V9

[jehovajah]

While I have no clue to what Complexionen refers to in Justus system there is nevertheless an Astrological association and thus a geometrical one.
http://digi.ub.uni-heidelberg.de/diglit/cpg832

The link provides interesting reading but I am not inclined to pursue it here.

[jehovajah]

A little context may help explain the origin of the germ of Justus ideas.

Geometry was taught in the main using a textbook constructed by LeGendre. Thus he had redacted several sources to construct his textbook. Thus a non Euclidean tradition was promulgated under Euclids name. Copies of the Stoikeia were rare but extant in Europe. They were studied by classicists who also were astrologers and Natural philosophers. Geometerscon the other hand studied Arabic sources and redactions.

Thus it was that the great farce was foisted upon geometers from about the 5th century ad when scholars thought the plan of the pythagoreans was to prove all propositions by preceding ones . The idea came to be put forward that the 5th postulate was a proposition( that is Not a postulate!) and required "proof" by deduction.

Such a proof was never forthcoming even up to 1800's when all looked hopeful and new. By the 1820's men likev Gauss exploded in rage and shame at our inability to prove such a fundamental " proposition" ! In the mean time Bolyai and Lobachewsky were claiming alternative consistent geometries existed!  The 5 th postulate was not necessary for geometry !

It was only much later in the 1900s when it was realised that these alternative Geometrie were well known from ancient times especially as spherical geometry and trigonometry and used by all seafarers to navigate the globe! Indeed the direction of Mecca was always calculated using such geometries.

The shame of geometers was profound. They sought to blame Euclid and to demean his work. But Euclid would not be suborned in this way, nor impeached. Those that read the Stoikeia realised how wrong geometers had been and how simple the mistake.

Men like Justus knew enough to look for alternative views regarding the Stoikeia. Along with Abel and Lie and a few other nature lovers they sought to find the truth in Nature and Natures Handiwork. Thus they restarted from first principles, rexamined everything and began to realise that Natures geometry was based on combinations of things, elements which, according to the new science had " chemical" properties that allowed the creation of new things, the transformation from one compound to another.

Why did geometry not reflect that?

In fact Justus realised that combinatorics did reflect this natural reactive combining, but you had to let go of narrow restrictive definitions and embrace a broader church of ideas and elements.

But this was not Arithmetic! Combinatorics was about counting and adding up things with certain specified properties. What Justus glimpsed was more general than that, yet it was useful to use the superstructure or scaffolding of Arithmetic to shape his insights as far as feasible or consistent with his material and its own impudent and impolite nature!

The analysis now stopped. There was no point in further analysis. Synthesis was the goal. And synthesis demanded the line and the plane, or rather segments of each in order to build.

The whole field now became Dynamic, as well as fresh and exciting. Stuffy propositions were no longer what was studied. Instead the questions became: How many?

How many different shapes can you make with 1 line segment(/arc segment)?
How many with 2 ? Are they really different? How do you decide that?

As the number of line segment grows so the fields of study broaden into crowded pens! This combinatorial approach suddenly seemed to make sense of geometry. Later geometers would note the similarity between these output results and the Progression of material in the authenticated versions of Euclids Stoikeia.

Euclids Stoikeia is not a book on geometry or mathematics, but on the philosophy of the Pythagoreans. That philosophy was to investigate nature and the Musai to the best of the gift they had bestowed upon you as a vessel. Thus drawn or drawing lines, planes and eventually mosaics and solidsv were the means to hand. They to declared the point as where synthesis begins. Once a point is reached no further analysis can be done. Now reconstruction or combinatorial synthesis  can be engaged in.

From this approach emerges naturally all arithmetic.'what is not fundamental is gleichgültig or commutative output results ? Shapes where commutative output results appear are rare symmetries. Symmetries can be found only rarely and yet our current physics and geometry highlight these atypical values!

Many of Justus Ideas and those of his collaborators are in the mainstream now, but have lost their pungency because no one knew where they came from and Why.
Klein was mainly responsible for making this approach and work Anodine and acceptable, but he downplayed the sectarian fervour it generated in his troubled times.

[jehovajah]

I have reworked the rough draft sufficiently to move on to the next.

It becomes clear that the early stages of Hermanns Ausdehnungslehre 1844 follows this poetic schematic very closely indeed. However the devil is in the details. Nevertheless this emerging companion entity as an a priori theory to geometry, analogous to arithmetic as far as it goes is clearly invoked.

It is not Arithmetic however, but something else that gives rise both to Geometry first and then Arithmetic as complexity increases.. We have taken to calling it Algebra or Group or ring Theory, and now Geometric Algebra, but Justus felt it was a new combinatorial Doctrine given by the Initiator of all things!

Thus modern combinatorics is not strictly the same as what Justus was promoting. In fact the nearest to it is the nerdy specialist topic of Crystallography. Yet even this topic today fails to promote the vision Justus is evoking here, I suspect.

Hermanns take on it is slightly but importantly different. Justus appears to have assumed mutual orthogonality as the god given norm, and along with that perfection in symmetry. Hermann took the more general view that orthogonality is not fundamental, but in fact very special indeed. Nature rarely exhibits it in crystalline form! Thus the Parallegram became crucial to his development of his fathers insights and life work and Space Theory.

It is also clear that the circle and sphere figure fundamentally in his work, but in a form very much in keeping with his times. This aspect of his work is in fact subsumed in the 1862 more perfect version, but I feel that was due to radical redaction by his brother Robert, also a publisher of several volumes on his fathers and the families conception of Ausdehnungslehre.

I have not read the 1862 version, but still think a great deal may have been lost due to these turn of events.

Ah! But it was ever thus
Ach! Macht es immer so!

[jehovajah]

"Umfasst" or the " comprehending enfolding" is not a new idea in philosophy. It is clearly a fascinating concept very similar to fractal levels, or scale free formations. But it also necessarily for Justus times heavily relies on the Geist immanent in Nature and Space.

We may have a religious paradigm that encompasses that, but if not the fractal geometry is a very utilitarian analogue.

The Platonic Socratic game, the theory of Ideas or Forms is a useful introduction into this way of thinking. In the game you have to chose to be either the imbiber of forms in space out there somewhere, or the conduit through which eternal forms express themselves in your immediate locality. Then you have to try to convince others that your position is the true one!

Unfortunately many do not recognise the deep purpose of this game, and some do not even realise it is a game! That is the playful manner in which Socratease conveyed fundamental insights to his neophytes!

So the comprehending enfolding turns out to identify the plane as a fundamental concept in space, and more importantly the oriented dynamic plane. That plane is conceived of as embedded or enfolded in space in wn
Himsically many orientations. But the point is: isbt conceived in the subjective sense o ly,or conceived in the objective sense? Or is there an interplay such that what is subjectively conceived is objectively realised, and Or what is objectively real is imbibed as a subjective experience?

Or is it a continuous dynamical iterative switching process?

Umfasst encompasses all these imponderable ponderables!

Nevertheless the oriented plane may have a circular,curved or straight edged boundary, or some combination. The boundary only has significance by its bounding rule.

Within that bound and the plane concept we may bind the line or arc segment, and within those we may bind the final concept, the point. Each of those is oriented within its bounding " space".

The more general notion of a surface requires a mixture if planes to describe or enfold and is necessarily dynamic.

Given that background the insight that Justus is here pleading for is the embedding of combinatorial rules for the elements within their bound spaces or rather within our perception or conception subjective or objective of such entities!

What these combinatorial rules are especially regarding lines and Figures he is making the subject of this booklet. But he is making a concise statement of a more extensive theory he has either previously published to an ignoble reception, or he was working on most of his life.

[jehovajah]

In fact boundaries are very significant for our( or at least my) mental processing of space.
Simply : a closed boundary provides me with a quantification if an extending magnitude( Ausdehnung) ,
An open boundary provides me with an orientation in and of space.
By the extension, projection and dynamics of these boundaries we develop , through quantification the logos Analogos properties associated with these bounded line knittings and figures.

It is by the sense of orientation that we divine parallelism, and the sense of quantised space that we divine greater and lesser, within and without and perfect fit or artios .

To the quanta we may add the distinctive names either of shape or of numeral . The numeral itself only gains significance as attached to a quantised figure and particularly as embedded into a sequence of order by greatness or leastness, or more generally order by pattern completing, forming or matching.

Such general patterns go beyond experience of extension into experience of intention and intensity in all sensory modalities..

All of this we contribute to and derive from boundaries in space..

So to focus only on 2 aspects like straightness( directness,trueness,goodness) and circularity( symmetry about a distinguished point) is clearly demeaning and impoverishing of our experience. Those that do so run the grave risk of becoming dead dry bones!  Yet there is an austere beauty even in such dead things!

But for me, I chose life! The dynamic spiralling out and in of natural behaviours, which indeed these dry bones serve an adequate purpose for to begin the full and fruitfully enjoyment of, with caution. The thorn bush yields indeed the most beautiful rose!

[jehovajah]

While Justus insight was new to him and his time the idea traces back to the Pythagoreans.
Quantum is Latin for a region of extending magnitude close bounded, but the Greek notion cannot be static ally so defined. Monas(accusative Monas) is defined dynamically as that entity which is placed down and called one!

That could be anything, and anything can be called one, and thus anything is Monas!

Monads are used to count and measure. Thus monads are Metra and Monas is a Metron.

We call,a system of quanta a quantity, but a sysystem of Monads we call a Mosaic after the place where these systems were found. These were found in shrines called Mousaion, houses that reflected the worship or devotion to the Musai. They were shrines of cultural significance archives of the arts and culture of a people. Those that served there were gifted with the Muses, the arts of entertainment amusement, culture, insight oratory etc. the secret knowledges of the Kosmos.

Within such shrines we're epipedoi, flat floors of mosaic patterns. On the flat walls of such institutions the epiphaneia were mosaics depicting representations of the Musai. Later these mosaics were found in rich peoples houses as epipedoi or epiphaneia, seeming decorations, but rather offerings to the goddesses of cultural pursuits.

But the Pythagoreans did not call them Mosaics , they called them Arithmoi. A monadic mosaic was an Arithmos, by which any scene was analysed into its many monads or parts, such Arithmoi comprehendingly enfolded the combinatorial doctrine of its parts. Ratio and proportion: that is Logos and Analogos are embedded in its form, from these Arithmoi Pythagoras said we develop the Geometry. Some mistranslated that as Gematria, and occulted the calculative part of their notions as Numerology or in Arabic the m'Qabbalah. The Kabbalah was demonised from calculation principles and methods to occult practices!

The enfolding of combinatorial doctrine has always been at the heart of geometry, the Algebra of calculus and calculation is at its core, and the trigonometry of geometry is essentially this combinatorial doctrine .

So what is different about Justus ideas? What new liberating insight does his presentation bring?

[jehovajah]

The topology ( literally the study of place, the rational expressions of topos ) was a study of the relationship between continuously deformable flexible space like objects. 

As a field of study it had no real provenance until the 1800"s when all other geometriesvwere in state of flux, especially traditional static geometry, sometimes confused with thevStoikeia. Differential geometry was the name given to a calculus approach to differentials, infinitesimals used to evaluate areas and lengths. However Newton had pioneered a different approach called dynamics, using his calculus called Fluxions.

A cross fertilisation eventually ensued which combined with revisionist approaches to geometry and demand from material scientists and engineers and architects led to an intense exploration of thin film materials and laminates. The work was to draw upon some of the principles in crystalmomics, but a dynamic version o the static version. This dynamic version was pioneered by adjusts Grassmann and his group of collaborators, but it was articulated by the Divines of their day, with little or no reference to their pioneering work.

Möbius and Klein took these ideas and gradually popularised them. Kleins Erlangen project popularised the new dynamic approach to geometry. Möbius explored thin film spaces, both for Barycentric properties and other properties.

The essential idea was "umfasst" , properties comprehendingly enfolded in space. In fact the Barycentric calculus showed how the boundaries of a thin film determine physical properties within the bounded space!  The bounded space was ultimately entrained to its boundaries! In fact we cannot determine the internal properties of space except by using boundaries as reference models. Thus our choice of boundaries determines what physical properties we can discern in the interior.

Why? Thisbisbthevessence of Justus insight: the combinatoricsk doctrine is enfolded within " space" by our choice of bounding lines! The best we can do is project fractal images of the boundary into the interior space of a bounded figure.

The combinatorial doctrine depends on the " fractal" projection of boundaries into the interior space.

Necessarily these projections are " rigid-like" . But though the fractal tesselations are rigid, the space they cover may be flexible. Thus the Möbius strip explores this issue: what if the space itself is twisted?

Unfortunately many still struggle with this idea. But given Justus idea of umfasst the combinatorics enfolded into the space, must twist with the space. The combinatorics must follow the boundary of the space.

If we join the boundary in one direction what happens?

Topology deals extensively with that, but my point here is does this in any way model real space? In fac it does model " molecular" ensembles very well if they join in a ring either by melding or by magneto thermo sono electro complex entities.thisbis crucial for understanding fluid dynamical flows from laminar flows to turbulence.

Space is not an emptiness. Space is something which we barely understand, but certain models we explore give us expertise in certain well defined circumstances . The Möbius strip is in fact scale free. We can posit such strips at all scales, and study the physical properties of fluid systems at different Rayleigh number( densities and viscosities) to see when this property is most evident. It is unlikely that the twisted space effect will be rare and polarisation of light deformations indicates it is very common.
<a href="http://www.youtube.com/v/M&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/M&rel=1&fs=1&hd=1</a>
<a href="http://www.youtube.com/v/M_p5NksDCCg&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/M_p5NksDCCg&rel=1&fs=1&hd=1</a>

[jehovajah]

Newton did groundbreaking research into fluid dynamics, but his model was flawed from the outset. Consequently he viewed fluids as resistive media only, rather thanmannextension of the elasticity of matter. Had he continued the practice of describing matter as corpuscular elastic materials ( no hard billiard balls as some like to say, but not Newton) then he would have perhaps derived the notion of Viscosity rather than the notion of Lubricity. In any case there were other more deeper flaws in his model which eventually became too complex even for him. He was unable to produce models and product designs capable of squaring with known data. However his point mass model seemed to work very well indeed. So well that he ventured that vorticity was not possible in space as a model of planetary motion. This was an idea expounded by Descartes whomNewton took a disliking to due to his disrespect of the ancient Masters.

Kelvin andvHelmholtz showed Newton to be wrong on this point , but there vortex kinematics is also flawed. In particular the Möbius and Klein surfaces we're not thought to be physically applicable.

We now know that they are fundamental
<a href="http://www.youtube.com/v/lQ2m&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/lQ2m&rel=1&fs=1&hd=1</a>
<a href="http://www.youtube.com/v/lQ2m-GSaX8E&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/lQ2m-GSaX8E&rel=1&fs=1&hd=1</a>

Space is not emptiness. The role ofvtwistsvin space are crucial at all scales. Trochoidal space  is the new paradigm for fluid dynamics, and it requires Justus umfasst concept for combinatorics in topological spaces.

[jehovajah]

The videos by Frederico Ardili are useful, but particularly lectures <a href="http://www.youtube.com/v/pCJNjW8kMIg&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/pCJNjW8kMIg&rel=1&fs=1&hd=1</a> , <a href="http://www.youtube.com/v/Ss9ukTUJlCo&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/Ss9ukTUJlCo&rel=1&fs=1&hd=1</a> and <a href="http://www.youtube.com/v/YpbbJC51yQk&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/YpbbJC51yQk&rel=1&fs=1&hd=1</a>.

While they are eclectic in presentation, I can discern strong echoes of the concerns and interests of Justus back in the very early 1800's.

http://m.youtu.be/Vg1e2FEpf9w

http://m.youtube.com/watch?v=Vg1e2FEpf9w
This video by Norman provides a brief historical overview of Determinants and systems of linear equations, but it completely misses the combinatorics underpinnings, especially Galois theory, early ring and group theoretical concerns especially by Abel. Gauss and Cauchy and others worked to understand when a system of Equations so called linear equations , were solveable , but the lineal equations themselves are what are usually skirted over.

These are called linear by definition because they are degree 1, however they are really Lineal because they derive from geometrical lines in the plane. Coordinate Grometry factors into this development as algebra and this isvwherebthe use of the words Algebra and Analysis become very messy.

Combinatorics thus appears as a murky muddled subject because it adopts the imposition of these names. Consequently combinatorics as a subject area was obscured in Justus Time and was subsumed into other subject areas . Meven now it is subsumed in ring and Group theoretical presentations.

Does it deserve to stand on its own as a subject?

The Verbindungslehre (combinations doctrine) is presented as a concise set of principles by Justus. In particular the connection to geometry through an arithmetical "structure" is important in defining the loose cannon word Algebra! This kind of lineal symbolic arithmetic is called Geometry by DesCartes, and employs what the Islamic scholars called Al Jabr, which literally meant "mind twisting"! This mind twisting took the principles of Arithmetic as applied to finite quantities and replaced the quantities by Symbols.

The interesting thing is that the symbols could be anything. In India , where the practice originated under the expositions of Arhyabatta, Brahmagupta and others ,letters,colours, animals etc all were used. Descartes simply applied the line segment in this way to expound his Geometrie.

Thus we can see how Algebra derives from Arithmetic, which itself derived from a close study of the combinatorial principles in space, commerce and construction. We may well regard these as principles of synthesis because it derives from the processes of synthesising products etc.

The design process of any construction necessarily includes the exploration of many permutations and choices. It is the Aesthetic choices amongst these that have led to the founding of many subject areas including principally Astrology, Geometry, Arithmetic, Numbers , Algebra, logic, later analysis, computation science, number theory etc.

Combinatorics thus is fundamental to all the scientific subject area, but it is also fundamental to areas like Mechanics, art, architecture, engineering, physics especially kinematics and chemistry. Because it is so fundamental and applicable so diversely there only ever seems to be an in subject discussion of it. Therefore to isolate it as a subject has taken time to find the right elements of study to exposit the notions through. For the Grassmanns that element was the dynamic oriented line segment  . This was the fundamental Ausdehnung on which to develop a Verbindungslehre, a combinatorial doctrine.

Once that doctrine is mastered its application is truly almost universal. The Ausdehnungs Größe derive from these knitting line segments and these Hermann fully exposits in his 1844 and 1862 writings. New also must include the insights and presentation of Robert, but it is clear that their work is itself a consolidation and extension of Justus work and Ideas on the teaching of Combinatorics.

[jehovajah]

If you are having trouble with YouTube please let me know.

For mobile devices of a certain age YouTube is not supporting the browser plug in. In that case you have to go to m.youtube.com/ to watch the video.

Combinatorics is what I call getting your hands dirty ! It is amazing how mathematicians hide this aspect of their notions away from view!

[jehovajah]

Randy Powells V9 lines everyway knitting Entity

[jehovajah]

magnetic intensities

[jehovajah]

This video shows tomBurnetts combinatorial Analysis of the V9 cyclus. To identify it as a group the modern set theoretic language is used, but in fact it is a permutation and combinatorial discipline based on modulo 9 process. Strictly it is a knitting of 9 creating elements. Modulo 9 means we can reduce any number of creating elements to a nine part fractal patterning.

The dynamic exploration is crucial. Removing the rotation dynamic makes this just a mess of numbers.

http://m.youtube.com/watch?v=
<a href="http://www.youtube.com/v/kxuU8jYkA1k&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/kxuU8jYkA1k&rel=1&fs=1&hd=1</a>

[jehovajah]

This introduction to Combinatorial exploration of 4 dimensions. Try not to get hung up on orthogonality. Yes it is " a little gem"!
http://m.youtube.com/rG6aIVGquOg
<a href="http://www.youtube.com/v/rG6aIVGquOg&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/rG6aIVGquOg&rel=1&fs=1&hd=1</a>

[jehovajah]

Combinatorial doctrine has turned my thought patterns upside down .

For example analysis, that is Auflösung as a direct transliteration from Greek to German, that is on top loosing , or dissolving from the top down , this is the precise methodology of analysis.  One starts from the top of any object and gradually " work" towards the centre of the object. We gradually dissolve the object from the top , bit by bit down to its central characteristics. We dissolve the crystal from its surface until its interior finally dissolves.

All these strong analogies are applicable to one form of analytical process or another. The combinatorial doctrine includes analysis as the precursor to get the " parts " for it to Synthesise! Thus analysis is arguably a fundamental part of combinatorial doctrine.

Synthesis then follows analysis and presents a surprising array of differing paths back to a synthesised whole! Thus naturally the combination( combining) of the parts is dependent on the permutations( per mutating) of the parts!

It becomes necessary to impose constraints or restrictions that are accessible or visible.  Hidden constraints must be exposed if the synthesised structures are going to be soundly understood, or our expertise is going to be soundly based.

Combinatorics thus deals with analysis then synthesis . The nature of these processes can be expressed by symbolic labels, a shortened rhetorical expression. Because it is an expression the label can refer to objects AND ideas.

How we synthesise objects actually informs how we synthesise ideas!

So if we move to the Idea called Geometry we can use labels to express the ideas in geometry. Then we can form an interplay between the ideas and the objects that we analyse into labelled parts and we can call the synthesis of these parts and ideas Geometry.

But we can do the same thing with any number of ideas! Combinatorics influences everything!

Frederico Ardeli stated there are 2 principles in combinatorics counting : the additive and the multiplicative.

At first these 2 principles seemed arbitrarily stated, but then I realised that combining things naturally gives rise to these " principles" . Some parts will be repetitive, some unique. The repetitive parts we can bundle , or group or ply or gather together into a synthesised object with a uniform structure. These have always been called by the Greeks multiple forms!

However this has not been communicated clearly and often this word polleaplassoi ( multiple homogenous forms) has been confusingly translated as multiplication!  The multiplication tables or number bond tables are incorrectly named. These are factorisation tables, introducing the analytical notion of factors! 

Indeed the factorisation tables organise the multiple forms sequentially, but we are done a mis service when they are presented as multiplication tables.

Multiplication is a much more general idea than we find in factorisation tables. In fact multiplication is a combinatorial principle based on the synthetic process and a kin to additive synthesis. As a general principle we are not tied down to the concepts bound in number bonds. Rather the number bond concepte rise naturally out of the combinatorial principles