Reference frames are intrinsic processing strategies for differentiating space. The first reference frame we experience is our own subjective points of view. However, before we get a chance to develop an understanding of that reference frame we are usually manipulated and socialised into a failial and cultural reference frame , and then introduced to he terminology at a later stage when we are well wedded into the cultural frame. People who do not follow this route of socialisation with regard to reference frames are classed as Autistic nowadays, before they might have been classed as thick or stupid, rebellious or disobedient and even possessed. whatever, their lives were ot made easy by not having a common cultural reference frame.
The same set of cultural and social prejudices inevitably exist in the sciences and mathematics derived from pragmatic experience, althought to listen to the propaganda you would think that creative thinking and innovation was the doyen of such matters. However, every human enterprise is the same, constructed along the same social and cultural group lines and paradigms and filled withthe same prejudices.
The first reference systems were astrological. They probably originated in the east African coastal regions when the Saharan was a great lake, an inland sea. They were called Tymes and Seasons and recorded the stars in the heavens in their relative positions,nusually as wall art. These skills and knowledge passed eastwards with the migrations out of Africa to the Arabian peninsula, along the coasts, probably now inundated, through to Persia and the Euphrates settlements and on to India and the Dravidian settlements along the Ganges. From their it seems a pincer migration from the Euphrates civilisations and the Dravifian civilisations spread it to mountainous people's who took it across the Russian steppes into northern China. Meanwhile a long migration eastward took the knowledge to the Indonesian chain and finally along the coast to northern Australia and southern China. Meanwhile a small band of Indonesian seafarers spread across the peloponnesian island chains to end up in south western Australia.
It seems that some of the north European migration followed the Inuit trails across land and ice bridges into northern Alaska and down the coasts Past massive ice sheets to southern America .
Thus in this way, knowledge of times and seasons in the astrological records was passed and updated around the whole world. However in certain civilisations the populations were so stable that large amounts of seasonal data, and astrological pictures of the stars were collected. These pictures and data were the motivation for a group of Magi to enhance there status by providing predictions, or forecasts. However this required accurate data, and so in Babylon and Egypt you find this more accurate data and charts being recorded and tools for measuring being developed.
The tools were simple ways of marking positions in the skies, and ranged from plumb line sights to sophisticated graded and scaled sighting tools. The data of the stars and planets plus their sighting marks was soon recorded on temple walls, papyrus and clay tablets, and stored in temples to the muse called libraries. This detailed knowledge did not travel as quickly or widely, and in fact it was only gradually spread by commercial and trade interests as well as religious and cultural needs of settlers..
It therefore took the vestiges of war and empire to enable this detailed knowledge to become widespread, and it was the Greek empire that provides us with much of our roots to modern astrology and astronomy, due to the Greek cultural drivers of hellenization, while inculcating the best a conquered country has to offer. Thus the times and seasons data became precise measurement data, that Ptolemy was able to decode and use to develop his Almsgest, the most accurate astrological method of his day and up until the Arabic empire. He included several innovations, including thr trigonometric methods of Hipparchus and the spherical trigonometric notions of chords and ratios with the diameter or radius. These chords the indian mathematicians took from the Greeks who hellenized part of India on the north, and they created the limb and the jamb. Through n Arabic mistranslation these became the sine and the versine.
It was the Arab empire that launched a centuries long calculation to calculate these sine ratios, which gave us as a result polynomials, from multinomialists, difference equations logarithms, and the binomial series expansion, along with the probability calculus and the differential calculus.
This astrological and trigonometrical reference frame was too cumbersome for some who wanted a direct way to refer to points in the plane. Descartes and his contemporary De Fermat began to use a simple coordinate system on the page. This system immediately took his mind out of arithmetic calculation mode into algebraic rhetoric mode. This was ideal foe Descartes who liked to spend his days meditating. To his surprise and delight he was sble to encode a number of geometrical ideas into this format and solve them comparatively simpy.
Although he did attempt to encode all of geometry with this method it was not so easy as time went on, for example the circle curve. Took time for cartesian idea to catch on, even with De Fermat helping to promote it. And it is not like Descartes understood how important it would become. It was printing, and the fact that Cartesian coordinates in the plane were so compact on the page which helped it to become so ubiquitous. Duerer but particularly Regiiomsntus with the complete solution of triangles made it de facto the way to go.
The introduction of misfortunes the numbers by Brahmagupta created the most enduring revulsion in the human psyche. However, these numbers received a colour change or two. We're taken to heart by the commercial and banking sectors and were given a spurious legitimacy by the Arabic algebraists. Thus they found their way by commerce and algebra into the Italian mathematical cultural scene where they contributed to the rapid renaissance of finding the roots of "difficult" that is mind bending problems suited to the name called algebra. The quadratic and cubic formulae and equations were the bedrock foundation of the later multinomialists explosion in Demonstrating mathematical prowess , but is was the curios formulary that Cardano and tortellini arrived at that threw up the odd and alarming sequence _/-1. This in particular gave Bombelli the incentive to write of his love of Arabic algebra, especially at a time when the pragmatic art was arithmetic. This enabled Bombelli to rehabilitate negative numbers into a respectable engineering context, but sparked Descartes to declare some roots as purely imaginary. The name stuck.
Along the way to this position, the Indian algebraists also exhibited a great love of the reciprocal, which, found in Babylonian tables had had a long use for loving certain problems. Also the Egyptian papyrus testifies to there usefulness, and thus the Indian algebraists took them into a purely numerical form that enabled them to represent intriguing problems about space. These reciprocals, together with the advice of Persian astronomers and spherical trigonomrtrists lead to the grudging introduction of fractions in place of ratios and proportions in many calculations. This and the healthy interest of Wallis and others lead to the first notion of a measuring line to evaluate the results of calculations. The measuring line concept was soon amalgamated with the Cartesian coordinate system, as it developed. The early cartesian system eas freer, as it had no convention of zero, jus ordains te and coordinate lines. It was the work of Wallis in particular that gave us the Cartesian system we are familiar with today. His measuring line daringly included negative numbers, and so neede an origin, a zero!. Wallis eas also the first to give a full algebraic description of the conic curves in this new system, thus single handrdly putting conic section makers out of business!
This Cartesian system gave wallis the chance to guess at what the imaginary magnitudes might be in terms of points in the plane! Bombelli called them adjugate numbers and along with surds Wallis found a position for them on his measuring line, but the _/-1 had to be off the measuring line, somewhere in the plane he thought insightfully. In any case under his tutelage, Newton used them without trepidation, and Newton's students De Moivre and Cotes made great headway with them prior to Euler!
Thus over time the Cartesian reference frame was deveLoped and culturally integrated and the negative and imaginary numbers given a tentative home.
The polar coordinate system was worked out before the complex plane reference system was mooted by Argand and Cauchy. But it was Wessel who went back to surveying to provide a logical basis for the flat plane as a directed magnitude reference frame, which explained the complex or imaginaries as rotations in the plane.
So as not to be left out of the picture Gauss used this paper by Wessel as a human shield for publishing his own doubtful ideas about imaginsries. Thus Wessel Cauchy and Argand and Gauss contributed to the notion of the complex or imaginary plane as a reference frame. Whenever challenged it became customary to misdirect by calling it the imaginary or complex plane, and to treat it formally as a mere thought experiment . However. Applied mathematicians and physicists were increasingly able to demonstrate real uses for this unrealistic reference frame, and eventually they could not abide the pontificating about the imaginary reference frame. They wanted to use a real reference frame thst did the same thing. Thus the hunt was on for a consistent reference frame and algebra that dealt with physical quantities.
Hamilton was convinced the imaginaries were the answer, they just needed logical foundations that overcame centuries of negative conditioning. Couples was his groundbreaking work that did this, and in this he replaced the term imaginary by the term complex, and the complex plane became a suitable reference frame for scientific calculations and modeling.
However others wanted a clean break from these curious constraints and thought that Gibbs vectors might be the way forward. Vectors were equally strange, and eventually scientists and mathematicians grew up, stopped wingeing, and realised the complex plane and vectors both indicated reality is more complex than they ever believed!.
The complex reference frame and the vector reference frame existed side by side as tools in the physicists toolbox, and then Hamilton discovered Quaternions, a complete reference frame for 3d space including rotations. Every body swung on the side of quaternions.
Grassmann published his Ausdehnungslehre; Gibbs and Heaveside eventually picks it up to steal a couple of ideas, but essentially to give spurious justification to his own, anti quaternion system. Thus the quaternion reference frame system waned in fortunes as the vector one grew in popularity.
Even the vector reference frame was superseded by Einstein replaced by tensor reference frames, with a view to achieving the most general non special description of physics in space.
The quaternion reference frame with its algebra is the most powerful system with rotstion we can devise which looks like our experience of reality. Clifford algebras just do not engage our senses in the same way despite their greater control over quaternions..
Quaternions provide us with a reference frame and the axis around which it spins!
The mathematical and symbolic description of quaternions is misleading. Hamilton strove to explain quaternions free of a coordinate system, and in so doing founded the essential group theoretic method. but Hamilton was at pains to point out that quaternions were the comparative ratio relationships of directed lines in space. Just as he had shown that imainaries were not imaginary, but rather complex relations between lines in the plane, relations which we somehow could capture y the strange processes of complex algebra, so we could capture the relations of lines that are directed in space.
We deal with such lines or ectors naturally. they are not hard to deal with or visualize. What is hard is to develop a good algebraic model of them which does not become too complex, and hamilton believed the quaternions were the solution.
How do we recognise a reference frame? By the simplest test that it identifies any given position . Now it does not do this uniquely, in fact it does this in a redundant way, as there are many ways to use the same frame to identify a unique point, and it is all these redundant ways that encapsulate the notion of relativity. However , for Cartesian coordinates we can identify a body only by speifying its positions within it. however if the body is rigid, we can identify it by one point, ususally a centre of symmetry but not necessaruly so. The important thing is that this point can be the origin of a Local reference frame.
We have to be careful of objective subjective paradigms here. All we need to understand is that either i am the reference frame or i give the reference frame to another object. But what einstein demonstrated is thar we always need both my reference frame and the one i give to an object. in this case i am calling that reference frame i give to an object a local reference frame for that object, my reference frame being the absolute one for m world!
The issue now is what if the object is dynamic, and it is moving in my reference frame and otating. how do i describe that?
Normally i creat another reference frame , which frees me up to take measurements with regard to this new reference frame. I can then derive equations to describe the data and the relationships i discover by measuring. The result is tedious amounts of work especially for the rotation. However if i create a quaternion reference frame, then i can measure the distances, but when it comes to the rotation i can define the axis of rotation vector , and then define the 3 vectors i, j, k from this axis and use quaternon elations to denote the rotation about that axis, thus simplifying the calculation load.
In such a frame i identify the cartesian point references in the local frame to the coefficients of the i,j,k vectors( q1,q2,q3 will be the quaternions hat rotate the model and these i identify by the quaternion product rules which include dot and cross products as well as multiples of combinations of the coefficients in a quaternion combination in general.
Now i can do this at any scale and for any reference frame , and apply the rotations in any pattern of sequence. in this way i can describe a dynamic rotating niverse with a disposition of quaternions in local configurations. This requires a lot of computing power which fortunately we have nowadays, The use of this facilty to creae realistic motion in CGI worlds today is truel amazing. What is also amazing is the ability of quaternions to generate fractal backgrounds, but it limitations are in its point cloud surfaces if the fractal is not a connected set.
There are other fractal tehniques for producing geometric structures for backgrounds, but the translation and rotation of those backgrounds can be done by quaternions in a quaternion reference frame
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