V9

[jehovajah]

Toward

The Physical Crystalnomics

And

The Geometrical Combinatorics

             _____________________





First Booklet.              1

[jehovajah]

For combinatorics put the words Doctrine of Combination.

Here Justus is drawing attention to the physical and the geometrical as the real basis and application of his alternative views!

[jehovajah]

Toward the physical Crystalnomics and  the geometrical doctrine of Combination 1829
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It gives a method of lines and surfaces  
       thusly  
        knitting to step with one another ,
       that  
       themselves thereout from self  
        allow to develop
       a crowd of the simplest and most regularly ruled shapes
       ,
which,
how out of a zauberlaterne( https://de.m.wikipedia.org/wiki/Laterna_magica)
 Herebefore held,
before the gaze  of the spirit  
to dive onto ,
with its clear eyes against the heavens in order to display,
and the darksome depths of the earth with its light to outwardly brighten
and throughseeable make.

[jehovajah]

Toward the physical Crystalnomics and  the geometrical doctrine of Combination 1829
-----------------------------------------------------------------------------
                                              3

Two or more unbounded direct lines in a plane can distinguish themselves thusly wholly in "alongside looking" their position, as also in "alongside looking " their directing, and there either the one , yet still the other, for itself tracked, is
One magnitude,  
thusly is therethrough  the possible quality of a" from everything" Magnitude Doctrine given, independent space like or geometrical combinatorial doctrine.

The combination roots and raises up Specifically  out of a knitting as unlike skilled entities
thusly,  how ,  like-skilled things here before, the magnitude goes.out of a synthesis.

The unshareable product of two direct lines from unlike directing is the intersection  point, from which point out of one line each was bounding the other line  in two,  concording to a side;
concording to the unbounded "running into against set " parts each the other "was causing to fall and split apart", which I name rays

Footnote
The label of Mathematic, out of  which the labels of magnitude and combination  come to be developed,
thusly how  they here given are,
follow, an offprint (it is by me) , "On the pure Tally  marks doctrine" , which was serving toward the programme curriculum for Michaelmas 1827, by considering the high school to Stettin as a "by considering position"  ,
how the whole of them
not detailedly, yet still as near as can happen,.
come to be developed

 This offprint has each time still not in the book trade form appeared, and I think therehere its essential contents guide in one of the later booklets to take on. Here I  satisfy myself to remark, that there the parallelism of the combination and the doctrine of Tallymarks detailedly  is out  loud declared.

                                                                                                                                                1*

[jehovajah]

Toward the physical Crystalnomics and  the geometrical doctrine of Combination 1829
-----------------------------------------------------------------------------
                                              3

Two or more unbounded direct lines in a plane can distinguish themselves thusly wholly in "alongside looking" their position, as also in "alongside looking " their directing, and there either the one , yet still the other, for itself tracked,
 is One magnitude,  
thusly is therethrough  given the possible quality of a" from everything" Magnitude Doctrine , independent space-like or geometrical combinatorial doctrine.

The combination roots and raises up Specifically  out of a knitting as unlike skilled entities
thusly,  how ,  like-skilled things here before, the magnitude goes out of a synthesis.

The unshareable product of two direct lines from unlike directing is the intersection  point, from which point out of one line each was bounding the other line  in two,  concording to a side;
concording to the unbounded "running into against set " parts each the other "was causing to fall and split apart", which I name rays

Footnote
The label of Mathematic, out of  which the labels of magnitude and combination  come to be developed,
thusly how  they here given are,
follow, an offprint (it is by me) , "On the pure Tally  marks doctrine" , which was serving toward the programme curriculum for Michaelmas 1827, by considering the high school to Stettin as a "by considering position"  ,
how the whole of them
not detailedly, yet still as near as can happen,.
come to be developed

 This offprint has each time still not in the book trade form appeared, and I think therehere its essential contents guide in one of the later booklets to take on. Here I  satisfy myself to remark, that there the parallelism of the combination and the doctrine of Tallymarks detailedly  is out  loud declared.

                                                                                                                                                1*

Sorry ! I pressed the quote button without realising!
However I did want to comment .
Justus begins with 2 unbounded intersecting direct lines under consideration.
Really we should and I do question this as a starting point. What is a line? What is a point? Or more fundamentally : what us a position? What is a directing?

You see by asking those questions that Justus relies on a common but naive geometrical understanding embedded within everyday language and experience.
Moreover he selects how he is going to use that general understanding.
He chooses how he will decompose the object under consideration and he chooses the names for parts . Here he defines The Ray.

Those who have read my thread and blog on the Fractal Foundations of mathematics( google jehovajah Shunya ) will know that the Greeks started not in the artistic or architectural schools , but rather the Pythagoreans based their philosophical exploration on the artisan the labourer who constructs and synthesises from the raw materials of our environment. And the fundamental materials and elements are firstly extracted from the environment!

Analysis refers to this human interaction and exploration of life living and constructing!  

Yes construction is a fundamental approach of analysts. Not only do the break down structures into elements they also rebuild elements into structures to ensure understanding, and to consolidate learning.

Euclids Stoikeia in that regard starts with the Synthesis of  spatial " actions" ! In that case a pointis any space like object we can not break down any further!
But that is not a "point", you might think. And you would be right to the extent that Seemeioon does not mean a mathematical- geometrical point! It means an indicator! This indicator is simply an object that we perceive as saying : no more dividing or analysis possible!!

At that moment we should realise it is time to synthesise.                                                                
What actions and instruments the pythagoreans acted with are well known the tools of the artisan, the mason the carpenter and joiner . Those craftsmen and crafts women, embroiderers weavers and spinners house builders are where we should start.
A point is a practicaln, pragmatic starting place for synthesis . A line is a practical arrangement and ordering of such " points" , such points are actively arranged , dragged, drawn, stretched into position. Innsuchnannaction two such "lines " do NOT share a point !

If they cross they do not cross through the same point! On the other hand a common point can be extended into both "directions " of such " lines" . That common point thus is a significant starting point. It is the Origin, the sprouting point for what Justus decomposes as rays.

These actions or active principles were well understood by the great Philosophers, and were embedded in Mechanics and Alchermy and Phusis, the system of opposites that underlies modern Physics.

And so Fractal geometry is not a creation of Benoit Mandelbrot, but a realisation by him that computational methods, methods of counting: these are not trivially about Arabic numeral signs or Indian Algebraic methods of symbolic accounting or Pythagorean forms and patterns of Arithmoi! They are about our human Logos Analogos sunthemata Sumbola summetria Theurgical response to every experience in our inner and outer worlds.

Drawing attention to clouds and mountains excited the dulled imaginations.

Geometric Algebra is the Grassmann synthesis method drawn from an analysis of the best thinking and philosophy of their times. It is in a " mathematical" discipliine format because it follows the Pythagorean synthetical method laid out in the Stoikeia of Euclid.

[jehovajah]

Norman Wildberger taught me about the affine, the projective and the universal hyperbolic geometry. But now Justus sets out a view of combination that precedes these geometries.

Firstly we have to realise that Justus was responding to a historical spirit of the times: revolutionary thinking and practice ; freedom, brotherly collaboration; and equal opportunity  to contribute at all levels.  The Humboldt reforms in Prussia were instigated by the emperor to make the empire self sufficient educationally.
Justus thus reviewed established " truths" like the Stoikeia and othervEuclidean works, the philosophy of the Aristotelians, the Arabs and the orient were all subject to intellectual scrutiny.  In part this was the general drift of the great French thinkers of the day. But Humboldt was a Prussian who clearly saw and philosophised and agitated on the condition of the Holy Roman Empire under Prussia. He called for a revolution in the educational system to develop self actualising people.

So Logic , Aristotelian , was believed to be the foundational organising principle. Teachers of logic. Had set out the major principles of the topic and it was generally regarded as reliable sound and the source of all intelligible reasoning..

Mathrmatics was considered as one of its chief products.

So where do tally marks  come from?
Norman gives some fanciful ideas about a recording system based on some symbolic marks. Justus believed he could account for them f.rom the geometrical elements of point line and plane.
  The logical structure of Aritmetic was Justus aim .

The general priiciple of  combining elements and naming the results was his methodology .

[jehovajah]

Justus believed logic was the essence of the analyis of human understanding of god and his creation. Logos or the rational entity in each man by the Geist or spirit in nature and natural objects was the foundation of understanding correctly. This was the common opinion of his day
Thus he starts with the logical point line , position and directing as his elements for synthesis. He then relates the logical lines to direct pencil , ink or sculpted marks made by an artist, or architect.
 The intersection of these artists lines upon analysis reveals the intersection point as a logical entity..,this point is Not shared. It belongs to one line as referenced by the other.

Thus we already have a dual ambiguity.

Do we worry about it? Pragmatically no, but logically it represents a logical ambiguity which logic cannot resolve! However Justus does not realise that because in his mind these logical entities were the same as the elements and terminology in a subject called Geometry.  

These broad subject boundaries of Logic, Geometry, Arithmetic were seemingly well bounded and definite, but in fact these boundaries melt away on close scrutiny.

Thus we usually accept the logical point , line or place/ position, as a metaphor for the geometrical point line or place /position. But vice versa we also accept the geometrical sense of point , line , plane/place/ position as a metaphor or simile for the spiritual/ logical/ mental status in a processing activity.

However , when we look with our senses in space( das Raum) we see nether point or line or plain , we only experience place!!!!( topos)

Consequently to assert that millennia of analytical activity of the ancients has established logic as foundational to our experience of space is clearly wrong!

The Pythagoreans understood this point , as did their renowned students, even up to Newton and Steiner, but it is perhaps Socrates andvPlato who famously devised the game called the theory or philosophy of Form/ Idea.

In this game you were asked to choose which to accept: forms are perfect in some other reality which we dimly perceive through imperfect forms in our reality; or we construct perfect forms in our spiritual experience from constant contact and empirical experience with forms in our everyday reality.

The former are often referred to as Ideas or Ideal, the latter simply as Forms .

In translation and conciseness we often lose the visual nature of these theoretical / philosophical positions. 

The game is to try to develop a lifestyle bases solely on one of thesev2 Assumptions! When you play the game you very quickly forget it is a game!! What you also forget is you have a choice at all times to adopt the other viewpoint, or even to start again as the other player.

But most importantly players never realise that they have more than just the visual sense to account for! Form or idea is not absolutely a visual construct!

Proprioception plays a fundamental part in our formulation and staging of experience of reality, and thus of form or idea.

To think that modern science and philosophy has discovered this fact is to be naive. The sophistication of thevPlatonic / Socratic theory of form/ idea belies that assumption.

Justusvthus chose the game position where analyses had constructed the forms and elements, not discovered them or uncovered them. Thus his whole point of view is synthetical: he attempts to show how we can construct geometry and arithmetic from these fundamental logical conclusions arrived at by analysis. In this he follows thevPythagorean school of thought. He plays the game as a Pythagorean scholar or Mathematikos !

We are at liberty to play it however we choose. However it is now clear that the Pythagoreannapproach has been extremely technologically successful, bringing practical magic into our everyday lives.
Evenso Hermann corrected hiscFathers approach in a reworking of Justus Ausdehnungs Lehre . It isvthisvreworking and extension that I have been studying for a while.

That is why I am translating one of Justus seminal works here to better understand Hermann Grassmanns reworking of it.

There are in point of fact versions of adjusts work developed by his other son Robert, but Roberts work relies heavily on Hermanns reworking. Robert in fact redacted Hermanns 1844 work to make it more accessible to Mathematicians in1861. Hermann was never entirely happy with hisc1861 version but that's another story.

[jehovajah]

Hermannmactually developed a method that took the logical ambiguities into a worse state!! They became the worst possible representations, but a process ofvHegels called the dialectic method resolved the difficultiesvandvambiguities into crystal clear domains of application.
Thusvhermann had to replace Aristotelian logic by Hegelian dialectic to make his fathers insights work in all the areas Justus believed it could.

[jehovajah]

Toward the physical Crystalnomics and  the geometrical doctrine of Combination 1829
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                                                                   ---- 4-----

Out of the combination of the rays roots and rises up the corner angle ,
Out of the combination of the corner angle the adjacent corner angle , the apex corner angle etc. As the product of the combination of three direct lines from differing directing can the three vertex come to be "alongside looked at".
Four of the like direct lines gives a concording account Tally  of four vertices etc . If one plainly thus makes the differing quality  of the space-like  positions by considering like directing at the foundational position of combinatorial developing , and then the product of these combinations with the circle everyway bound, thus reaches one to the indicated aggregate from figures, of which magnitudes everyway held in space entity the Geometry
investigates,
and alongside which the Geometry
their propositions develops.

One goes now on the betrodden path further and everyway searches it, planes with one another to knit together, in the aspect, the differing body-like Forms to develop, which through their side against side by bordering  come to be built a representation of , thusly therethrough comes one very directly into the case,  no more with Clarity   to be able to overlook all rooted and risen up  everyway held in space entities !
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Footnote
A developing of these space-like combinations in the planes have I everyway searched to give in a little script, which in year 1817 has come hereout under the title: " Space Doctrine for public primary schools, 1st part: plane space-like knitting doctrine ",  
the content list is there for the first elementary instruction in the Geometry, whereto it itself presentationally centralises, (to come to be edited)
and I have had to possess in the imagination thereby children from 7until 9 years and teachers who do not have a great foreknowledge,  


the "therethrough necessary" "come to be  methodica"l out of one another settings,  and the chosen language form are pleasing the originating motivation
that the contents label only from the paedegogical, but never from the expertise like Side has come to be tracked,

 I am conscious, unattended to by me the expertise-like process
 was at least the successor of the propositions and forcefully affected the personal exercise,
to have thereout through that pedagogical  hindsight nothing to pretend ,
what  also, by considering so simple a thing,  completely nothing was difficult

[jehovajah]

Unexpected combinatorial conundrums in my everyday life, finances and health have preoccupied me over the past few months . But I will be continuing this translation as combinations of circumstances permit.
Meanwhile Prof Wildberger is an excellent "fix" if you need stimulation about these fundamental issues in computing.

For those of you who have read thefractal Foundations thread, I wonder if you recall Manipumè ?
Hermann thought that the thought patterns would eventually become the core subject of study for philosophers andvMathematicians alike, but this is obscured by the computer science label we put on this foundational endeavour .

[jehovajah]

One of the fundamental aspects of Natural Philosophy in the early 1800's was the observation of measurable quantities and their relationship to observable matter.
Counting and measuring is part of the fundamental Logos Analogos response.. The Pythagorean school pointed out that counting and measuring was fundamentally a spaciometric response. Investigating all manner of spatial forms they demonstrated the applicability of the Notion of Form , the Denkenformen to a visual mosaic patterning of space.. Then they went further and showed the essential pieces of a visible form as described by artists also were a complex mosaic that held within their structural spatial arrangements Logos Analogos representations.

It is this kind of reductionism to mosaical essence that guided philosophers in the 1800's
The observation that many Logos Analogos pairs were constant in all kinds of varying situations drew out of people like Lavoisier and John Dalton and Mendeleyev the conviction that God had set a Law in nature regarding combination of the elements of these " mosaics".

Thus Justus use of the term Hieroglyphs in relation to discovering secret laws of nature that would profit mankind was not a whimsical fancy, but a serious proposition , postulated before his fellow natural philosophical researchers on the basis of the Combinatorial patterning he was observing.
Like the three chemical pioneers mentioned above, the Logos Analogos sensibility that trained Pythagoreans have guide and shape thinking: both logic ( induction, deduction and dialectic) and observation could be systematically structured and investigated by these patterns in the mosaic of space or the analysis of counted or measured Data.

When such a ' Law' was surmised (hypothesised although. Newton hated unfounded Hypothesis) it was put forward as a theory, that is as an idea from the Gods, or Musai,

Today's elaborate theories have cut themselves off from these Natural Philosophic ideas and replaced them by a divine Madness called Mathematics! Logos Analogos is not Mathematics. It is something altogether more straight forward being simply counting and measuring and comparing.

It's simplicity is seen in many cultures artistic endeavours both in dance and artisanal wotk( architecture pottery basket weaving etc) .the music of the dance brings us right back to the inspiration of the Musai, the whole basis of the practices of the Pythgorean school .
One has to realise that Aristotle subtly distorted our understanding of the Pythagorean school of thought, but Plato also was a novice in this school. Euclid and Euoxus were masters. Recognised as Mathematikos by the Pythagorean school. Arisyotle was not. Rather he set up his own school with the help of the Greek And Macedonian emperor Phillip and his 2 sons Phillip and Alexander.

The categorisation that marked out Aristotles compelling contribution was in fact a symptom of his autistic traits. The Pythagorean school had a much looser categorisation system as found in the Works of Euclid, and thus more Natural or Muse led. 
Kant represented the ultimate extemporisation of the Aristotelian system. Hegel however went a great way toward combining the Pythagorean system with the Kantian or Aristotelian categorisation .

The philosophy of Hegel is the one that Hermann chose to utilise while Justus his Father is definitely Kantian even if he disagrees with Kants assertions about the origin of Mathematics.

The foundations of Mathematics is this Combinatorial apprehension, but then Mathematics is shown to be a poor implementation of this combinat orial basis! There is so much more that this combinatorial doctrine can and has given rise to .

[jehovajah]

Most of us have a rudimentary exposure to a subject called Geometry. Evenso we think we know what it is.

How surprising then when you find out that our ancestors were not so sure!!!

I find that the Pythagorean school of thought defined geometry in terms of something else. They represent a European school of a far more ancient Indian and Egyptian and  Babylonian traditional wisdom  the many stories of its founder are clearly philosophical counterparts to the Christian stories about Jesus. And research shows that the Jesus stories have a surprising Asian and Indian traditional scholarly documentation outside the Gospels.

So the Pythagoreans searched depply into how you and I think and perceive . They conceived of the notion of magnitude. And the notion was based on objects . Then they analysed these objects into the barest essentials, the minimal that one could perceive and recognise distinction!!!
Visually that was a dragged line that in any number of ways marked a surface: gramme in Greek.
Kinaesthetically it was a shaped mound or block , auditorially it was a duration of a sound, the frequency of a sound , the intensity of a sound.

The richness of their sustained application of thought to the analysis of our perceptions is hard to describe. And yet we consign them to a disrespected backwater subject called Geometry.

The combinatorial insights Justus is discussing here represent a growing awareness and appreciation of the Pythagorean teachings and investigations, but of course they are not directly referenced as the sources. At the time it seemed hardly necessary. The few scholars knew whereof the studied and spoke, but modern subject boundaries have obscured this direct link and followed the Aristotelian model of patronage . An Aristotelian scholar had to survive on patronage. Thus any claim that bigged him up over his rivals was economically justifiable. In this way devoted students often severed the link to the past heroes who inspired their revered teacher.

This video powerfully illustrates how poor we have made this rich legacy from the Pythagorean school

http://m.youtube.com/YUFs_1vKYlY
https://m.youtube.com/watch?v=YUFs_1vKYlY
Enjoy and ponder at the combinatorial simplicity .

[jehovajah]

Toward the physical Crystalnomics and  the geometrical doctrine of Combination 1829
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                                                                   ---- 5-----

 This difficult quality becomes pointedly  every way minimised , if one at the place  of the planes only direct lines has in the space,  infinitely times more  if these all  go through one and the same point .
But it is the position of a plane and all parallel planes with it  through a direct line given which  onto the same planes is vertically dropped,
And one can therehere  place in the place of  the combination  of the planes the combination of direct lines, which all drawn through one point come to be, with fore part to introduce ,  in which one every time through whimsical points, of the same , thereupon vertically dropped planes lays, and the rooting and rising up shape then closely investigates.

The above comes to be to reach from afar, itself to make an approximate label  from the  indicated parts of the Geometrical Combinatorial doctrine , which from all magnitude concordings is independent, and which I would like to name of the half part  the unmixed geometrical combination Doctrine .
From this I think in successive booklets to give to the readers an outline 

Under the unending like crowd of possible account Tallies, and the possible positions of the from common points out going radii or rays, which to their verticallly dropped onto planes act as agents, is the simplest case, for which rays a complete boundaried  shape roots and rises up, the that indicated one, by considering which  three not lying in one plane direct lines through the point alongside taken come to be, and to this case seems already everything to grab around., what to us the Nature besides such forms is serving , which from the plane surfaces are boundaried, even if one  the entity through a thusly like closely to concording  combination away thereout of derived shapes develops .

[jehovajah]

Slow as i am at the moment These sculptures show some procedural significance.
Technically they are iterations of the complex exponential combinatorial Ausdehnungs Grosse, or a Fourier Transform for 9 frequencies if you like.
Pictorially the model a rotational wave form in 3d

[jehovajah]

Norman Wildberger is currently posting in his Universal Hyperbolic Geometry series no.s 47 onwards on the platonic solids. The content of Stoikeia book 13.

As a culmination of the philosophical preparation of the previous 12 books it seems remarkably bereft of non trivial results!

The occurrence of symmetry, rotation and permutation, duality , regularity and  expression of only 5 of the possible flat shapes is a salutary lesson: do not think more of yourself than you ought!!!
A deeper thinker might realise that these simple and obvious observations must somehow be fundamental to the whole philosophy. Thus if the result is these  properties then they "must" be at the foundation of the topics. Thusly one is drawn back to the examination of foundations and to searching out how these properties are layered into those foundations.

I have spent a long time exploring the hypothesis that a fractal  topology underlies all,of so called Mathematics. Thus these properties become founding principles at every level and at every scale. .

For this to be the case no philosopher can fail to point out that this is not a justification from nature, but an imposition of a conclusive model onto natural situations and circumstances AS we begin to think, define and describe them!

Furthermore in Justus time, mans psychological and philosophical imposition of such categorical taxonomies was well understood and formed the study of reason and reasoning called latterly Logic, but then
Rhetoric!!!

Certain idealists could not accept that this was the product of the human mind and human logic! They sought to assert that the origin of this Divine simplicity was God. That this understanding was a divine transcendent revelation woven into the fabric of reality by God .

Thus the divide was between those who sought evidence of the Divine in mathematics and also in all human expertises, and those who expressed a logical designed architecture to applying human ingenuity , that is by constructing the necessary tools and models at every level to gain expertise and control over ever finer details. The construction principle was by Induction, not by deduction only.

The culmination of this kind of insight was exemplified by Hegel, in contra distinction and evolution of Kant and Kantian Idealisms.

The foundation then is seemingly based on trivia, but profoundly determining the structure of the whole edifice! And the insights Justus had and was preoccupied with was Combinatorial structures that naturally arise out of primitive elements, and the natural symbolic labelling that we linguistically choose to name and keep tabs on ideas, states of combination, descriptions of specific combinations etc.

At this level, the labels we choose naturally help or hinder combinatorial descriptions, structurings or prescription of  sequences of change . The other revelatory insight was the classification of crystals!!

These ideas did not easily sit together, but naturally supported one another and guided Justus into many natural insights into the structure and arrangement of materials physically, chemically and biologically!

It seems strange to realise how radically different this concoction was, and yet how essential.

Justus did not work in isolation, but with others like Abel,Lee, and definitely Jakob Steiner. The synthetic combinatorial nature of mans efforts and constructions was their hope of better founding human knowledge on Natural processes.especially as so clearly delineated in crystal forms.

The foundations of group theory and ring theory draws on these pioneering efforts and researches as well a Galois reasoning from a combinatorial standpoint.

So what I find here is a 19 th century conception of the Natural Order of things, but with a divine fanaticism driven by irrefutable crystalline forms.

But now the labelling takes on a form. To be useful and to describe details It had to be arrsnged and arrayed.

It is this array of symbols on the page that captures combination, sequence and combinatorial status!

This array becomes the symbolic representation that is studied, and from which patterns are deduces and induced.

The most naive array is a line sketch of the relationships. For the dynamic changing situation a sequence of sketches. But quickly symbolic arrays show their worth and compactness.

It is this compact array of symbols that has shown itsel to be a worthy companion to the naive line sketches, and to be in many ways mentally prior to the line sketch.

And yet it is limited by its arrangement and compactness. To avoid the misleading shortcomings of these naive symbolic representations one has to accept that a naive sketch or array may represent many points of view of the same real relationship, or indeed one array may represent many real relationships!!

It is therefore the accompanying set of constraints and test conditions that are vital to the utility of the compact arrays.

Devising such a set of constraints and conditions and designing a suitable typeface of symbols, and associating it to a style or stylised sketch of the real relationship is what fascinated Justus! But he went one step further, he grounded his systematic method in the eternal irrefutable forms of Natural crystals .

 For always a crystall has had the same shape and the same dynamic origin. The crystals are so consistent they are used to identify materials. It is this grounding identity that founds Justus combinatorial system. And it is this symbolic pattern that allows certain similar crystalline structures of differing materials to be gathered or typed as the same or related.

It is on this foundation that years of study, hunches and insights have lead Justus to glimpse an undeniable true array. The invariant array .

The invariant array captures stillness and dynamic symmetrical rotational or cyclic permutability. Of all the things studied we can only certainly know that which is dynamically invariant.

And yet Pythagoras and the pythagorean school  explored this symbolic array so thoroughly that they were able to state : the Arithnoi precede all Geometria!
The mosaic is prior to all counting and measuring.