V9

[jehovajah]

How do I analyse an object or an idea in order to get the central characteristics of it, and secondarily to facilitate the Synthesis of the analytical results?
Secondly
How do I synthesise the analytical labels, data and notions to recreate the whole original, and perhaps to improve on the original design?

These are the fundamental questions of Combinatorics..

http://m.youtube.com/MInlPBLOaro

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

This is what fractal combinatorics can achieve!

[jehovajah]

At the time when Justus began to work on his ideas about combinatorics, the academic world( also known asnthe theological seminaries) were a twitter with excitement about the hieroglyphs of Egypt. Napoleons expeditions had brought back " ancient" wisdom encoded in strange symbols that experts could say nothing about, who could crack the cide and reveal the secrets if Egypt even the philosophers stone fir alchemists or the elixirs of eternal youth!

This public interest and buzz emboldened and inspired many to attempt to decipher thevHieroglyphs, including Goethe to name one famous interpreter. But alsomHegel and other dour philosophers were enlivened by the notion that secrets could be encoded away in figures and symbols and pictures arranged in patterns. They made the mental leap of analogy to the waybNature encoded her secrets!

So here Justus makes the same analogy, but nit by hubris or hyperbolic rhetoric, rather because of workmanlike results he was getting from pursuing and developing a combinatorial doctrine!

The results were all the more remarkable in that he came to his present course of study and intrigue by a series of lucky connections!. Each circumstance drove him on to an ever widening research endeavour bringing him at the last to the consideration of Crystallography.

Now if ever there was a subject of dry bones needing the reviving of the power of God, as in the valley of dry bones innEzekiel, crystallography was it!. Yet in this climate of deciphering hieroglyphics, the deciphering of natures crystalline firms seemed almost a divine occupation! Add to that the in time appearance of an authorative publication on Minerals pertinent to his study and you can see why Justus utilised the inmVogue social effort to decipher the hieroglyphs as a powerful call to his colleagues to help decipher the messages of nature in .

Unlike Goethe justusvwas not claiming to understand the hieroglyphs, but rather to point out that he had hit upon an idea which was being applied to crystals in a way that might also crack the code for hieroglyphs!

This code breaking method was combinatorics " born again" infused by the spirit of Nature to a new and fruit bearing life! Combinatorial principles and ideas and experiments or experiences informed and developed an expertise that analysed the outer shapes to predict the internal characteristics of a crystal.

This method was bit by bit yielding astonishing results! .

Central to learning any language is its presumed structure. There may be no written grammar, but everyone who speaks recognises distinct patterns. Patterns are the bread and butter of combinatorics. By understanding these patterns the core or central patterns emerge around which the sub patterns lie as interesting anomalies, particular transformations, regular declensions etc, and the conjunctives that join these parts together in patterns become identifiable.  In addition, since actions are mostly observed in space, the geometrical and dynamic verbs structured by prepositions have a precise geometrical instancing.

It is this combinatorial structuring that is so fundamental that few ever realise it exists! It is this methodical exploration of these structures that Justus will be communicating in this booklet.

[jehovajah]

There is analogy and there is Analogos!

By this I mean that booksv5 and 6 of the Stoikeia, reputedly the work ofvEudoxus, deals specifically with analogy as Analogos . And by this Analogos of aliens and combinations of lines as figures and combinations of lines as knittings Eudoxus lays out combinatorics of the greatest sophistication. Indeed we have hardly advanced beyond it in our declensions of ring and group theory and so called Algebras.

Found within its definitions propositions and exercises are the Core ( eigenthum) ideas, notions and relations Of Combinatorics.
I t is falsely said, in my opinion, that the attempted coup among the Pythagoreans was due to unity being revealed as insufficient to denote all measurements, that upset tha apple cart. ButbEudoxus shows the nonsense of that anecdotal myth. In book 7 the formal introduction of the Proto Arithmoi reveals a deep classification and apprehension of approximate tallying, tallies that do not articulate, that is join up Combinatorially in neat bundles with precision fit( 'rt*) a rhythmical flowing together. Except that in their " kind" they are as fluid as the " exact" tallies( artios) and thus an Analogos exists between kinds of tallies, and such kinds were called Protoi, after protos being the first position. Protos as a position in Eudoxus Analogos scheme for combinatorial analysis replaces the specific idea of  "1". The idea of 1 is cardinal( although that should properly be "ranked" lower than ordinal!') whereas protos is an Ordinal designation: first.

We seevthisvreflected in the fundamental definition(!) of Monas in book 7 the definition is Ordinal and dynamic. It is the most intriguing definition I have studied( google jehovajah Wordpress Monas) and it generalises the specific idea "1" .

The definition combines tallying and measuring in one sentence relating counting to the activity of selecting out by means of a Metron laid down onto the thing to be quantified, and declaring that action and that Metron in that action to be " 1".

To say that this was not understood in the times of Eudoxus is to be Naive. Rather Europe in particular has had to climb back out of a period Known as the Dark Ages when much rational insight was lost to its scholars and theologians, and dangerous mythologies were purveyed for political and monetary gain. The Kings ofvEurope and the Popes were In the Thrall  Mammon!

Thus from book 5 onward the combinatorial doctrine of the Pythagoreans is expressed , and the philosophy of the Pythagoreans is contained within the Mosaics of these combinatorial patterns and the Logos Analogos scheme of encoding relationships and constraints on their designs and discoveries.

The segmented line as a fundamental encoding and decoding structure is introduced in book 2 where the encoding practices and rules are simply laid out in the definition of the Gnomon and the plane figure! Here we see the fundamental of Justus lineal combinatorics thoroughly admired as a lineal Algebra. But it was never thusly presented when I was taught " Geometry" as a 9 to 12 year old. And while the geometrical representation of polynomials of degree 2 was given as an exercise in consolidation, nowhere was it set into a larger combinatorial doctrine driving all computation!

Thus when i was plunged into axiomatic theory at university level i was ill prepared, Ill at ease and clueless as to what the whole rag bag collection of tricks I had been taught to memorise really represented, and how I knowing these things would be able to cope with the new challenge of a strange and bewildering rambling, and the inwardly vying  approaches of conflicting boundaries within the subject called Mathematics.

As Hermann lamented in 1861: why is this combinatorial doctrine taught in primary education but not in higher education!

He was lucky because Justus put it as core to the mathematical curriculum of Stettin, where he was educated, but very few others adopted it as their approach at prinpmary level. Nobody until Clebsch adopted it at higher level. Even then it was misconstrued by Academia who could not believe that the low caste Grassmanns could have pioneered such a fundamental revision of all they thought they knew!

We shall see how this panned out for the Grassmanns, and I think there is a great story to bevtoldvabout how the Key players of the time manipulated the patronage opportunities for their own ends, often burying new ideas and claiming them as their own.

[jehovajah]

One of the combinatorial distinctions arises when you conceive of rationals as fractions. That is to say the ratio of metronised quantities in the logos analogos scheme of Eudoxus.

One may work the scheme using whole objects or fractions of the whole, where it becomes apparent that homogeneity, one of the fundamental strictures of the method was very important . Unless the units/ Metron is the same the counted answer is scaled by a factor of commenurability.

Studying this behaviour lead to the experience of reciprocals. Reciprocals are known to be factors tha give unity. It is this relationship that Newton utilises in his method ofvFluxions by Moments. Newton used this to establish his Fluxions method. He did not set " infinitesimals" to 0 but rather observed that they vanish into their initiating value as the products of reciprocals behaved. Most of Newtons clearing of change effects was due to considering befor and after the desired position of the event. He felt it was an an unsafe method to either set infinitesimals  to 0 or to ignore them, but approximating to them was valid and valuable. More importantly dividing by a reciprocal normalised this variation to variations near 1.

These are basic analogical combinations which respect the combinatorial rules in Geometry. By removing differential from geometry, the insights become misleading or flawed.

The idea of a limit acknowledges the behaviours of reciprocating products but fails to harness the power of reciprocals.

[jehovajah]

Combinatorics overview.

http://m.youtube.com/7kcO8EYY7xs

<a href="http://www.youtube.com/v/7kcO8EYY7xs&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/7kcO8EYY7xs&rel=1&fs=1&hd=1</a>

Note how combinatorics is now a branch of modern mathematics. It was not considered a part of mathematics ib Justus Time!


The basis of the subject is not explained by Norman, because he does not apprehend one. However Justus contention is there is one and it is these line knittings and plain figure constructions.

The translation of the Foreword to the Nature patronage booklet by Justus continues carefulli.

[jehovajah]

Getting closer to redacting draught translation of Justus' Foreword .

If you are waiting to read the result, and you are a German speaker or translator can I implore you to assist the thread by helping out with your own or collaborative translations?

This particular tract is fundamental in my opinion to understanding Hermanns version, but also early ring and Group theory of the Abelian type, with insight into Eulers groundbreaking contributions to combinatorics and synthesis and Analysis of space and kinematics and phoronomics( stereography) of space as well as crystallography and early magnetic theory prior to Maxwell and Helmholtz.

It is an open thread on the combinatorics of the V9 structure so feel free to contribute your views .

[jehovajah]

Toward the physical Crystalnomics and  the geometrical doctrine of Combination 1829
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In the developing of the combinations concording to forms and in the geometrical combination doctrine are now the elements to the creating whole of the supremely manifold shapes given , 

and it was in practice,
 related to the S.4 conducted alongside you  Circumstance , 
 related to the combinations of the planes with the inner representation building  power
 the difficult quality to follow, 
of which planes
upon the meditations
 was guiding to the plane( situated) lines to substitute as their agency.

Another circumstance guided from here out furthering. I had specifically to the geometrical combinatorial doctrine a singular exercise sent ahead, around the pupil Scholars in space to orient, and to this end the main directions concording to the Dimensions of space, the between directings between each two, and the "Middle" directings between each three main directions laid down.

The aim was immediately for the space-like extending magnitude, direction and kinematic motion, a firm underpinning basis to achieve,  whereupon these ( as projection axes) were relating , and these can become everyway stood- like besigned .

In the entity
 I was tracking these directions as agent from the planes, 
and these was bringing upon "like"  away displacing from the Middle point, found I thuslylike,
That the 3 perfectly concording shapes of the regular system, which I  knew out of Hauy's Physics, were created whole through the 3 systems from directions 

This then was conducting wider to the preoccupation  with the Crystallography, 
in which I was hoping thuslylike     to find, and work-like was finding,
the necessary  pointing fingers of Nature to the restricting, 
 Related to which fingers 
sprawling  combinations
concording themselves to all sides in the immeasurable space 
.

Here considering must I once again bring it to your attention as a fortunate circumstance ,
That plainly sometime around the time, as this remarking was becoming made, the first Volume of Moh's fundaments of mineralogy   appears,  which striking presentation I am  very much everyway obligated to thank

There against, must I sorrowfully bemoan,
That I do not have by me the off print of the royal  Prussian academy , in which  the  works of professor Weiss are enholden,  in order to be able to everyway fashion.

In the entity I now from the combinatorial Standpoint everyway followed out my centralising property alternative view, 
with the  prepared to order Means
 the those indicated output results  were uncovering themselves bit by bit  , from which an overview  shall come to be shared here.

Out of The said thing one  will easily extract , 
what is   to be expected in this writing 
specifically no new inner decorating, observings and Measurings in the Mineralogy , 
whereto, alongside the Means and the Opportunity
it was lacking to me
Importantly,  nothing what out of the expert experience is inwardly (unconsciously)  loaned, 
and plainly thusly,
 least a critique of the until here slivered off achievements.

There against I wish my reader therethrough to over pull,
partly from the empowering quality of a 
Either not relate tracked  
or of until here not "full bake" edited 
branch of mathematic, 
partly from the great importance and significance of the same,  
That would come to be shown 
their applying to the construction  and clarification of a complete  array  from natural appearance entities, 
and how they consciously matched to the keys to the hieroglyphs, 
the same enhold, 


Because every external shape, 
which to us the Nature proffers', 
can and must as the Expression  of an everyway hidden interior, 
as if a hieroglyph 
come to be tracked , 
which to us the "into running against" holds the nature, 
to which the into cyphering burdens us
 it burdens us as a personal exercise from way off .

And  this encyphering is consciously certain from still great interest for the human kind, and is everyway communicating upon  every case  a great prize , as  if the prize of the Egyptian hieroglyphs!

It pleases , of all things to appear  unexpectedly  ,
That
A new branch  of the pure Mathematic, 
being developed
without for itself,
and in its,
expertise like togethermelding format,
laid before the public here in its appllying onto a branch of Natural philosophy
That therefore 
the beginning not with the beginner entity, 
rather with the entity what the conclusion should be, 
comes to be made.

 Many circumstances have everyway  alongside allowed myself into the entity of it  here
One time I was furrowing, 
the "time of old"  was becoming  
of a pure theoretical Speculation under 
the masses of the yearly appearing  books half baked no remarked upon quality bestowed
and the applications onto technical and nature contents not loitered for,
 or yet to  have in order to overlook ; 
ad infinitum the author  could  fill the lack of  his Writings through nothing in the Literary World of recognised names  . 

Thusly, this organising sorting and fetching appears around  basic things, 
there the  from hereout resulting  12 years space doctrine, 
which in its first parts was enholding the Elements of the planar Geometrical combinatorial doctrine of my alternative view, concordingly systematically ordered 
from the expertise-like side 
Is valued By a never approaching proving ground, or tracking ,
from the pedagogical side
imagine if it  has found some alongside opinion  
and has  given self opportunity to other exercises   .

It,  to the great part, pleases to have this its ground reason  in the form of the book 
Of which book  
Relating to the former mathematical instruction: was appointed for  children from 7 - 10 years, 
relating to the volume: was cloaked  in question and answer , 
and was with somewhat  pedagogical remarkings worked throughout (the text), behind which concordingly the against expected position of the matter
 never anyone an expertiselike prize  could everyway moot 

even if my alternative view is concording to no methodological work of solid Value 

Even if it  is not immediately a great  cleaning up of the elements 

And thuswith allows to make Claims upon a expertise like tracking .

If 
 in and  of the Entity these sortings organisings and fetchings also had to allow themselves to be sidelined 
thusly 
 other circumstances out of exercising of a doctrinal label
did yet make the general geometrical and phoronomical  combinatorial doctrine 
for now impossible.

The personal everyway holdling in space of the Author belonged not for the public scrutiny, 
the  things of mine only seldom a free hour placed to me and openly  were tearing  myself in pieces, 
before the results of the performance of the work could come to be fixed onto monthlies and quarterlies  
Irrefutable  
from the same things far away.
Thusly 
that all fruit the same everyway  lost went, 
and some content many times is begun,  before it was to me reaching, 
  off to achieve an unrestricted performed result related to it.

On the other side of the entity, 
but  the content was appearing to me  thusly important,
 that I myself everyway responsibility held, 
the achieved performed results not to go under, 
Nor it thereupon  to allow to arrive on stage,
 if   the Fate sent thing would be in another  position 
in order Myself to everyway set 
to publish it Rather to the Public scrutiny . 

The itself hereout developed counter claim, which to me self destructing was threatening  must  come to be  sidlined, 
and so enclose I myself, to the thought forms alongside the processing of a melded together doctrinal label to give up as a result ., 
and the content in singular offprints  to the languages, to bring, 
how allowing and off tearing  also the shape was pleasing to Be, 
in which  shape, thusly, it must step upon stage before the Public scrutiny  .

Alone of these offprints an overview must go forth  of the whole, 
without which,
 by considering the new quality of the Content 
those not Relating to everyway standing,
 At  least  its relating onto the main content 
could come to be not  to grasp notationly 

Concordingly some various miss launched research attempts 
 in form of an offprint 
to push together such an overview for one entity of the read journals for Mathematics or Nature Purchasers  , 
and by considering the sorting and organising and fetching, that  some entities  in their masterly aspects strive against,
or over stepping the openly somewhat pedantic staked off  boundaries of the surrounding  area and contents table , 


no "taking on board " would i like to find,

 i can at last to the inner conclusion , 
The  "this content  troubling " works in unrestricted  booklets 
To self hereoutresult,
But This one an overview of the whole,
 in which random one specialised content everyway woven is, 
to send forth out.

This overview has the appearance of booklets 
around a year everyway worn looking, 
and was for myself the main difficulty, 
Yes, it thereby considering
 thereupon came alongside, 
the important quality and applicable quality of a pure  expertise to show, 
which until right now is developed half baked nothing melded together.

How If one  any one,
who in a completely strange town comes,
would like to willingly orient,
 and him the main part, 
To this flighty  entity that precise one, through wandering allowed
and   around it a label ,
from their core acting property would like to give, 
randomly a specific content which is sent thereto , 
here out heaved, 
and it subjugated the special observing entity,
 but holding the precise everyway standing of the future before, 
how time and opportunity would like to output result it.

Thusly wish I the reader in the field of combinatorial expertises to orient,  
and I have chosen relating to the specific contents, the completely accounted for shapes of the ruler measured systems .

Alongside the material for the remaining booklets of this binding, 
thusly how for multiple succeeding bindings  it does not mar
unattended by the author by the Nature of the matter measured, 
until right now this chronicle has no companion worker alongside , 
from the future "something  quarter yearly" should appear
a booklet from 6-8 arcs, of which 4 make out  a bound piece  , 
thusly directly is selected inwardly an advantageous  on-taking of the work 

The author reckons then thereupon,
 that concordingly  by and by,  
companion workers will find    themselves

Quite specially would it come to be to enrapture him, 
if the one or the other , 
from the "here to the language  brought" contents
 would come  be gripping outwardly,
 self performing would be working further, 
and the matter to be would be  doing

The table of contents of the first binding  communicates   out of the Title . 
It will "pulled before" mannerly be devoted to the geometrical  and phoronomical combinatorial doctrine and there application to Crystsllography .

The second binding , if the writing should such an entity outwardly live,
 indeed would itself come to be not restricted any more onto one thusly special content
in and of which the Author thinks therin his alternative view of the general combinatorial doctrine, and its applying onto the Nature patronage,
 especially onto the Systematics, 
thusly how related to the labels of the Mathematic, 
from which arising  it has reached to this aspect , 
to develop .

The distinguished Character of this chronicle, wherethrough it from other ancestrally related tables of content goes completely off, 
comes to be
That it alone is devoted to the elemental Mathematic , and itself is setting the greater shining upon and better  alternative ordering of these elements to their main purpose.

Also the methodology of the instruction, 
so far as  it  thereupon grounds itself , 
It would  come to be  not shut   out from itself .

From the artform
 how  it  is remembering therethrough at the same time for the nature patronage  to work out,  
the layed out before you  booklet gives  an outreach Sample  .

Stettin, the 13.April 1829.




Grassmann

[jehovajah]

Commentary on the foreword.

This has been a long process of redaction and meditation squeezed into a very busy personal life at the moment.

At times it felt like few rewards were urged in the text, at other times the ground literally shifted beneath my conceptual feet, and my worldview changed.

The prospect of ground breaking insights has been replaced by the more accurate expectation laid out in the foreword itself.

The title itself indicates an appeal to mathematics and to nature patronage. We may personify these two addressees. Mathematics is asked to consider a new branch or root to itself, nature patronage is asked to consider an underlying combinatorial structure to everything.

What is nature patronage? It is all those people and groups who are patrons to nature in all its forms , but the translations highlight nature study or natural history as subjects of study.
http://www.lincstrust.org.uk/whisby-nature-park

The patronage is a group of interested parties and individuals who like some thing enough to be willing to pay for it to happen or to exist. For such a group experts as advisers procurers and distributors and valuers as well as displayers are also a necessarily part of the scene. To such a diverse group Justus alternative doctrine and view of combinatorics really applied to minerals and crystals..

Nature was characterised, indeed classified simply as animal, vegetable, and mineral.
From the time of Pythagoras if not before taxonomies of our knowledge have been made . The formats have been different, but essential structures have remained the same. The monads- Henads taxonomy or systematic classification of knowledge, the  Tai Jitsu, the alchemists earth wind fire and other, were all taxonomies or systematic arrangements of knowledge.

However it is Aristotles lexical and grammatical taxonomic practices which have come to dominate western thought through the Islamic scholarly traditions and preferences. Thus we essentially have an Aristotelian view of the categories and classifications of nature. The only major revision to this tradition is the less well known Hegelian taxonomical classification.

We do not exclude taxonomies of Being that is ontologies, but for this comment it is the taxonomical traditions related to nature that Justus addresses, and specifically the mineral taxonomy.

The point about a systematic organisation of knowledge is it is a fractal structure. Thus working out the application of his combinatorial doctrine to minerals will " generalise" to other parts of the taxonomy , and indeed to higher levels of the system. Thus Justus was appealing to mathematicians and to those who loved nature enough to categorise it , to fund the labours of the taxonomists and to print their taxonomies!

This is the meaning of Nature patronage for Justus.

It is also clear that Justus deliberately aimed his views at primary aged children in order to grow a group of companion workers in his new view of the ordering of nature and mathematics. One of these students was his son Hermann who took a grip on his fathers work and took it beyond where even Justus could conceive.

Justus admits his work is too big for him alone to complete and that his achievements were only thin slivers of what potentially was to be had , but he pressed on convinced he had something equally as important to mankind as the deciphering of the Hieroglyphs!

[jehovajah]

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


The Contents Guide of the first Booklet

------------------------------------------------------------------------------
                                                                                                                                                 Page
Possibility of an  on all magnitudes doctrine ' ultra dependent geometrical combinatorial doctrine _____3
Combination roots and rises up out of a Knitting as unlike artforms together______________________3
Intersection points, ray, corner angle etc, three corner, four corner etc as products  of space like combinations if one lays at the foundation, thereby considering,  the differing quality of the  directing  ___________3,4
Differing quality of the position  as foundational position  of geometrical combinatoon _______________4
combinations of the planes become directly very difficult_______________________________________4
The position of a plane and of things gathered together with  their parallel planes is appointed through a foundational position ___________________________________________________________________5
Are the lines like and vertical onto one another, thusly roots and rises up a System of shapes, of which we denote the regular System______________________________________________________________6
By rooting out from the Means according to a given stance of the body- stand point of Crystalography and Astronomy _________________________________________________________________________6,7
Placing  of the  elemental agency  and the caused planes alongside the room everyway indicated_______7
Label of the Radius constructor, or agency. Elemental agent_____________________________________8
Piecemeal progression to the planes on their agencies _____________________________________\\_8,9
Combination of these kinematical entities– Phoronomical combinatorial doctrine ___________________10
Combination of running into against set agent is excluded

[jehovajah]

This link is explicit and for adults. It deals with the effort to decypher  or encipher  ancient Egyptian imagery and text.

There is a combination of Cyrillic or coptic script and hieroglyphic symbols. ,

The interest in hieroglyphs is still to this day a great motivator for academia

http://m.youtube.com/LxSAeTQMTOY
https://m.youtube.com/watch?v=cQwQYV3CoRo
If first link is broken

[jehovajah]

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


The Contents Guide of the first Booklet

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                                            _xx_
  Fore-running  Construction of the completely perfect appointed Shapes      

                                                                                                                                            Page
Presenting of the combinations without repetition ____________________________________________11
The Die, the rhombododecagon,the octahedron ___________________________________________12,13
Alternative ordering of the Elements and of the thereout derived from complexions is able to serve to represent a body in differing positions ________________________________________________14

  Quick  Review on the general Combination Doctrine

The development of combinations concording to classes,  in which one all elements tallies,  is against the central- like Sense, of the Combination Doctrine __________________________________________17
Tally Marks and Combination Doctrine  placed together ________________________________________18

Development concording to Forms. Presenting of the possible forms of the complexions out of the elements bcd  b'c'd ____________________________________________________________________________19
Methods of developing the complexions, which are  held under every form– elements ringform like written and read away from left here round ._______________________________________________________20
Placing the accents_ main placing, nearby placing ____________________________________________21
Developing the bin-ions without repeating concording to the given  rule____________________________22
Developing the complexions of the form _________________________________23
Developing the complexions of the form _____________________________25
Developing the complexions of the form ____________________________27
Hemiedry , besigning the differing artforms of the same things___________________________________28
 Combinatorial knittings and reckonings ____________________________________________________31
Combinatorial addition  and subtraction besigning____________________________________________32
The inconcordant quality of combinatorial multiplication combinations of the Crystalography of which to besign ______________________________________________________________________________33
Graphical representing of the space-like complexions__________________________________________36
Representing onto the spherical surface_____________________________________________________37
.[/i]appointing of the place of a whimsical set together radius constructor through the intersection of two normal  circles_____________________________________________________________________38
Representing in the plane—projections_____________________________________________________43
Highlighting of the Repetition exponents ___________________________________________________46
Combinatorial and  analytical  expressions everyway  likening___________________________________46
The Repetition  points/ exponents are for construction  and  Calculation
          Coefficients _____________________________________________________________________47
They are continuous complete tally marks___________________________________________________48
Like artformed, valued  ancestrally related complexiions________________________________________49

                                               —XXI—
Position of the Tilted edge. Besigning the projections of the same onto the sphere____________________50
The tilted edges corner angle is the outward completing corner angle to two right entities for the corner angle for the agency____________________________________________________________________52
The radius constructor of a tilted edge is by simple shapes the combination out of the constructors of the side surfaces__________________________________________________________________________53
Ancestral ly related thing empowers from the rad. constructor of a vertex__________________________54
Geometrical interdependence of the projections onto the spherical surface__________________________54
Planar corner angle of the labelled surfaces of a shape _________________________________________55
By means of the Globe can the personal assignments of the Crystallogrsphy come to be loosed ------------55
Interdepending the sphere in the plane _____________________________________________________57
The complexions are centrally the same form, only space-like permutations_________________________57
Shapes of 3 dimensions, systems, ranked arrays______________________________________________59
The complete gathered like valued Complexions always give a simple shape________________________60

Main assignment: if 3 themselves adjacently  half running all over lines in space are taken to one side, and the possible positions and every way holding themselves of these lines, and  for every simple and set together shapes to concord to, which out of the Complexions of the differing  forms , roots and sprouts_______________________________________________________________________________62
The representing of the complexions of the regularly ruled system grappled, which is of the remaining systems beneath itself___________________________________________________________________64
Shapes of 2 dimensions _________________________________________________________________66
Developing of the simple Shapes of regularly ruled systems _____________________________________67

Everyway holding out in space personal test relating to the of the mineralogy entities_________________68
In the regularly ruled systems are the repeating points masses of vertically dropped coordinates. — trigonometric functions– the nature knows not our corner angle measuring methods – tables for simple and rational corner angle functions— space-like schematical presenting._______________________________69
The complete concording shapes of the regularly ruled system-— the throwing   Die Accepted hammers out the isographic projection ________________________________________________________________71
Main planes, space octants, positive   negative -______________________________________________73
The Dodecahedron______________________________________________________________________74
The Octahedron________________________________________________________________________76
The combinations or the repetitions are the foundation of all remaining entities; thusly also the out of their herebefore arising shapes—the Axes give the Agency.__________________________________________76

[jehovajah]

https://m.youtube.com/watch?v=cQwQYV3CoRo

Hemihedrary is the main reason that vectors have the design on the page that they do.
When one has a completely symmetrical object one can refer to one half of it and still retain all the information . The half is used to symbolise the other symmetrical half .

From this Justus designed a shorthand that is the basis for Hermans notation in the 1844 Ausdjnungslehre.

[jehovajah]

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


The Contents Guide of the first Booklet

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                                                 –XXII—

                                                                                                                                               Page
Singular or pyramidal axes; binary or prismatic , ternary or rhombohedral axes;____________________77
Besigning of the tilted edges concording to the axes between which they lie _______________________79
The everyway varying  simple shapes of the regularly ruled  System______________________________80
Form unsharing  of means space-like developing of the shape b³c²(Pyramidal Die)__________________________________________________81
Geometrical and Combinatorial developing __________________________________________________83
Form ; pyramidal octahedron _____________________________________86
Form ; Leucite shape _________________________________________88
Form , tetra tilting edge octahedron________________________________89
It enholds all remaining shapes of three dimensions in itself.  Hemihedral shapes____________________92
Besigning of the shapes. General concordingly voted besigning of the complete shape, and every single flat surface alongside the same shape_________________________________________________________95
Table of the fully tallied shapes of the regularly ruled system____________________________________98
Table of the Hemihedrial shapes of the regularly ruled sytem ___________________________________99
Crystallographic characteristics of the Cubic iron pyrite _______________________________________101
Logical Not regularly ruled systems _______________________________________________________103
The Prismatical system. Evolving of its simple shapes out of those of the regularly ruled system _______104
Individual  therein coming before us shapes_________________________________________________105
Crystallographic characteristics of the  heavy  barium spar_____________________________________109
The Pyramidlal system. Evolving of its simple shapes_________________________________________110
Individual shapes of the same ___________________________________________________________112
The rhombohedral system. Equal valued Complexions; evolving the same_________________________113
Singular shapes of this system___________________________________________________________116
Different systems of simple shapes. Equal valued complexions out of the shapes of the regularly ruled system evolved_______________________________________________________________________120
Parallelism of the tilting edges , and thereupon founded developing of set gathered together shapes____122
The agency of a surface , which with two others,  builds a representation of parallel tilting edges, is out of the agencies of those surfaces Combinatorially set gathered together_____________________________123
Developing of a set gathered together shape of the Prismatic system_____________________________126
Developing of a set gathered together shape of the unregularly ruled system. Relating  of the same onto other  elements, and out of finding the means of the repeated points_____________________________132

[jehovajah]

Many YouTube links no longer seem o work! I will try to correct these, but it is YouTube that keeps changing the links and restricting platforms , I am afraid.

[jehovajah]

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


The Contents Guide of the first Booklet

------------------------------------------------------------------------------
                                                 –XXIII—
                                                                                                                                                  Page
General Magnitude Developing

By calculating of binary  agency __________________________________________________________137
Out of finding the means of the everyway held state of the elementary agency . Discordant quality , which is not completely sidelined ________________________________________________________________141
By calculating  of the ternary agency , if thr elements of the shape are completely given _____________145
If simply the Leaning corner angles of the agency are given____________________________________147

Magnitude developing for the shapes of the regularly ruled system

Concurring of the position of a singular agency concording to the Pyramidial lengths and widths_______151
Concording of the corner angle, which  two agencies build a representstion of with one another________152
Applying onto the individual  body of the regularly ruled system, if one out of repeated points the corner angle , and seek the repeated points out of the corner angles   _________________________________154

                                 Hypotheses

Aggregating status/ condition markers— taking to one side closely over Cohesion and Contraction______161
The outer-like shape is an Expression of the inner shape/entity space ____________________________162
It has to be performed concording to 3 direction skills_________________________________________163
The representing Skills set have to be tracked to become as vertically onto the rooted and rising up surfaces_____________________________________________________________________________164
The growth of the crystal happens through "offsite  changing"  contraction and expansion_____________166
The "by building a representation" skills can combine themselves . Complete "alongsides these " combinations_________________________________________________________________________167

Closing Remarks

The right kind of developed Combinatorial Doctrine must a general applying of the magnitude doctrine onto the natural phenomena every way share the means__________________________________________171
The "Synthesis" of the spirit finds itself in the nature everyway worked out________________________172
The uncertain knowing of the central character of the combination doctrine is the ground reason , that thusly one  has known unil here less  uses  to pull from it___________________________________________178

                                           __XIV__

Developing of the Complexions concording to Forms,  is important especially for the dynamical point of view of Nature____________________________________________________________________________177
Where in nature  rational every way held out entities appear, hold they themselves firm always inner part of the boundaries of the consonnance interval_________________________________________________178
The chemical formullas tracked as combinatorial entities___________________________________[___181

Everyway holding out in space entity of Mathematician and the nature researcher__________________183,