This topic since Hamilton and Gauss has sometimes been presented as quadruples and triples, based on the cartesian ordered pair, and triple. There is no cartesian quadruple, so for Gauss to be thinking of Quadruples needs some explanation. It is simple to understand if like Hamilton and Rodrigues you think of the ordered bracket as not containing numbers but measures in a specified direction, that is as axis "vectors". The problem then becomes how do you relate these vectors in a way that makes sense geometrically and preserves the field properties of Arithmetic?
This is of course not always possible as Gauss showed and Hamilton demonstrated . But what Hamilton like Bombelli showed was that the Algebra was still useful! The Dominating idea of Arithmetic, both in India and the west was commercial applicability. This was because most patrons of mathematics were merchants. Similarly geometry was dominated by "Rules" and "rulers'" wishes to construct and quantify securely.
In a very real sense this may be a reason for the dominance of sequencing in numbering and counting: secure knowledge of exactitude!.
In passing the geometrical basis of mathematics and symbolic representation drew attention to properties of space not really distinguished before. Polynomials and their graphical representation began to highlight "smoothness", and scale and continuity and discontinuity. In addition polynomials of higher than signal 1 have always graphically demonstrated the folding of space near a point and expansion away from that point.
When polynomials of more than one parameter were explored it became possible to represent surfaces and volumes directly graphically and to visualise thes deformations. This is a recent thing but of course has its roots in the study of polynomials of more than one parameter.
Where Benoit Mandelbrot fits in this development is in the confluence of computers, geometry and Algebra, and mathematics of scale and error approximation and statistics.
Originally posted by author:
Originally posted by author:
Alice, angry now at the strange turn of events, leaves the Duchess's house and wanders into the Mad Hatter's tea party. This, Bayley surmises, explores the work of the Irish mathematician William Rowan Hamilton, who died in 1865, just after Alice was published. Hamilton's discovery of quaternions in 1843 was hailed as an important milestone in abstract algebra, since they allowed rotations to be calculated algebraically.
Just as complex numbers work with two terms, quaternions belong to a number system based on four terms. Hamilton spent years working with three terms – one for each dimension of space – but could only make them rotate in a plane. When he added the fourth, he got the three-dimensional rotation he was looking for, but he had trouble conceptualizing what this extra term meant. Like most Victorians, he assumed this term had to mean something, so in the preface to his Lectures on Quaternions of 1853 he added a footnote: "It seemed (and still seems) to me natural to connect this extra-spatial unit with the conception of time."
As Bayley points out, the parallels between Hamilton's mathematics and the Mad Hatter's tea party are uncanny. Alice is now at a table with three strange characters: the Hatter, the March Hare and the Dormouse. The character Time, who has fallen out with the Hatter, is absent, and out of pique he won't let the Hatter move the clocks past six.
Reading this scene with Hamilton's ideas in mind, the members of the Hatter's tea party represent three terms of a quaternion, in which the all-important fourth term, time, is missing. Without Time, we are told, the characters are stuck at the tea table, constantly moving round to find clean cups and saucers.
Their movement around the table is reminiscent of Hamilton's early attempts to calculate motion, which was limited to rotatations in a plane before he added time to the mix. Even when Alice joins the party, she can't stop the Hatter, the Hare and the Dormouse shuffling round the table, because she's not an extra-spatial unit like Time.
The Hatter's nonsensical riddle in this scene – "Why is a raven like a writing desk?" – may more specifically target the theory of pure time. In the realm of pure time, Hamilton claimed, cause and effect are no longer linked, and the madness of the Hatter's unanswerable question may reflect this.
Alice's ensuing attempt to solve the riddle pokes fun at another aspect of quaternions that Dodgson would have found absurd: their multiplication is non-commutative. Alice's answers are equally non-commutative. When the Hare tells her to "say what she means", she replies that she does, "at least I mean what I say – that's the same thing". "Not the same thing a bit!" says the Hatter. "Why, you might just as well say that 'I see what I eat' is the same thing as 'I eat what I see'!"
When the scene ends, the Hatter and the Hare are trying to put the Dormouse into the teapot. This could be their route to freedom. If they could only lose him, they could exist independently, as a complex number with two terms. Still mad, according to Dodgson, but free from an endless rotation around the table.
The Raven is like a writing desk because they both have blue-black Quills!
Space the final frontier of mathematics, or rather spaciometry!(both readings).
I have come to an appreciation of notFS that i did not start with. NotFS is perceivable as space, but not an arid Newtonian space but a dynamic form that everywhere impinges at different scales through relativistic motion, relativistic motion transfer and relativisric equilibrium. And where it impinges on me is precisely where i interact with it on all sensory levels and beyond those by scale to levels of interaction i call quantum and universal .
These mysterious levels of interaction are truely beyond my comprehension, forcing or more exactly energising me to accept them as essentially existing but unknowable, but excitingly and surprisingly apprehendable: something i can devote my feeble and mortal life to with the fullest of satisfaction.
That i see iteration at all levels is a symphony of the essential harmony between the processes that govern me and my form and the inherent processes within notFS. My response , the Logos Response, reveals to me the apriori informationinherent within notFs which i uniquely both discover, uncover, and recover as well as create and transform cause to exist in a certain form and relativity and initiate into my consciousness and thereby uniquely store and retain it as a transformation of notFS. These transformations i call the set FS, what i know and have modelled from interacting with notFS.
Therefore randomness and complexity are not attributes of notFS, but of the set FS. This is not to say notFS is not random and complex, but that i cannot know if it is, i have only reacted to it at certain stages in my life and development by calling it so.
The better reaction i feel is to think of it as containing all possibilities, and thereby to approach it positively and exuberantly, all the days of my vain and futile life!
In my approach to notFS i intuitively utilise iteration(unavoidable) and iterations of signals convoluting them into a format called by me measure. These measures proportion notFs and spatialise it giving perspective and ratio and pressure maps in all sensory systems, utilising signal interference and thus inhering a notion, an intuition of iterative pattern making on the basis of an equilibrium pattern of forms called sensor meshes. These evolved forms of sensor meshes represent a transformation from biological molecular conformation networks, responding to basic quantum electromagnetohydrodynamic motions in notFS.
Thus the logos response generates my spaciometry and i distinguish my spaciometry by language forms which provide me with the basic mathematics of my computationally constructed spaciometry. Mathematics is language development and language development is mathematics.
Therfore i sought to rechristen mathematics as manipumæ.
I guess we just do not remember. Few of us do recall the first experience of consciousness because it is in the womb. But the first thing we iteratively measure through our sensory mesh is the dynamic motion of pressure and sound. And then when we are born we are overwhelmed with the dynamic motion of vision and taste and smell and gravitational pressure.
In some animates the sensory mesh has evolved to provide apriori motor and spatial ability through "instinct", which of course is a genetic neural programme or operating system enhancing following and copying behaviour through chemical and sensory stimulation. Even plants follow the sun!
So we use our sensory mesh instinctively in and iterative way to capture the dynamic nature of notFS. How we organise this flux of information is the proper study of computation, but mathematically,by manipue we use relativity based on radials and meshes and we sense equilibria in these meshes by rotational, translational, orthogonal and proproceptive sensors.
Originally posted by author:
So far i have found that few people are properly studying cymatics
but fluid dynamics and high energy plasma physics have already linked trochoids to standing wave phenomena at distinct energy levels. So i will watch the developments with interest.
One question i have asked myself recently is how is light refracted around an edge in a thin slit interferometer, and i suspect it has something to do with trochoidal motion in 3d when light is polarized.
How ironic i must say
if god turns out with dice to play
or that we shall come to see
that god is us , that's you and me
and that is not to say
that we shall live forever and a day
but that the world does come about
by how are processing turns things out!
Originally posted by author:
As usual lots of things to research on the significnce of Hamilton. Time and of course Leibniz views on it with regard to motion, randomness in which Benoit makes a welcome appearance, Quaternions through which i hope to place in my mind Hamilton's mileu and how he presented his discovery in different fields of maths and physics, starting with his battle with cartesian coordinates and the introduction of versors,and his application to generalized coordinates,
After that Hamilton is subsumed in the general Lagrangian mechanics and distinguished as a Extension field in a more general study of classical mechanics and lighting the way to quantum mechanics and Feynman path integral mechanics. Hamiltonian mechanics can now be approached as a "worked over" reformulation of Hamiltons notations and inventions, generalising them into other fields with dry and self important names which totally disconnect the general reader!
We need to look at the Legendre transform, but after that it becomes all greek to me!
An overview of vectors is instructive given the history discussed before in an earlier post. It is interesting to note that Newton's contribution to vectors is little recognised, being overshadowed by his Principea.
the impact of compound interest in the development of Calculus i have noted before, and hidden in that is Newton's contribution to vector application, through Descartes diferrentials and tangents.
It is rather beautiful to me that i have these thoughts pass through me , hurrying on their way to some genius who will fully decant them like some special wine. i have not the "time" to write of the rather wonderful things that come to me, like the conversation about the probability measure that lies within the Lasgrangian exposition, or the fundamental difference that degrees of freedom make to an exploration of notFS without the hindrance of so called dimensionality, all conversed as if Richard Feynman had taken upon himself to share his thoughts in his inimical and charismatic way.
The notion that the motion field is a discontinuous function with outputs from minus to plus infinity, but including 0 which being an improbable case has a region of applicability, outside of which the probability increases but discontinuosly,
the rather wonderful notion that the trochoids are a fundamental group of functions which could be substituted into a Fourrier equation describing systems, or that a quaternion with all its coefficients being trochoids would produce a rather beautiful description of so called chaos and randomness, or that the general trochoid is not that of circles but rather those of irregular spirals…
or that the notion of god has a very low probability,but the existence of an absolute gud remains a possibility! You see nothing can be ruled out because we cannot rule everything in..
To see and hear these things as if in some deep absorbing conversation is truely wonderful, hopeful, and inspiring, because i am glad that i am not some deterministic computational machine with randomness slowly being squeezed out as we find out more and more, but rather a trochoidal maniac, spiralling interestingly throughout space out of control, but seemingly able to mitigate that sufficiently to Enjoy being alive!
Any dynamical system based on 3d trochoids of a circle are amazing, beautiful and moving, but they do not gow or collapse, they oscillate. For this reason i feel that the more general trochoid needs to be explored. In this case a dynamical system will have the possibility of infinite growth or infinite collapse, with a small region of oscillatory like stability.
However at this stage 3d trochoids of the circle with variable radii are good enough to have a look.
Originally posted by author:
What distinguishes animates? Certain unique abilities or forms. But in what way are they unique: binary,or discontinuous but ordered ranking, or continuous scaling of similarity?
So binary, either it is different or it is the same eg does any other animal cook by controlling fire?
Now discontinuous but ranked, it is different by some measure or value that compares against a list or sequence of important values. Things re ranked according to the list, eg do other animates utilise fire?
Finally a continuous scaling o similarity, every thing thst is similar is distinguished by another measure that is scale which has a continuous property, which means i can find a scale value for every distinction i want to make eg do other animates utilise temperature variation?
So from that while warming my tea in the microwave it struck me how that act right there demonstrated that heat was the rapid movement of atoms relative to there molecular and aggregate bindings.
So as some put it the kinetic motion of molecules and atom is heat. Feynman described it as "wobbling". But of course we know that "particles" are regions of space that have spin, therefore it is not unreasonable to expect this wobbling to be trochoidal, and not harmonic oscillation.
The difference this makes is that at certain levels of heating the regions would break apart impulsively rather than elastically as one would expect from a smooth curve oscillation like a sine wave. In phase diagrams this would be saying that a solid could turn straight to a gas if trochoidal wobbles of the extreme kind dominate, and to liquid if sine like trochoids dominate.
Originally posted by author:
Loxodromes are also curves on the surface of an object, usually a sphere but also on a cone where they take on the name vortex or conical helix more readily. So one way of thinking about the Trochoids of a vortex is to think about the trochoids of a loxodrome.
Loxodromes form a nested sequence of curves as one moves toward the centre of a sphere along a radial that intersects a loxodrome. Imagine as the sphere roles along the path of a loxodrome how the trochoids form a nested set of twisting curves at an oblate angle kind of tracing out a 3d set of twisting cycloids,sinusoids and helixes.
The thing about a cone is that it always rolls around in a circle if it rolls on it conical face, thus the trochoids have one easy order but the loxodromic trochoids spiral in or out depending on the surface spiral(loxodrome). This kind of dynamic system is used everyday by British Rail in its wheel bogeys. Feynman relates an interesting tale on how this was an entrance question to joining the fraternity of physicists at his college.
Now on the subject of vortex rings. I guess it has not escaped attention that a stable vortex ring is an ideal candidate for the science fiction idea of "force" fields. If we could generate them in a descending cone of expanding vortex rings each stable ring on top of another spinning anti to it, it may make a viable mosquito net? :dink:
Originally posted by author:
Many of us work on the premise that if something is old and venerated it is profound. But here ps a conjugate idea, because it is profound it is new and up to date.
Grammar of course has a mathematical structure. Before i found the Logos Response i did think that maths derived from Language in this way,giving structure, syntax, order and parsing to it through the notation and algebraic rules for their use. Now i think they are in act aspects of the same thing. Therefore if an expression has a conjugate in a algebraic form there ought to be conjugate to language forms.
Thus x+y has a conjugate x-y, therefore subject + predicate should have a conjugate subject – predicate. So what might this – predicate mean?
There is a big fault that primary teachers have fallen into with fractions i think. Hardly anyone understands proportions and proportioning as a reasoning structure for exploring geometry. Instead of its preeminent place it is reduced to skulking in the backwaters of historical development of fractions. The Lesser has replaced the greater i am afraid, seriously damaging our engagement with mathematics. The moan that usually accompanies the mention of fractions is not to be taken lightly, for at that time millions of children's hopes of enjoying what up until then had been a wonderful subject are dashed beyond salvation it seems.
If there is one subject that can usefully be left to Advanced level it is Fractions, and then as a historical footnote in the development of the numberline concept.
It should be replaced with proportion and proportioning, thus allowing a creative mix of mathematics art and music and dance to be brought together into the maths curriculum. From this we may derive future Albrecht Duerers or even Da Vinci's, and make sense of the spaciometry of our world in all faculties of understanding.
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Ladies and gentlemen….Sir William .. Rowan…Hamilton! Applause please!
So i awoke wondering why my education in vectors had been soooo lamentable?,Why Cartesian coordinates had so constrained the field of mathematics? Why Descartes had had such an influence on mathematics and science through it, after all it is really a simple reference frame, aand when he introduced it nobody was that bothered.
It slowly dawned that it was the personalities, the camps, the warring groups in the grand game, the nationalities: that in mathematics there was as Hamilton puts it a "mathesis" a way of doing mathematics that was doctrinaire nationalistic, patristic and encrusted with anachronism and tradition.
Because of this english maths suffered a loss for over a century due to the Leibniz Newton farago, important insights were overlooked in the case of Grassman because they were not part of the old boy network, and they discouraged outsiders..didah didah didah… Nothing new then.
Rodrigues and grassman what they would have given to have ccess to the web!
Hamilton is a great Irish figure in distinction to the italian, german and general European figures in our part of the history of maths. Because of the web mathematicians of all ages and abilities can get together and critique create and contribute without the old guard control!
So this new frontier in mathematics won't lead to chaos…because wikipedia has shown that effective democratic controls can be put in place, and minority or special interests can set up their own group, without the need for this centuries long hostility and browbeating.
I know that evolution means it is inevitably going to be involved somewhere along the line,but we invented gods to control this sort of thing, so we should use them!
So Hamilton almost singled out the area of Algebra as the new Messiah for mathematics and the sciences due to the success that had flowed from it due to Bombelli, Cardano,Euler Napier,Gauss, Riemann,Newton,Laagrange and Laplace,, on and on as the simple cartesian coordinate mathesis, method tied together all those in the game of maths,both assisting and frustrating mathematicians in what they wanted to think about, to measure and to manipulate.
Even today the myth of number is used to convey concepts that are not related except algebraically. The emergence of a dynamic applied geometry was masked by a clinging to the number mathesis,myth and method. It is amazing to look back and see how mathematicians and physicists struggled to establish a proportinate measure concept of space, whichGreek and earlier mathematicians had in their geometries, and which was common up to the time of cartesian coordinates and beyond.
Newton did not have to have a vector algebra to deal with vector quantities, or dynamic situations, but he did have to invent a new mathesis for dealing with dynamic situations geometrically. It was based on Descartes, but dynamic not static, that is why he called it fluxions.
Didn't he half get into trouble for it! Berkely Lambasted him later. Fortunately for the shy,autistic Newton he was in a respected position because he was right and bright, and the plague had killed off a lot of other contenders! Still he delayed publishing until asked to by Hooke, because of the criticism and prsonal attack he would be subject to.
In those days it was no joke to be besmirched as Galois indicates. You defended your honour with your life! No reason to disturb the frogs and toads then, by troubling the waters unecessarily!
Descartes methods included small differences called differentails later. Newton studied these extensively and through them and compound interest formulae found the binomial series. With this and the fact that differentials were used to algebraically study tangents through Proportions he was able to develop fluxions as a way of compounding tangents to give a curve solution to a dynamic system.
A differential is thus a "compounding sum" of tangential proportions and gives a curve. But along with tangents areas under curves were being studied again by small differences related to tangents. The small differences of these areas under the tangents could be Aggregated directly and they became integrals and were seen as and shown to be the inverse of the mathesis or method of "tangeation".
The difference between compounding and aggreation was not thought that significant, and yet it is a systematic use of vector addition using the parallelogram rule. It was probably hidden by the infinitesimal numbers or fluxions as Newton called them! These were everywhere evident to Newton because he had developed the binomial series and could see them vanishing away in the limit in the series! But what is overlooked is Newton also regarded them as the result of the parallelogram rule, without which he would have quantities but no direction. Newton needed both quantity or magnitude as they distinguished it then,and direction to trace a curve path by tangential envelope.
So by fusing cartesian coordinat geometry and euclidean geometry with algebra of proportions Newton created fluxions for dynamic systems.Leibniz came to it later but for geometrical purposes and without the binomial series which he did learn from Newton.however he did publish both in differential tangential calculus and integral tangential calculus before Newton, giving no reference to their correspondence or collaboration. That was the basis of the dispute.
So sir William Hamilton was not in a glass bottle when he did his maths degree, and he read and corresponded widely.Whether Grassman and Rodrigues were known to him is question, but in ant case no one claims that he stole their ideas, rather that all mathematicians were looking at how to tackle 3d dynamics. Newton for all his brilliance was constrained to 2d by rhe mathematics of his time, and in any case geometry dealt perfectly well with 3d.
So between Newton and Hamilton a generation of mathematicians enamoured with a more symbolic approach as opposed to a geometric one grew up almost disdaining geometry for its lack of algebraic rigour!
Hamilton puts it succintly, they sought a deeper truth than apparently plane and solid geometry could give, there being no advance in it for thousands of years! Plus no one could make their mark in geometry, that belonged to Euclid! Riemann was giving it a good go though.
It is very simple: Bombeeli showed that there was an algebra that was prior to arithmetic, something that had not been realised until he stated it in his treatise. In addition he had shown that it made sense of the "mene" the √-n . Because of him polynomials were generalised and gauss proved a general theorem using these "imaginaries " as Descartes called them. The imaginaries lead to an explosion in the interest and development and applicability of algebra, in which fiels Hamilton was greatly interested,especially as it applied to geometry and space. Because of their notation and their history Hamilton saw the calculus and the differential geometries as a powerful algebra applying to the real world the secret of imaginaries! But coordinates mucked things up!
Much as Cartesian coordinates werethe greatest unifying simplistic idea, they also constrained the imagination and thinking.
Hamilton studied optics because he hoped to understand how rays (or vectors) behaved in the world. Once he had done that he had the technical framework to be able to dismantle cartesian coordinates. Maybe he studies Moebius as well as Malus,but regardless he had a geometricl structure linked algebraically to positions and directions in space. from this he sought a deeper connection with the imaginaries which are involved with rotation and therefore reflection equivalents.
Hamilton's theory on the Algebraic Couples prepared him to be able to do his task. It is to be noted that Hamilton was in a club of Algebraists who were linked to the sciences, along with his friend Graves.
That the imaginaries inspired algebraist is seen in Hamiltons reference tot the doctrine(mathesis) of imaginaries and graves seminal but rejected work.
So i woke to find that these things called vectors by hamilton, both the line and the coordinate system were conceived by him to promote the extraordinary efficacy of Algebra as a valid and useful part of mathematics over arithmetic, and a a gateway into a deeper understanding of the world around us.
Hamilton's dream has been realised but we are not party to it because some have made it their aim to promote themselves,and to keep us in the dark. Would that we all could see Hamilton's rays of light, and his notion of the vector as unencumberd by coorfinates,freeing the geometry to speak to us of the best form to represent it in.
And now we have computers we can perform vectors as naturally as putting knect pieces, or Zome parts together.
Originally posted by author:
For me the natural matrix for mathematiccs is geometry. Everything proper and useful springs from geometry, within geometry is motion form, surface relationship, size of all sorts : tensors ,scalars matrices and vectors; magnification, scale and affine transformations,symmetry of all types of measure and measurement tools, and tools of construction with methods of construction including neusis and arrangements of all sorts including aggregates ,bundles ,structure groups and basic operations of addition ,joining grouping and subtraction and division and separating and disaggregation.
In our response to notFS we intuitively create the geometric background,and the measurement imperative with tool design based on fractal patterns and behaviours. So why did Number become so important? i think that the historical importance of number comes from a different subject to mathematics and its matheses. I think that Numerology and gematria have been confused with mathematics and took on an inordinate influence due to astrological and religious beliefs or faith systems. Certainly the number theory part of gematria is important, but not more significant tna anything else and not as useful as measure and measurement.
it was said that when pythagoras found out that some measurement could not be given an exact magnitude that he was dismayed, but that Eudoxus saved the day by developing a theory of proportion. At that time numerology became a seperate and minor field in mathematics , but no one recognised it in Greece.
Measurement is such a natural given in geometry that it is no surprise that vectors were identified and utilised by some in greek thought at the time, although not called vectors. There does not seem to be any early concept of vector in China, although i feel polar coordinate were natural to the chinese through the taijitu in the i ching and astronomical methods.
So really no one saw the need to develop an algebra of vectors until Bombelli , who did not of course se them as vectors but as adjugates to numbers or measurements. Bombelli was an engineer and saw them as measurements plus an adjugate . The adjugate was neusis but of course he would not have realised that, he focussed rather on the form "mene" the √-n.
What Bombelli had found was magnitude and neusis: magnitude and a kind of adjustment of direction of the measuring tool.
Later moebius and others in defining a vector as a magnitude and a direction left out the neusis, the "leaning toward" by adjustment and so missed the rotational element of a vector as well as the direction . Of course Euler took Bombellis work and restored the rotation but not the magnitude or the adjustment to direction.
It was not until Argand that the direction of the "number" became significantly linked with the magnitude and the rotation. I do not yet know what Grassman did in his Ausdenslehre the study of extension, but the full notion of Bombelli is really only repeated in Hamiltons work, which includes all the motions that should be ascribed to a vector definition: a measurment that has variable direction through rotation and translation.
Thus a +ib takes its full meaning, as being the translation part of neusis and i the rotation part and the measurement is given by √(a^2+b^2).
Bombelli by giving rules formed what later became the algebra of "complex numbers" but now it should be recognised as the algebra of 2d vectors, which Hamilton extended over the hyperreal measures so called.
By regarding gravity as a pressure field or system and looking at the ocean depths as an analogy the simple question why do things float, in particular why do gas bubbles that originate at the bottom of the ocean rise relative to a gravitational pressure field, if such it is?
The observation that gravitational pressure causes a condensing motion is utilised as a starting point to think about this curious behaviour of a bubble rising when it originates at a place where the surrounding water pressure field is high but not uniform.
The motion indicates that the system is not in equilibrium despite the bubble presumably having a relative internal pressure equal to its surroundings.
Algebra I feel now has grown up or at least the cell division for growing up has taken place.
One cannot form algebra without analysis,so within every playful algebraist is a keen analyst and observer.analysts as you might imagine are quite a weird bunch of people even for mathematicians, so it is good to keep the quality I think grounded and not abstracted into isolation.
As usual analysis distinguishes itself from algebra and algebra from geometry and maths from physics etc… But really the relational links are greater than the relational disconnects because the same human workers founded the mathesis or doctrinal methods in all. It is the doctrinal part that drives the separation.
I pretty much think I have a proto structure forming in my mind for a modern ontology of manipume based on spaciometry adult algebra! and calculii of various sorts with arithmetic hopefully being an example of an early example of this taxonomical structure.
Algorithm. When I use the word I tend to use it in a more general sense as a specification of a set of motions that are sequential and iterative through a branch node specification which is based on the test operate test exit cybernetic principle. The algorithm is a kind of predicate to the entity it develops if you think of the entity as a sentence describing its motions.
This rather general gobbledy gook means that I can pretty much use algorithm in most transformation cases of interest and can make analogies with differing systems and within systems.
Analogous thinking is probably one of the first recorded types of thinking we human animates described, but of course it is a feature in all animates by degree. How very self similar and how fractal!
Originally posted by author:
Originally posted by author:
C-C-C-C-C-C-C-C-C-C-C-C-C-COMBO BREAKAHHHHHH!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! O0
its about time somebody else posted in this thread. (google 'the bloody board')
Thanks James. Had not realized how intense it had become, while trying to work something out in different posts!
Just re read a bit and see typos and mis communications all over, so will try to tidy up. Hopefully got a render of a 3d hypotrochoid to post done in quasz, so that should break things up more.
Some of what I write may be right some wrong. That is not a problem to those with there eyes wide open, but I make the statement for anyone who may be tempted to think I have got it all right!
For example I have only just read a short biography of Descartes !! So that puts things in perspective. Descartes did think he was better than anybody else before him in describing reality, but that was just Descartes and I am sure he was an admirable fellow otherwise. We all have a delusion of grandeur anyway and we all certainly are hypocritical!
Originally posted by author:
Thanks Herrmann, great illustration of the concept.
I have no hesitation in saying that Bombelli preceded Descartes in using an orthogonal coordinate system.
"The linear representation of powers, the use of the unit segment, and the representation of a point by “orthogonal coordinates” are some of the noteworthy features of this part of the work"
This is not to diminish Descartes, but to draw attention to the fact that coordinates are not of his sole invention,and very likely he was influenced by Bombelli's Algebra, and that of the italian school.In any case triangulation for positining a point had been a long established geometrical application, so positioning a point even in 3d was not the importance of the coordinate system of Descartes.
Bombelli as i noted previously used a kind of set square tool to find roots of equations expressed algebraically by neusis. Although these were geometrical constructions it would not reveal the complex conjugate pairs without a representation in the plane of the ordinates of measure and sign.
Bombelli clearly did not have the influence of Descartes through influential friends and contacts, but the boy done good for his time and influenced algebra in the popular voice for nearly 400 years.
As with all these things nothing is ever final, and my research has shown that algebra and analysis were one and the same until the differential and integral calculus overwhelmed the majority of "lesser" algebraists and their analytical abilities. Intellectual differences/snobbery i think lead to those analytical algebraists who could do diff and int calculus differentiating themselves from those who wanted to analyse other things!. I also found the distinctions made between analytic and synthetic geometry.
Nevertheless analysis is still a subset of a wider algebra, no matter how good analysis thinks through its proponents it is.
Descartes in his co ordinate system, using them algebraically thought of the reference frame as being whatever it neede to be: static for fixed points,moving or at least having a linear velocity attributed to the parallels for moving points etc.
So the coordinate reference frame was itself an algebraic notation, which could denote measure or direction of motion or axis of rotation or point of rotation, or comparison of measures quantities etc,
Descartes influence was not mainly his coordinate system, but his method of algebraic analysis and this is called a praxis , a mathesis or a doctrine of philosophy on Algebraic method. To Descartes his coordinate system were a tool of analysis ,for the carrying out of his method, and represented the simplest analysis of geometrical forms that he could conceive, and thus was the starting point of his form of analysis. From this start he would add in anything that naturally presented itself as being necessary in the analysis , but only after exhaustively manipulating the simpler analysis to see if it was sufficient to achieve his goal.
Thus Descartes praxis was governed by necessity and sufficency, and by this he hoped to arrive at solutions economically in terms of analysis, for as the Kohleth says" of the making of books there is no end!"
To this end Hamilton seemed to desire a similar influence on method through his quaternions, but this was at a time when Cartesian philosophy was a waning influence and the defense against the skeptic was anathema to the very progress of science.
What Hamilton achieved is all that he could in this time a recognition of his inspiration to mathematicians and scientists around the world for shaping the field of vectors in a useful way.
Manipume¨ i would say is devising methods of proportioning, whether by algebraic analysis or happy intuition, playfully arriving at some solution to some engaging problem, and thereby being delighted!
Originally posted by author:
By analysis found out basic orbits are trochoids. Have not worked out an interesting 3d formula yet but this is a start.
I feel i have come to the end of this research, and by end i mean telos: that i have found what i was looking for even though i could not enunciate it; conclusion: that i have come to a conclusion or set of conclusisons which resonate with me and are of a fundamental explanatory nature to me; transformation: that i have come to that point, moment, place, space where transformation is occurring from who and what i was to what i am becoming.
For me there is no beginning or end as commonly put, just transformation between states, as commonly put.
I feel that i have fallen out into a space that is a dynamic magnitude,
A stillness that is not still and a quietness that is not quiet.
I know the expansion is balanced by a synchronous contraction
That a skew term is what is needed in our "transforms" as well as an "opposing rotations","opposing reflections" and "opposing magnifications" operators generalising the notions of translation and rotation, reflection and magnification so they can occur simultaneously and in opposing "directions". This is so that i can in general fold or tear a piece of paper, or curl up into a ball and go to sleep and explain it by a symbolism that accurately describes it.
I know that since Eudoxus mathematicians have known that the natural numbers are named proportions and are intrinsically scalars; that there is no real entity called a number, but that we have devised this as we have with all our tools and measures as fulfillment of our inate intuitive desires to measure the incomprehensible magnitude which we realise we are in and a part of.
I know that topology is the enclosing concept of all geometries, but i prefer spaciometry, and in our spaciometry we have struggled to piece together a proper appreciation of the magnitude, that it is not just proportional but dynamically proportional, and whatever else dynamic means it means motile in every measuring reference frame tool we care to construct to analyse it.
I know that the reference frames should not be used to delude ourselves that they are inherent within the magnitude, but rather they are tools we have created to explore and analyse the dynamic magnitude, which i will now call dynamic space( DS).
That our reference frames for DS are analytical measures which are not to be ordered or compared to numbers but to define kinds and styles of measure from scalar to tensor, including of course vector and matrices. These styles of measure have their own utilitarian function in our analysis and together begin to form fundamntal algebraic concepts of the spaciometry.
I know that it has been fun and that manipume¨ is a catchy tune!
Originally posted by author:
Finally :toast: i think that DS is aperiodically iterative, by whatever measure we may construct to measure iterativity and or periodicity, and therfor produces aperiodic fractals of which i am one.
That to measure this DS the spiral/vortex shell form is necessary with the spherical shell being the "unity" of this topological group.
We will have to develop our topology of closed and open motile forms, developing suitable analytical measures for this fun thing to do and checking whether we can transform between the two in some algebra that makes sense of what we are looking at spaciometrically, always checking against the spaciometry for sanites sake!
For me trochoids must form some irreducible component part of this mix, but the trochoids of the open form not the closed form, and the simplest examples of an open form to me are the conical and spherical vortices found in fluid dynamics.
Descartes by the way had a theory of vortices, but it is Hamilton who perhaps has lead the way to devising a suitable analytic measure which is directly spaciometric. I will have to look at clifford et al to see if any advance has been made in anlytic measures of this sort. Penrose's Twistors come to mind,but i am not familiar with them.
Of course i am not going to stop posting interesting tid bits
Well i think they are of interest to me and i need to get the words out.
Eudoxus is of fundamental significance to all of western culture as is Theodorus. I know that the chinese nine chapter had a system of proportioning but i do not know much of it yet. Eudoxus howevever is the protypical weights and measures guy! He established the scales of measure and the theory of proportion and proportioning for greek society and western culture.
I do not yet know which cultures valued the void as a cosmogeny besides the Indian and the Egyptian, nor which revered unity as the cosmogeny besides the zoroastrian and the Egyptian Atun dynasty and the bhuddist philosophy school and of course Judaism which is an offshoot of zoroastrian ideology in one sense. I suspect the Greeks valued unity or at the very least the pythagorean school did.
So when Theodorus elegantly showed that there was no fundamental unity it threatened a whole lot in Pythagoras mind , because he portioned the magnitude of space as a multiplication of a fundamental unity. Thus the natural numbers have always been proportional, but in a scalar way as multiples of unity. This is why they were distinguished as Integers that is proprtions of unity,scalars of unity.
Eudoxus restored pythagorean equanimity, but they wanted to hide the information Theodorus had brought to their attention, i guess, so maybe he did not survive the social bomblast he caused. It was left to Archimedes to refer to his work and his proof of the non existence of a fundamental unity using the so called pythagorean theorem or fundamental relationship between integers.
What Theodorus showed was not understood or welcomed, but it was rediscovered and gradually embraced by western culture through Archimedes, who tamed its consequences by establishing the archimedian rule of magnitudes, a pragmatic approach to ratios and proportioning based on Eudoxux which essentially was: deal with what is necessary and sufficient when proportioning. So really it was the indian culture and its love of the void that found no harm in their being no fundamental unity,and allowed for infinitely large and infinitely small, with limit placed only by human perspicacity and endurance to iterate.
Why did Archimedes revive part of this social outcasts work? because of its obvious and beautiful linking of the circle, a respected "perfect " form to the spiral form. Archimedes had found a utilitarian function for the spiral and only Theodorus had a mathematikos on it until he started on the subject. It is to be noted that Archimedes eventually defined spirals in terms of a ratio of motions which bypassed the need to use the surd roots in its description and allowed unity to be used again to define even a spiral.
The spiral also provided a link between direction and measure and established the carpenters rule as the first "model" of a vector, and a great aid in neusis for Archimedes, as was the spiral. Archimedes using the spiral as a neusis path and a rule, likely to be a set square, found several ways to trisect an angle accurately. He also used it in the approximating of π.
Pi was allowed because the circle was believed to be a product from the gods and therefore magical. It was hoped that it would reveal its secret integer ratios to the devoted, instead as it turned out it ultimately revealed all so called numbers are baseless: they have no fundamental unity. It also revealed that numbers are scalars, that measurment is a vector action that trigonometric ratios are fundamental to any analysis of any measure of magnitude, and that they are scalars less than unity, therefore the basis for the notion of fractions; that the pythagorean theorem is a metric for all measures of space, and that right angled triangles are fundamental to any decomposition of the magnitudes of space. The circle also supports a ring group algebra of unity and sign, and vector bases,and the generalisation to the sphere does the same.
The unit circle has taken a pole position in Euclidean and noneuclidean geometries and is a fundamental analytical tool. It has also been used analogically to establish Algebras of non euclidean spaces and geometries and the development of dynamic vectors and algebras like complex algebras, quaternion algebras, clifford algebras, Lie algebras and musean algebras.
So where is the analytical tool for spirals? i believe the sphere the cone and the radials have to be combined to form this type of tool, and that its fundamental components are the trochoids of a spiral which is simply the trochids of a dynamically radially expanding circle or sphere or alternatively the trochods of a sphere projected onto a cone or of a circle onto a line angle tangential to the circle, or rather a rolling expanding circle between the line angle. This is not a clear description i know but only animation and exploration will make it clearer even to me.
As many have remarked Bombelli's wild idea has come up trumps.