress took them on as an import.

Here, in the forum I have tried to focus on what might be relevant to Fractalers at the same time as learning from the masters here. But my research interest is mathematical physics, and the foundation of physics.

Again I did not know what I did not know.

So I have meditated on rotation as the fundamental physical dynamic for years . I wanted twistors to be a fundamental depiction of " Vortices" . I wanted mathematics to reveal the foundations of reality as vortices!

Again I did not know what it was I was hoping for.
Fundamentally I had to apprehend Shunya . When I did I started on a journey of clarifying natural philosophical foundations to all the sciences but particularly Astrology.

I found that Astrology is the foundation of Mathematical Physics, and that Astrology was taught in the west by the School of Pythagoras. The mystery of Pythagoras is most likely resolved in Tibetan and Indian subcontinental traditions, but Sumerian, Indian and Egyptian traditions play a vital part.

For all these cultures the sphere was the ultimate magical form closely followed by the cube and the cone. These perfect ideals represented and represent mans mental and thought form interaction with Shunya, that is "Everything" . Outside of these forms was the wild chaos form of the trochoidal surface, typified by the serpent.
What controlled and defined the trochoid was the sphere and the circle. Thus the magic circle was the great defense against spiralling chaos.

The cone revealed aspects of this chaos that were stable and enduring even if filled with destructive power! And the cuboid was mans approximation to perfect uniformity. All of these arose from the spheroidal forms found whole or shattered in Shunya .

These are the Mannigfähltigkeit of Shunya, the manifold folded properties of space-time  or aether as a dynamic fluid entity .

I started withbTwistors and ended ith Grassman Fourier series forms or Grassman Twistors. In practice I have been restricted to Quaternion Fourier forms, and yet they have proved to be candidates for the iterated function that delimits how points move under iteration .
The bounding condition plus this delimiting function reveals how internal and external limitations have very important effect on outcomes.

The question of how did we get to this fortunate position of having an application that oes this omplexity iteration of our derived formulaic expressions is a story that reminds us that
On Oct 14 2010 Benoit Mandelbrot died trying to urge his colleagues to recognise the importance of his work on fractal topology .
It is this topology that takes the work of earlier experimental philosophers and makes it amenable to and contiribuive to combinatorial and computational science..
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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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