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 Author Topic: V9  (Read 7872 times) Description: The general group structure on a circle with a specific example 0 Members and 1 Guest are viewing this topic.
jehovajah
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 « Reply #90 on: November 18, 2015, 08:59:29 AM »

In his Maths foundation series Norman Wilfberger has shown repeatedly how an array notation can be used to represent combination, composition snd Factorisation . These founding principles inevitably allow an algebra to be described as a calculus. .

AnnAlgebra is just a symbolic arrangement of arrays on the page. These arrays may be elemental or complex. The combination of these rays is then given some expression and significance.

Itbtakescacspecial expertise to design and define a composition of these arrays, and it is herevwherebthe Factorisation process is key. When an object or relationship is analysed, the analytical products may be called elements . However they are also called Factors!!

Factorisation has always been an Analytical process. It is this analytical process that describes subtraction and division.. It takes a clear understanding to apprehend the unity behind the words analysis and factorisation!  Such an expertise is often lacking because of an unfounded belief I multiplication!!

It took me some time to translate the first parts of the Ausdehnungslehre 1844 , bu eventually I understood what Hermann Grassmann was founding and constructing. His understanding of analysis/ factorisation is profound. I am tidying Justus work really to see if Hermann learned this from his father or whether he originated this particular distinction. I already suspect that this notion was more widespread than thevGrassmanns, but not so systematically or so powerfully expressed , organised and utilised. But without a doubt Hermanns Ausdehnungslehre constitutes a breakthrough formulation of the expertise at a time when Algebraists( as symbolic Arithmeticians) were glimpsing a fundamental structure to arithmetic as expressed in ring and group theory much later. The most noted algebraists were Hamilton and Boole, so it is praise indeed when Hamilton recognised Hermannmas his master!

Norman develops this combinatorial compositional system several times, but only a few real relationships have a ring structure associated with their combinatorial description.

The combinatorial basis of any calculus holds the key to its usefulness as a calculus. The arithmetic model is the foundational calculus that we find most useful . Any other combinatorial system that closely mimics that arithmetical system is formally called an Algebraic system. The use of terminology like this screws with your brain, but essentially it means we can count and measure real relationships even if they are not extensive magnitudes!!  So temperature is not an extensive magnitude, but by naively representing it as an extensive sketch or line segment we can apply a combinatorial system like the Ausdehnungs Gr廲e to it.

Andbthisb1-1 correspondence is crucial to the effectiveness of the whole symbolic system as adjusts goes on to find out.
Norman repeatedly explores these issue from many angles, but he is unwilling to give upon mathematics! Hermann and I recognise the inevitability of the demise of Mathematics, especially with the rise of computational systems that can model so many extensive and intensive magnitudes.
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jehovajah
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 « Reply #91 on: November 18, 2015, 10:21:02 AM »

I cannot rest without remarking that Newtins Principles for Astrologers is a powerful Combinatorial system which applies the fundamental Pythagorean Geometreea to the counting and measuring of the celestial bodies in transit.

That it is based in a fractalnTopology is undeniable, for the very system is based on Galileos observations and reasonings as expressed in his application of Pythagorean and Aristotelian Geometria to the observations made through his telescopes. He was a professor of Mathematics at a time when Mathematics meant a qualification in the doctrines of the Aristotelian and Platonic systems based on the Pythagorean teachings from Euclid,Eudoxus Ptolemy and the Academies of his era.

The confusion introduced by Aristitle was not enough to obscure the invariant principles of the Pythagorean school, but enough to denigrate their source, and clear thinking application. After all it was Apollonius and later Nicodemus that brought the Pythagorean school to its high state of utility, but the Aristotelian peripatetics and their Islamic scholarly converts that confused the 2 systems.

It should be remembered  neitherv Plato nor Aristotle ever were qualified as  Mathematikos! But Euclid and Eudoxus we're both masters of the Pythagorean traditions, as indeed was Apollonius.

The brilliant searching mind if Newton was able to discern these distinctions and to pay homage to the Forefathers of his craft. Many if his contemporaries never grasped these traditional combinatorial systems within the Stoikeia.

By paying close attention to Galileo Newton constructed a combinatorial system equivalent to a lineal Algebra. It took Lagrange to bring these methods out in a form that Hermann could later synthesise under his Masterwork the Ausdehnungslehre 1844, flawed as it is.

Despite his brilliant devotion to Newton and Aristotle, Hamilton was not able to approach the generality of Insight that Hermann, free from Academia was able to synthesise from his lights .
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jehovajah
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 « Reply #92 on: November 28, 2015, 02:59:51 PM »

The combinatorial topic that inspired this thread is the Vedic and Bahai numerology based on the arithmoi 8 and 9.
These nets and mosaics are related to the topology of the circle, the topological form or division that describes the arcs and spheres of the planets, and stars of the ancient astrologists.

The topology is advanced by modern topological considerations onto the toroidal form in order to extend the meditation to a 3d form.

Initially the topic concentrates on the surface mosaics, but some have attempted an " axial " distribution to extend to other pattern formats.

The axial system is misleading, as the mosaic does not describe a fixed form but rather a dynamic process of oscillation or frequency movement. The patterns capture stable Fourier wave patterns at certain frequencies.

The doubling. Circuits are arrangements of topological forms into a geometric tesselation, that capture the spiral nature of growth and decay, which is exponential .

The application to electromagnetic wave patterns naturally describes this kind of dynamic growth. Sinusoidal wave forms are a mathematical/ physical myth, that misdescribed a natural growth and decay process in a fluid dynamic .
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jehovajah
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 « Reply #93 on: December 04, 2015, 10:58:25 AM »

http://www.yasni.de/justus+gra%C3%9Fmann/person+information

Enjoy!
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jehovajah
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 « Reply #94 on: December 28, 2015, 03:09:35 AM »

This continues Normans careful construction of an Algebra. We see how the motivation to mimic the combinatorial structur of the Arithmetic of the Arithmoi is divined or distilled via the judicious use of symbols and or labels!
This is called Algebra from the Arabic Al gibr! This means the twisting . In the vernacular it refers to the " mind fluff!" involved in this distillation process. Most of us can relate to that! But actually it refers to the spiralling thought patterning required to distill the essence of any calculation!

Calculation is derived from the notion of calculus using a small stone to represent or symbolise an entity and ultimately a quantity of entities. By extension it also represents a quality of an entity by an intensive magnitude experience.

The Arabic m"Quabbalah , is simply the performance of these calculations , thus the spiralling rearrangement of calculii, that is the combinatorial ordering of these representative stones as well as the sequencing or patterning. These are the origin of Sequences and Series.

Of course this is usually translated from the Yiddish or Semitic as Kabbalah.

The spatial arrangement and sequencing associated with these calculii, means they represent spatial Metrons . Metrons are topological forms used to measure space. The measurement of space is called Geometree, in the Greek, but the the process is called Katametresee .

Thus the Arithmoi are geometrical forms , better topological measures organised into patterns  and used to measure space !!

The representation through calculii shows arithmetic to be a symbolic representation of spatial measure and arrangement and thus the Combinatorial structure we impose or divine on or in space. This arithmetic is thus already a symbolic Algebra. But using this as our foundation we have worked hard to develop alternative symbolic Algebras.

Because ultimately we want to do Astrology and astrologers devised the topological forms in the ground and in mosaic patternings to comprehend the patterning in the sky above and to apprehend invariant measures. By this means the planets were highlighted; we strive to make our alternatives conform to this seminal Structure and distillation of spatial measure: Arithmetic

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jehovajah
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 « Reply #95 on: December 29, 2015, 08:01:37 AM »

One of the issues regarding mathematics so called is it's abstract nature. However it is a set of thought patterns( after Hermann Grassman) that are extracted from some activity by someone who is developing an expertise.

Thus an artisan or a musician or a farmer or a physician will develop their expertise by abstracting patterns of behaviour, performance and thinking. It is this practiced arrangemnt and combination and or sequence of attributes that form the structural basis of any expertise.

This video shows how an expert, in conjunction with the equipment and tools of their expertise attempts to impart the specifics of any procedural or performance behaviour.

The Doctrine of Forms as Hermann posited as a possible replacement for the defunct term Mathematics , attempts to impart these skills as thought patterns through the written medium. Of course we now have the video medium which more precisely enables an expert to communicate and demonstrate.

I make no secret that Norman Wildberger is my number one choice when it comes to unravelling all things about the Grassmann expertises.

The translation of these works and papers, particularly of Justus and Especially Hermann are a valuable exercise in re orienting the Natural Philosopher within all of us, but especially those gifted enough to apprehend abstract thought patterns.

I invite all interested in revolutionising our primary level education to contribute to this thread in terms of translating and commenting on the original papers as far as we have access to them .

Looking forward to making more progress in 2016, and thanking you all for your support and interest in 2015.

Vielen Danken !!
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jehovajah
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 « Reply #96 on: January 02, 2016, 11:11:45 AM »

This is an experimental post and will be edited
I am trying out Dragon Dictation! I have to find a way to progress my meditative translation

Quote
The here out inward springing personal exercise can as the most important personal exercise of the indicated part of the the geometrical combinatorial doctrine come to be tracked with which we here occupy ourselves
Three of their positions, according to to the given items of a point out going radius are concording to even if in a single plane they do not lie a bodily spatial corner or a spherical triangle . Gives One to the corners which they build under themselves (the sides of the spherical triangles) concording magnitudes and to the radii themselves concording outwardly holding entities thus steps One through one how through the other out of the field of the unmixed geometrical combinatorial doctrine which from all magnitude outwardly held entities independent should be here out and each in this relationship possible differing qualities guides onto an array of "away lead" shapes. We would here at the nearby with the simplest of all thinkable cases begin coming to be rather to find that it onto conscious mental attitude all remaining entities already in itself begrips. This case is specifically the one by considering which which the three lines as  concording to the three dimensions of space are going on to one another perpendicular

Translating

The personal exercise hereout springing to mind can be tracked as the most important exercise of the indicated part of the geometrical combinatorial doctrine, with which we occupy ourselves here.
Three radii, their layout according to the given items of a points outgoing radii, even if they do not lie in one plane, are concording to a bodily spatial corner or a spherical triangle.
One gives concording magnitudes to the corners they construct under thmselves( the sides of the spherical triangle ) and to the radii concordingly outwardly holding magnitudes, thus one steps hereout through one, how one steps through the other; out from the field of the unmixed geometrical combinatorial doctrine, which should be independent of all magnitudes, all outwardly held magnitudes
And in this relationship each of possible differing qualities guides onto an array of " away leading" shapes!

We would begin here at the nearby with the simplest of all thinkable cases, to find rather, coming to be that it begrips already in itself onto conscious mental attitude all remaining entities .

This case specifically is the one by considering which the three lines, as concording to the three dimensions of space, are going perpendicularly onto one another
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hermann
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 « Reply #97 on: January 03, 2016, 08:14:45 PM »

Hallo Jehovajah,

I wish you a good new year.

I did some programming the last days one of the results is the following table.
http://www.wackerart.de/mathematik/big_numbers/fibonacci_numbers.html

May be you like it.

Hermann
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jehovajah
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 « Reply #98 on: January 06, 2016, 03:19:12 AM »

Lol! Hermann !
Amazing what computation can achieve now in the palm of your hand .
I am sure you realise the numerical strings are of little significance or practicality, but like Pi demonstrate our systematic method for capturing large permutations .

Justus Grassmann would make sense of these as crystalline structures , Hermann Grassmann would make sense of these as n dimendional forms where n is some fibonacci number.

But you can also see that our physicists do not venture above 11. Dimensions! So no physical theory can accommodate these Arithmoi or mosaic structures .

The most useful interpretation would be as video frames of a Dynmical structural crystal form , but I do not think even that kind of morphology would utilise beyond the first 200 Fibonacci numbers, in permutations .

Their is the aspect of intensity , the intensive magnitudes rather than the extensive ones. Here the potential to capture colour , greyscale and dynamical changes in these , plus textural changes may utilise possibly up to the 400 th Fibonacci, but that is a pure guess.

Another possible use is in generating secure codes?

The pure numbers are therefore of little interest, but what they could encode is of interest
Xxx

Happy 2016 to you and your family and great success in our work and hobbies xxx
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May a trochoid of 去逸 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 #99 on: January 07, 2016, 07:34:03 AM »

Incredibly google translate now translates photographs of text using my phone app.
While this is not a good translation it may help me to speed up my translation habits. For example I am still one finger typing this when I could use dragon dictate!
Oh well! Old habits die hard! Xxx

Typo warning; it could improve my typo rate ... Or not!!! Lolxx
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May a trochoid of 去逸 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!
hermann
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 « Reply #100 on: January 08, 2016, 08:22:15 AM »

Hallo Jehovajah,

the table of the Fibonacci Numbers is a test for the software I have developed during the christmas holidays.

In summer holidays in 2014 I started developing Ada-Software for generating HTML-Pages for my Geometric Algebra Page.
http://www.wackerart.de/mathematik/geometric_algebra.html
One of the resulting pages can be seen her.
http://www.wackerart.de/mathematik/geometric_algebra/several_base_vector_tables.html

The Ada-Software did only some basics works, mainly converting Ada-Matrixes into HTLM-Tables giving me a tool to convert Ada-Datastructures into HTML-Pages.

I have done some improvements on HTML-Software allowing me to visualise results of the software.

A nother issue lies in the use of real numbers in mathematics and in software development.
Writing 1/3 is far more elegant then writing 0.33333333333... !
So a long time ago I desided, that it was far more elegant to use rational numbers in software development then real numbers.

I also fully argree with Norman Wildbergers critics on real numbers, infinit sets and the mathamatics taught at our Universities today:
http://wacker-art.blogspot.de/2016/01/why-infinite-sets-dont-exist.html

For this reasons I used the following package for implementing rational numbers.
But using only standard integer numbers as generic parameters for instance of the generic packages was not very satisfying when I tried to approximate Pi or e.
So bigger Integers are required for further progress!

On may prime number pages I also came very soon to the limits of the integer values used by JavaScript:
http://www.wackerart.de/mathematik/mersenne.html
Also generating big tables for displaying results became a challenge for a software developer:
http://www.wackerart.de/mathematik/gilbreaths_conjecture.html
This Tables may challange your browser:
http://www.wackerart.de/mathematik/gilbreaths_conjecture_structure.html
http://www.wackerart.de/mathematik/distribution_pattern.html

Hermann
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jehovajah
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 « Reply #101 on: January 25, 2016, 10:28:56 AM »

The history of Indin Mathematics.

Your pursuit of Numerical symbols is admirable. The use of computational programmes is good to emphasise the true discipline to which so called mathmatics belongs.

But now the above video confirmsvwhatbi have long suspected: the language grammars of India fundamentally shaped what we came to call Mathematics!

The Pythagorean school undoubtedly connected to India and other esoteric temple societies.

Justus in sifting through combinatorial notions directly connects language to forms and from forms to counting and eventually Arithmoi or mosaic forms later called "Numbers"..

The Pythagoreans were not phased by irrational numbers, because such numbers do not finalise. They approximated as did the Indians.

These Ganitas or rules are very practical . These rules were known to Newton,Wallis and other classical scholars through the Arabic or Islamic scholars.

Many lies have been told about Indian Mathematics

The Greek Cannon of Mathematics I think is a historical fiction. The Islamic scholars favoured Aristotle who was a very autistic taxonomist! He was not a Pythagorean but a Platonist offshoot. His work is a collection of dry bones!
Euclids work is more extensive than the Stoikeia, and Aoollonius is reputed to have made corrections and improvements. The Pythagorean tradition is very Indan! The Aristotelian one is the rigid dead Skeleton of a prolific genius who often got it as wrong as he got it right!

These tables of relationships, matrices or arrays code a practice of measurement and counting and combining forms. This combinatorial base is the foundation on which topology is based.
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May a trochoid of 去逸 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 #102 on: March 21, 2016, 02:33:08 AM »

The shame is that Justus and Hermann and Robert were publishers whose works achieved limited circulation! But Peano acknowledges that he read a translation of Ausdehnungslehre1844 prior or in tandem with his own work!

Thus all these " axioms" are thoroughly discussed in the Einleitung of 1844
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jehovajah
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 « Reply #103 on: May 04, 2016, 04:16:14 AM »

Magnitude and Quantum are 2 experiences we seldom find defined.
Magnitude is an experience of extension or extensivity in a notion of space . Sometimes we use this notion metaphorically to describe experinces of intensity ( including density) thus a moving object has an intensity we call speed but an extensivity we call velocity
The extensive magnitude is continuous and indefinite.
Quantum is an experience of magnitude which is discrete distinct and definite because it is finite.
Discrete items can become continuous through contiguity. If discrete items do not touch then they are not cnontinuous as a group

Group contiguity is how counting and measuring differ. By placing Metra contiguosly we can measure an object into constituent of the smaller type . These are mosaics .
If we shatter an object it flies into discrete pieces or quanta, but these quanta are not: contiguous,uniform/regular or uniformly oriented!

It is the combination of these quanta and the composition of them in space that We seek to record by our Begriff or notational system of labels and handles.
The combinatorial system Hermann lays out is specifically dealing with contiguous quanta. As a consequence the combinatorial action brings quanta into contiguity.
The compositional action arranges how each quantum is contiguosly related to another, and this is principally by relative orientation or relative rotation.

Thus if I compose quanta I am generally specifying their relative orientation and if I combine quanta I generally spr ivy their contiguous relation. The labelled outcome result of these two processes is the lineal sum: another line ; or the lineal product: a flat figure usually a parallelogram but not necessarily so.

To get a parallelogram from an orientation process clearly involves more than just relative rotation about a point. Thus the complexity of composition should never be underestimated, and it should be no surprise that commutativity is not a fundamental compositional expectation! After all, we do not expect to build a brick wall using commutativity! We expect different outcomes if we compose quanta differently!
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jehovajah
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 « Reply #104 on: September 11, 2016, 10:32:45 AM »