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I highlighted a confluence of notions in a piece that for me ties Grassmann's Analytical Method into an more easily accessible bundle, and opens up the way to some terminology to explain how to build general reference frames.

http://jehovajah.wordpress.com/jehovajah/blog/2013/05/07/the-v9-group

My first topological space is a flexible but inelastic line. In 3d space it is a flexible string of minimum diameter, also inelastic: a spiders thread!

The metric on this thread is curious. The thread can be cut in 2 places to produce a segment which will be defined as a metron. The Metron will be chosen as a monad.

Now the Algorithm that is measuring process on this thread is called Euclid's Algorithm, or the Highest Common Factor algorithm, etc. I think it is actually a method devised by Eudoxus and explained from Book 6 in Euclid through to Book 7.

This algorithm is actually a fraction making process, a factorising process, a fractalising process.
The thread is continuous. The segment is discrete. This thread can measure continuous and discrete lines. It is contiguous.

The fractalisation creates smaller and smaller segments. The algorithm shows how to swap metron for metron  until we are satisfied or exhausted. When we are satisfied or exhausted we may call the metron a monad and define our metric by it, by counting: a naming process that relies on sequencing and memory.

The recording of these sequences, the memory of these sequences is also our fundamental notion of "time". Time or Tyme is an old fashioned word meaning to record the positions of the planets and stars. The vast Babylonian tables and records are the real basis of our notion of time, but the sequencing and memory and recording is the model of all our processes of sequencing and tabulation.

The notion of length and orientation have yet to be defined.

Grassmann defined a topological space on a scatter of points. This was truly novel. The metric was defined on a process of constructing a bisector and selecting a midpoint. You have to go outside the topological space of a scatter of points, or you need to embed the space in a larger spacee to do this . Grassmann did not necessarily specify this detail for dialectical consistency, but hey what the heck!

I was thinking that when i get onto V18 which i will, i wont be able to represent it on the fractal generator. But then i thought i would have 3 sets of orthogonal vectors which could constitute 3 modes of behaviour!, with the overall behaviour being a combination of these 3 modes.

That means i can choose 18 radials in quaternion space and fix the quatenion description of those. and then formulate in terms of those.
Then Quasz will sum them using the underlying quaternion math to give me a combined picture.

I think i tried to do something like tha for Kujonai, but i did not get how to include all the information in his system until now.

"+"	aI	bII	cIII	dIV	eV	fVI	gVII	hVIII	iIX
aI	a2II	abIII	acIIV	adV	aeVI	afVII	agVIII	ahIX	aiI
bII	baIII	b2IV	bcV	bdVI	beVII	bfVIII	bgIX	bhI	biII
cIII	caIV	cbV	c2VI	cdVII	ceVIII	cfIX	cgI	chII	ciIII
dIV	daV	dbVI	dcVII	d2VIII	deIX	dfI	dgII	dhIII	diIV
eV	eaVI	ebVII	ecVIII	edIX	e2I	efII	egIII	ehIV	eiV
fVI	faVII	fbVIII	fcIX	fdI	feII	f2III	fgIV	fhV	fiVI
gVII	gaVIII	gbIX	gcI	gdII	geIII	gfIV	g2V	ghVI	giVII
hVIII	haIX	hbI	hcII	hdIII	heIV	hfV	hgVI	h2VII	hiVIII
iIX	iaI	ibII	icIII	idIV	ieV	ifVI	igVII	ihVIII	i2IX

I have chosen to multiply the coefficients and add the axes labels modulo 9 to form a product for this group.

"x"	aI	bII	cIII	dIV	eV	fVI	gVII	hVIII	iIX
aI	a2I	abII	acIII	adIV	aeV	afVI	agVII	ahVIII	aiIX
bII	baII	b2IV	bcVI	bdVIII	beI	bfIII	bgV	bhVII	biIX
cIII	caIII	cbVI	c2IX	cdII	ceVI	cfIX	cgIII	chVI	ciIX
dIV	daIV	dbVIII	dcIII	d2VII	deII	dfVI	dgI	dhV	diIX
eV	eaV	ebVI	ecVI	edII	e2VII	efIII	egVIII	ehIV	eiIX
fVI	faVI	fbIII	fcIX	fdVI	feIII	f2IX	fgVI	fhIII	fiIX
gVII	gaVII	gbV	gcIII	gdI	geVIII	gfVI	g2IV	ghII	giIX
hVIII	haVIII	hbVII	hcVI	hdV	heIV	hfIII	hgII	h2I	hiIX
iIX	iaIX	ibIX	icIX	idIX	ieIX	ifIX	igIX	ihIX	i2IX

I have chosen to multiply the coefficients and multiply the axes labels modulo 9 to form a product for this group.

I am going to do V18 here

"x"

aI

bII

cIII

dIV

eV

fVI

gVII

gVIII

iIX

j*-I

k*-II

l*-III

m*-IV

n*-V

o*-VI

p*-VII

q*-VIII

r*-IX

aI

a2I

abII

acIII

adIV

aeV

afVI

agVII

ahVIII

aiIX

aj*-I

ak*-II

al*-III

am*-IV

an*-V

ao*-VI

ap*-VII

aq*-VIII

ar*-IX

bII

baII

b2IV

bcVI

bdVIII

beI

bfIII

bgV

bhVII

biIX

bj*-II

bk*-IV

bl*-VI

bm*-VIII

bn*-I

bo*-III

bp*-V

bq*-VII

br*-IX

cIII

caIII

cbVI

c2IX

cdII

ceVI

cfIX

cgIII

chVI

ciIX

cj*-III

ck*-VI

cl*-IX

cm*-II

cn*-VI

co*-IX

cp*-III

cq*-VI

cr*-IX

dIV

daIV

dbVIII

dcIII

d2VII

deII

dfVI

dgI

dhV

diIX

dj*-IV

dk*-VIII

dl*-III

dm*-VII

dn*-II

do*-VI

dp*-I

dq*-V

dr*-IX

eV

eaV

ebVI

ecVI

edII

e2VII

efIII

egVIII

ehIV

eiIX

ej*-V

ek*-VI

el*-VI

em*-II

en*-VII

eo*-III

ep*-VIII

eq*-IV

er*-IX

fVI

faVI

fbIII

fcIX

fdVI

feIII

f2IX

fgVI

fhIII

fiIX

fj*-VI

fk*-III

fl*-IX

fm*-VI

fn*-III

fo*-IX

fp*-VI

fq*-III

fr*-IX

gVII

gaVII

gbV

gcIII

gdI

geVIII

gfVI

g2IV

ghII

giIX

gj*-VII

gk*-V

gl*-III

gm*-I

gn*-VIII

go*-VI

gp*-IV

gq*-II

gr*-IX

hVIII

haVIII

hbVII

hcVI

hdV

heIV

hfIII

hgII

h2I

hiIX

hj*-VIII

hk*-VII

hl*-VI

hm*-V

hn*-IV

ho*-III

hp*-II

hq*-I

hr*-IX

iIX

iaIX

ibIX

icIX

idIX

ieIX

ifIX

igIX

ihIX

i2IX

ij*-IX

ik*-IX

il*-IX

im*-IX

in*-IX

io*-IX

ip*-IX

iqIX

ir*-IX

j*-I

a2I

abII

acIII

adIV

aeV

afVI

agVII

ahVIII

aiIX

aj*-I

ak*-II

al*-III

am*-IV

an*-V

ao*-VI

ap*-VII

aq*-VIII

ar*-IX

k*-II

baII

b2IV

bcVI

bdVIII

beI

bfIII

bgV

bhVII

biIX

bj*-II

bk*-IV

bl*-VI

bm*-VIII

bn*-I

bo*-III

bp*-V

bq*-VII

br*-IX

l*-III

caIII

cbVI

c2IX

cdII

ceVI

cfIX

cgIII

chVI

ciIX

cj*-III

ck*-VI

cl*-IX

cm*-II

cn*-VI

co*-IX

cp*-III

cq*-VI

cr*-IX

m*-IV

daIV

dbVIII

dcIII

d2VII

deII

dfVI

dgI

dhV

diIX

dj*-IV

dk*-VIII

dl*-III

dm*-VII

dn*-II

do*-VI

dp*-I

dq*-V

dr*-IX

n*-V

eaV

ebVI

ecVI

edII

e2VII

efIII

egVIII

ehIV

eiIX

ej*-V

ek*-VI

el*-VI

em*-II

en*-VII

eo*-III

ep*-VIII

eq*-IV

er*-IX

o*-VI

faVI

fbIII

fcIX

fdVI

feIII

f2IX

fgVI

fhIII

fiIX

fj*-VI

fk*-III

fl*-IX

fm*-VI

fn*-III

fo*-IX

fp*-VI

fq*-III

fr*-IX

p*-VII

gaVII

gbV

gcIII

gdI

geVIII

gfVI

g2IV

ghII

giIX

gj*-VII

gk*-V

gl*-III

gm*-I

gn*-VIII

go*-VI

gp*-IV

gq*-II

gr*-IX

q*-VIII

haVIII

hbVII

hcVI

hdV

heIV

hfIII

hgII

h2I

hiIX

hj*-VIII

hk*-VII

hl*-VI

hm*-V

hn*-IV

ho*-III

hp*-II

hq*-I

hr*-IX

r*-IX

iaIX

ibIX

icIX

idIX

ieIX

ifIX

igIX

ihIX

i2IX

ij*-IX

ik*-IX

il*-IX

im*-IX

in*-IX

io*-IX

ip*-IX

iqIX

ir*-IX

This Thread on Grassmann will be of significance as i develop the Group structures.
http://www.fractalforums.com/new-theories-and-research/geometric-algebra-geometric-calculus/msg61550/#msg61550

The geometric algebra discussed in this thread will be Hermann Grassmann's Which is not so much an Algebra as it is a Method of Analysis and synthesis.

What i am partially tabulating here is a deconstruction of Der Raum into a 9 axial AND an 18 axial system. However, the generality of Grassmann synthesisi means i am also describing any system with 9 or !8 oriented lines.
Curved lines can also be extended to .
The key is the relationship between the inner and out product  as defined by Grassmann.

This is the beginning of a series on finite field, that gives a fundamental insight into what these tables are about.

http://www.youtube.com/watch?v=yekCUuBYSGs&feature=share&list=EC3C58498718451C47

Notice the topology of these finite fields.

http://www.youtube.com/watch?v=bGY3oUY-Ikw&feature=share&list=EC3C58498718451C47
Norman demonstrates a fundamental "proof" that a finite field like F₇ has power for each element in the field that takes it to 1 and that exponent or power or logarithm is 6 in F₇.

The method of this proof relies purely on procedural behaviours  and not at all on number properties as claimed. The key observation is that the procedure of " multiplying" a row of elements gives a result that is a rearranged row. This comes about because a process of finding the remainder after dividing by 7 gives a result that is one of the 7 possibilities.

Therefore it is purely a procedural result, not a numerical one. Using that observation allows us to rearrange the notation, and then to apply the process of factorisation and bilateral division.

Once this is done the proposition is easily shown to be true.

As Norman pointed out, the one use of the prime definition is fundamental to demonstrating the general uniqueness of this rearrangement. Do not miss that part where he shows a = b! Realise that this is not Algebra, rather it is Rhetoric! Try and get a sense of the difference by repeatedly watching it .

Also, avoid the temptation to call rhetoric Logic! This not logic but rather rhetorical assistance to rearrange your perception of the processes involved.

The last in this quick series

http://www.youtube.com/watch?v=n3dGwu07kKI&feature=share&list=EC3C58498718451C47

Marko Rodin's original research conference on his V9 group extended to 3 d.

http://www.youtube.com/watch?v=2pXI0b4P2oA&feature=youtube_gdata_player

I will get round to completing the tables.

Analysing the research being done on the Bahai faith 9 circle as opposed to the Vedic 8 circle which is more consistent with a rectilinear approach, i realise now that Rodin, in transferring it to a 3*3 rhombus and arranging the numerals as he does is picking out only 1 of many arrangements of these numbers in the Rhombus.

That it ties in with certain magnetic and electric properties is not fortuitous, but it is not as a result of the number 9

|The mnemonic value of the system is of course extremely valuable in focussing meditation and attention, but essentially plays the same role as the binary, octal and hexadecimal numbers in their applied fields.

In fact one proponent of this application of the modulo circles has developed alternative powerful Electromagnetic coils, but of course the mystical nature of Rodin's mathematics is essential to its powerful appeal.

I have completed the table for V9 . The issue of how to represent it is covered by principles in this video

http://m.youtube.com/watch?feature=plpp&v=MkNfQtINEjo

Basically I choose 3 Quaternions that have normals in different directions and use these and the orthogonal Quaternions as a basis of 9.

The mutually orthogonal vectors in basis of the vector part of the quaternion are not totally orthogonal. They are orthogonal in sets of 3. These set of mutually orthogonal vectors are rotations in and of each other.

For V9 , regardless of whether I use a. Combinatorial + or a combinatorial x process, I require 9 vectors in the unit sphere of the quaternion.

I can use 2 methods to pick 9 independent vectors for a basis , that is a set of axes.

Exp(i*( 2*pi/9)*n) for n = 1,2,3,4,5,6,7,8,9

This gives me 9 roots of unity in the plane circle. They are vectors in different directions and so independent. It is not important to set out a constraint that the be orthogonal, but I have to consequently accept that any sculpture is likely to be distorted or deformed.

Of course the 9 vectors have to be distributed around the sphere, and I use the following processh to do this manually.

First I assign i,j and k their own independent great circle. These great circles intersect at the pole and antipope. 2 of the great circles are possibly orthogonal at this meet, but the third is not. The details of orthogonality I have left to the generators implementation of spherical surfaces in relation to i,j and k.

Now I spread the formulae for the 9 vectors along these 3 great circles sequentially, by cycling through the vector factor in the formulae.

Thus i goes with 1,4,7;
J with 2,5,8;
and k with 3,6,9.

The coefficients are then obtained by collecting like terms from the table.
However we have 9 coefficients and only 4 loop variables, so some compromis will have to be made. Possibly some fractions or factors of the four .

The second method involves using the direct exponential quaternion form
Exp(0+2*pi/3* n*i+2*pi/3*m*j+2* pi/3*l*k) where n,m,l= 1,2,3 independently.

These 9 Quaternions are unit Quaternions and serve precisely as vectors. In this case I probably only need the 4 loop varables to generate the coefficients with some adjustments. The possibility is that the generator will do most of the calculation itself

In thinking about how to define a point
http://jehovajah.wordpress.com/jehovajah/blog/
I became clear that V9 could represent a tetrahedral rhomboidal reference frame, both lineal and curvilineal.
In fact, in many of my experimental fractals this form appears in ghostly shadow, if the coefficients are right.

An insight today , written in the Light thread came with a number theoretic component. This was due to the strong association of the octal numbers , that is a group under addition modulo 8 with the periodic table.

Due to this structure we build our model of the atom on hydrogen( actually it is an adjusted carbon 12isotope nowadays) . This model was exposited as a period of 8.

This has always been unsatisfactory, as a glance at the periodic table shows. The huge area of rare earths and Lanthanamides occupies a floating position somewhere in the middle.

https://m.youtube.com/watch?v=r81hUk5qe5w

It would seem that using a period of 9 might be a suggestion, especially if the isotopes of elements are included in the table.

The spiral nature of the table is obscured in the modern layout but historically it figured greatly in its development.

The Bahai faith versus the Vedic faith is mirrored in the standard model versus the Rodin, Russellian models of these fundamental counting structures.

The base 10 model might be suitable for commerce, but it seems base 8 and base 9 are better for physico chemical behaviours

These next 2 videos demonstrate how to construct an exterior Algebra on a set of oriented line segments. While he over complicates it by constructing a tensor space topology, he explains every step.

To be honest this is not a good introduction for beginners, but it is presented in detail .

I claim, that if you understand Hermann Grassmanns 1844 Ausdehnungslehre, you will get a better handle on what this man is building, and how it relates to what I want to construct with nine Oriented lines.


http://youtube.com/watch?v=-6F74TH1i_g

The second in the. Series is equally as abstract, but he actually constructs a basis for an exterior algebra on 4 primitives. This would be suitable for a direct 4 dimensional space like the Quaternions, but in fact this Algebra is a hexidecimal,based one! I did say he over complicated it!

http://youtube.com/watch?v=F7DjnV6kw90

http://m.youtube.com (http://www.youtube.com/)

http://www.gaalop.de/wp-content/uploads/134-1061-Zamora.pdf
Having identified this as a candidate geometric algebra for " bubbles" , it occurs to me that particular solutions or views may topologically fit a toroidal with one, two or three holes.

In any case this is a 9 dimensional algebra which I am still constructing......slowly! :D

The history of ring and group theory

http://youtube.com/watch?v=VSB8jisn9xI

Vortex Based Maths and it's influence on wave Mechanics etc.

http://youtu.be/sK_ydsCn3Ko

http://youtu.be/lDomfhKBqIc

I have left it in link form to relieve pressure on older machines .

One of the beautiful things about Normans range of interests is his frankness and no BS approach. Consequently things which I only suspected as sn undergraduate Maths student , oh 42 years ago he is explicitly stating as the case. That is there is a big cover up of certain fundamental difficulties with mathematics. At the same time the Hegelian resolutions are also available!

One thing I would point to as a reason why Norman has difficulties which he has to resolve is precisely because mathematicians hate too much Philoophy! Thus they do not know that Hegel even exists, generally speaking! Most still use Aristotelian logic without realising there is another way!

http://youtu.be/Y3-wqjV6z5E
http://www.youtube.com/watch?v=Y3-wqjV6z5E

The relevant section in this masterly discussion is how modulo arithmetics underpin our number field concepts. Thus Vortex Base Mathematics is not as it appears to some as insignificant mumblimgs( sorry Marko but you do ramble) it is about the fundamental dynamics if our decimal system, and the topological spaciometric representation of the Arithmoi that underpin it.

Why use a sugar cube when you can use a doughnut, right?,  :dink:

May be my little program is helpful for creating the tables in HTML or LaTex.
http://www.wackerart.de/mathematik/geometric_algebra.html#rainy_days (http://www.wackerart.de/mathematik/geometric_algebra.html#rainy_days)

Hermann

Thanks for the contribution, Hermann .

I will look through the programming code to see how to adapt it.

It will be good to be able to do this stage automatically.

The next stage of combining products would then be more accessible.

Hermanns products and combinations are only one possibility. However, because the combinations are derived from the products of geometers, astronomers and Astrologers of his time his choices have immediate geometrical/ spaciometric applications.

However his thought process is a unique pattern worth learning. At the level of creating new combinations and new interpretations of products one is on ones own. Musean " numbers" are a case in point. Charles Muse has done the tables, but few of us even see any relevance to say physical phenomena at this stage.

Hermann on the other hand provides a course or doctrine into this kind of thought pattern and how to apply it. This is the main reason why I am not rushing to complete the tables,which are repetitively boring  and have a prospect of not being applicable to any current fractal generator.

The Newtonian Triples I was able to apply to a quaternion reference framed generator, but the twist in the presented image is due to the orientation  of the axes relative to each other. In the quaternion block i,j,k are assigned the orthogonal triple, but in the Newtonian triples the axes are not orthogonal. Hermanns line segment method gives us the freedom to create 6 axial directions where the 3 axes are equally spaced on the spherical surface.

However I knew the 6 roots of unity are defined in a circle, not on a sphere, so I was not able to understand let alone verify that the map from the  roots of unity labels to the  3 dimensional space axes labels was going to give a "true" representation, and whether I could defeat the embedded quaternion Algebra by imposing on the embedded programming.

When I get to Hermanns Rechtsystem I hope this aspect will be clearer to me. But I already intuit that the 3 dimensional set up in the programming space is not what I thought it was, an expression of reality, but rather it is a system of evaluation and a commonality enabling us to agree on any axial direction in space.

Hence by freeing the orthogonal system from any product association, we can establish any axis direction through it, and then define the axis products at that level.

All very good untill you try to look "backwards " at how these calculations evaluate in the orthogonal system. Doing that begs the question: what are the products of the orthogonal axis  elements!

That is the bit that requires some understanding. Do we as I suspect ignore that question, or is there some root of unity model that not only produces a rotation result at that level, but also maintains the relationship at the desired level?

My suspicion is no.

To design the Newtonin triples I used a general group product table with the required modulus. There were several abelian choices, but only one seemed to match the quaternion product rules. I then abandoned the quaternion product rules and used the group product structure, assigning a fixed label to each product to ensure I had a closed set that matched the orthogonal labels.

I can write about it now, but then I could not grasp what I was doing, I just followed my intuition.

It is important to me that Hermann constructs the. Product rules from his line segment tables. This means that the roots of unity represent a construction process not some mythical " numbers" out there, but rather a thought pattern process in me, subjectively.

Applying this to the definition of spaciometric orientations is the unusual part. Until Hermanns work I was left with the definite impression that that was " cheating!"

There are a lot of fundamental misconceptions I had to let go to understand where Hermann was coming from and how muddled our thought patterns actually were/ are!

Some seemingly outrageous claims made by Marko Rodin for the Rodin coil during his Vortex based Mathematics presentation now make sense,

http://youtu.be/4gHhAA-9o_8

http://www.youtube.com/watch?v=4gHhAA-9o_8

The spatial distribution of modulo 9 is demonstrated in this video

http://youtu.be/64gIN_mPOrw

http://www.youtube.com/watch?v=64gIN_mPOrw

The diagonal Spaciometry actually means the whole space can be rolled up onto a torus with the addition parallelograms spiralling round the form.

Using just the 9 digits we can form this pattern within the decimal system. This is therefore a modulo 9 pattern Within a modulo 10 system. The result is the modulo9 pattern spiral within the modulo 10 system.

I am glad I waited for the results of researching Hermann Grassmanns thoughts because my surmising was off target. It was close but now with a richer everyway standing of Grassmanns Method i understand these Caylry-Grassmann rank arrays as elements for product designs.

These product designs based on Normans Grouo theoretic explanation, mix the additive and multiplicative knittings in a way that isolates both systems from each other. To use them I now have to decide on subjugation order, and then use that to decide on an axial representation.

All of this I was doing intuitively bumbling along, but now I can use Grassmann speak to describe it more clearly!

The cyclus group 1,2,3,4,5,6,7,8,9 is the group that controls rotation for a 9 line segment system. The way the permutations work I will now be able to work though Grassmann Style, but it is already clear that Randy Powlells Abha vortex based Mathematics is a credible version of this system, and the uses to which it has been put are a creditable model well capable of modelling Magneto Thermo sono electro complex behaviours!

Of course, being a parallel world time traveller,  it's basic knowledge to us that the universe is indeed represented by 196883-dimensional field, the monster group.

It's no coincidence that modular functions like j-invariant are related to the monster.

It will be a long time yet, but soon people will think beyond the number line and think in terms of fields and multi-dimensions.
Without this it's impossible to understand how anti gravity and multi dimensional time travel is achieved.
The universe is a multi dimensional fractal you see.

I found this channel on Comninatirial theory, so it has encouraged me to take this course and to translate Justus Grassmanns Combinatorual Doctrine in this thread.

The V9 group or ring I will analyse following Hermanns example in "The Quaternions place in the Doctrine of Extending Magnitude "thread. This means that I start with product design , and then establish the general pattern of an extending magnitude that fulfills that product. This method , using the associative design step ensures closure , and it is closure that establishes the " group" or " ring" members or elements.

http://youtu.be/pCJNjW8kMIg

http://www.youtube.com/watch?v=pCJNjW8kMIg

Thanks for contributing to the thread flexiverse.

I hope you will elaborate on your point re the humongous group and other points . With regard to dimensions I have written a lot on the uses and misuses of that concept, especially the science fiction interpretation.

I have a view on the dimensionality of space, but it was only when I stumbled onto Hermann Grassmann that I found out that I had been living in Wonderland like Alice. The rabbit hole goes deeper!

As I begin the process of implementing the V9 group Grassmann style, I have to admit a certain sense of satisfaction and sense of growing understanding of what I am asking Quasz to do, and how to ask it.

The Cyclus group, I realise is a for next loop equivalent!

In modern computing we have come a long way with regard to the for next loop, and it is now a much more general and flexible notion than the cyclus group can ever be!  Using the for next loop in its various implementations( while, until , if ..) we can perform many more product design actions than Hermnn dreamt of!

The quaternion output results can be directly implemented by the cyclus group constraints or likenings or by the algebraic constraints Hamilton set out for label switching. The results will be the same because the behaviours are homologous.

Everything that Hermann has highlighted has had a direct bearing on or implementation in computer, syntax, coding and function or product construction.

The realisation dawns that the real field of Hermanns method is in computer programming , and not just for Maths programming either, but across the whole range of modern computer programming including Wolfram Alpha and Mathematica.

[f=z+i*(2*pi/9+0.0)
z=e^f+e^(f*2)+e^(f*3)+e^(f*4)+e^(f*5)+e^(f*6)+e^(f*7)+e^(f*8)+e^(f*9) +2.01*c

 A crude implementation, rendered in julia mode

[f=0.1*z+i*(2*pi/3+0.0)
g=0.1*z+j*(2*pi/3+0.0)
h=0.1*z+k*(2*pi/3+0.0)
z=e^(f+g*2+h*3)+e^(f*2+g*3+h*1)+e^(f*3+g*1+h*2) +2*c

another crude implementation. Sculptures in julia and mandelbrot mode

These crude implementations are half baked! I have not yet included the "running into against set" elements .  V18 is just that format.

However, the cyclus group running backwards controls this aspect. The fact that I implement using "—" is due to the conventional Axes format. The for loops within the app include the negative of any lie segment already, so the real issue is how much direct control do I want over the automatic set up.?

Strictly speaking these products are more synthetic knittings than subjugating ones as implemented, because I have not yet introduced the " squared" table version.  What is sculpted I now understand as Chladini forms representing harmonic and anharmonic patterns.  

http://youtu.be/5kYLE8GhAuE
http://www.youtube.com/watch?v=5kYLE8GhAuE

Since my introduction to the Mandelbulb effort, when suddenly I had to decipher what this Iconic"set" was, and what  complex multiplication was, and what a fractal was, and hat a fractal generator was, I have always understood the sculptures S carvings of the quaternion block( inQuadz) and a kind of " negative" of the dynamics.. Here however is the most simplest and relevant conception of these sculptures: they are Chladni forms!

To me that is satisfyingly ignificant because it means the Mandelbrot set and the Mandelbulb definitely have an associated sound!

From the shape alone my guess is that it sounds like a gun shot explosion!

The other aspect of this relationship would be that they both have an atomic and or molecular analogue, some element or substance that they represent,

And finally they image the magneto Thermo sono electro complex which fractally distributed permeate what we call space!

Toward the physical Crystalnomics and  the geometrical doctrine of Combination
By Justus Günther Grassmann
First pamphlet,
 Stettin
By consideration of
Friedrich Heinrich Morin
1829

[f=0.1*z+i*(2*pi/9+0.7)
g=0.1*z+j*(2*pi/9+0.3)
h=0.1*z+k*(2*pi/9+0.0)
z=e^f+e^(g*2)+e^(h*3)+e^(f*4)+e^(g*5)+e^(h*6)+e^(f*7)+e^(g*8)+e^(h*9) +2.0*c

A  variation of a crude implementation.

One of the striking patterns in the work and reasoning of Hermann Grassmann is the pattern of 3. This makes V9 a very interesting group to work with. In addition Hermann starts with the product and then identifies the fundamental elements that fulfill the product.

Quasz is set to make easy implementation of a tally mark( in this case one derived from a continuous measure rather than a discrete count) and an arbitrary  line segment , combined as a synthetic knitting .

This is often called a 4 dimensional object, as if it was outside our every day experience. In fact it is not . Think of a brolly and you essentially have a meaningful concept of a quaternion!

V9 can be thought of in a similar pattern . It would naturally sit in a 10 dimensional space, and exhibit complex vortex patterns.

Toward the physical Crystalnomics and  the geometrical doctrine of Combination


The  Foreword

The developing of the  entities,
 as a (complete) collection  in the nature observed crystal shapes,
 upon the combinatorial-like way, that brands: pure Mathematical way,
already appears to be to me, by itself,  a contents list of the greatest interest;  
but if
this developing from the hereforward emerging of a new, until now not processed mathematical Discipline
is slid by ,
which discipline hereforward invokes and Concords
through a pure inner Synthesis,
and
through  this constrained centrally accumulating construction,

and a parallely travelling to it schematical representation invokes and concords,
the shapes  ,

and which discipline lays their near melding together onto the most clear format for the eyes ,

Thusly this interest  would like to become
yet assigned importance,
raised up for the mathematician and the Nature researcher!

It itself opens a new Field.

Having read ahead I do not know if I want to commit the whole of the Vorrede to translation. In many ways it is similar to Hermanns Ausdehnungslehre, but admittedly a more plain verse style and accessible. In addition Justus is less intriguing than Hermann.

But experience dictates that if I miss something out i will only have to come back and do it again!

I have watched several mathematical presentations recently and find them very obscure and off putting. There is something about Normans presentations that are so appealing. Even when he is hurrying through pretty dire obscurity say in differential geometry I still learn something. Not so of many other presenters!

However with regard to combinatorics Norman has not done a full series on that topic so Fredrico is the best I have found, even if he is not so engaging a presenter.

It seems to me, and I have written on this topic in my blog, sequences and sequencing is fundamental to my interaction with space. Thus combinatorics of extensive forms is crucial groundwork to any topology and thus geometry and Astrology of space. On the flip side: these notions provide a framework for modelling intensive magnitude.

I already see the emergence of the Ausdehnung Größe in Justus development if Verbindungslehre.

I have not got a clue what he is on about beyond the word Fractal and spira!

However this is relevant as an example of what Justus is referring to.

http://youtu.be/xopBPi_Wcfk
http://www.youtube.com/watch?v=xopBPi_Wcfk

What is missing is what I call the Shunyasutras. Because the constraint is regarding space packing the straight line dominates and obscures the circular arc.

The sphere is supposedly not space filling but I think Apollonius gasket disputes that. Tessellating spacebwith the sphere emphasises the fractal nature and veracity of space and the fundamental rotation in space at all dynamic levels. It explains the deformation propagation characteristics of space and the observed propagation modes of reflection, refraction and diffraction, as well as polarisation, absorption and emission of deformations of space. It allows space to be modelled as a trochoidal fractal fluid dynamic that expounds a model of turbulence.

And of course it is my contention that it is a model of the magneto Thermo sono electro compex entities in and of space.

Toward the physical Crystalnomics and  the geometrical doctrine of Combination 1829
-------------------------------------------------------------------
It itself opens a new Field of the spiritual developing,
 it yet not allows itself to overlook  extending magnitude and fruitable quality  of the Field,
but it  entitles to the most important expectations, that already now confront us in a concording neighbourhood of the same expectations:
 the representations of Spirit at the same moment as representations of Nature.

For this pure expertise,
which in general the space-like or geometrical combinatorial doctrine can come to be named,  
specifically,
are the Shapes,
which to us the Nature in the crystals proffers up,
only the resultant of the analysing of a singular free like comprehending personal exercise!

The  claims,
which in this declaration lie,
are all kinds of things
from thusly out-of-the-ordinary artform,
that only the material itself it   everyway  was pleasing to finish with rights.

But plainly this is also it,
 through which I myself hold everyway encumbered responsibility ,
they, howwholly with unified  striving against,
in their complete biting sharpness,
express out.

 I was believing to permit nothing to everyway give to the  material ,
 I was believing to have to bring to it itself my personal feeling
to the sacrificial offering to have to bring !
 In the flow  of writing of each artform,
which in un-overlookable crowds daily henceforward pressed themselves,
  the work-like new and fostered entity  
becomes  itself only  easy to overlook,
if it not
impudent and impolite ( cheeky and churlish) henceforward presses
and  to make itself "empowering" knows.

It processed itself here therefore around the handy entities,
and the valid quality of a new self standing mathematical expertise,
Whose goal it is,
simple and set together shapes to henceforward shout out.,
and their  melding together to show,
and which makes the claim,
all crystal shapes in a single personal exercise to comprehend!

If this comes to be publicly known,
thusly comes to be the criticism to the Author over his presentation,
and what he therein selected to fillet,
 indeed not like- valid, but yet from the most meagre  assigning to be.,
there no one can feel it any more as it itself,
how least it is until now slivered therefor,
and how  least of the least it  is
Here in the stand to give!

In which I overhere the criticism of the knower quietly expect ,
I do  yet hold
myself, in everyway encumbered with responsibility over
   the rooting and rising up and piecemeal  developing of my alongside aspect of
       the "here treated" content,
    thusly  how over :
     the contents table,
     Sharp-cut around and form of the before you lying content
     and the succeeding   booklet,
 myself in all shorthand to expound,
Themselves towards non finite times out of this   rooting and rising up
would like to output results

some unifying  not completely unimportant moments
 
to the appraisal of my endeavourings .

How I
through the involved working with the combinatorial doctrine
piecemeal to the super generator  ( of conviction) am come ,
     that  the until now notation of mathematic too strict  be,
     in which it rigorously taken the combinatorial doctrine has,
      which according to its  whole character  undoubtedly relates to the
        mathematic,
not with comprehending enfoldment !
have I
   partly at many places of the before you laying script (S.3,S.14-19)
   partly already for 12 years in the first volume of my space doctrine,
    which volume the uneveryway mixed geometrical combinations
    in the plane is  comprehendingly enfolding,
 expressed( S.10-12 the Foreword)

But Out of this outwardly widened label of mathematic outputs itself a completely other progression of Combinatorial doctrine.
Because if it the moment of the unlike quality Is, (inequality)
which a synthesis to a combinatorial entity makes,
thusly thereout follows, that also the companion ordering,
Distinguishing partitioning, nomenclature of the complexions in relating onto the same thing
 must happen/occur,  
and that thereby considering the tally mark of the elements
only in so far as keeping at the back of ones Mind
everyway serves
as it subordinates itself to that moment,
that brands, so far as through the tally mark of the elements
 a differing quality  comes to be henceforward brought.

This conducts then with necessary quality onto the developing concording to Forms,
and this shows itself as the completest forms,
yes singularly satisfying forms,
 thuslydirectly from complexions with unrestricted repetitions
 out of a finite crowd from elements
the discourse is.

Plainly this moment of unlike quality ( inequality) places the combinatorial doctrine in direct comparative situation against the arithmetic,
and travels it from the same in which, what to it innermost centring action property is,
completely not there(los),
thusly that it a centring activity property, from no other dependent expertise comes to be,
the plainly-of that comparative situation comes to be
but to wayfare
  a conscious parallelism with the Arithmetic comes to be claimed
and related by ancestry
synthetic entities become placed( cf.S.31 sq.)

Out of the outwardly widened label then follows further,
that also space-like contents,
which a differing quality allow to be  placed,
 for the combination  come to be needed,
but only in so far as can build a representation of a separated entity
or un everyway mixed entity  
with afar off self standing and centring action property part of the combinatorial doctrine ,
so far as this differing quality yet from that magnitude everyway arrangement independent is.

This remarking ,
that  Must itself allow to be created whole upon this way  ,
   the central like material of the geometry
    in lines-everyway-knitting entities and figures,
everyway bound with the  remarking,
guides towards the geometrical combinatorial doctrine,
how such a doctrine in its first initiating grounds in the above named writings
as a fore preparing ( a priori) to the geometry is presented,
and I have pulled myself over,
 siding to the entity many times ,
that only therethrough
can come to be   realised by work,
the henceforward stepping entity
    which in the geometry tracked lines-everyway-knitting entities and figures  
       the   seemingly arbitrary and surprisingly strange entities accepted,
        the stricken companion ordering of the geometrical propositions
                 as a natural and self necessary entity from the Initiator Apprehended,
and
  a complete settling of the claims of the everyway standing entity
alongside the System of the geometry .

The learning from nature and the vortex structures of nature is crucial to Justus mind set and in fact underpins his conception of the value of his combinatorial views.

http://youtu.be/eCJxhxDMIb0
http://www.youtube.com/watch?v=eCJxhxDMIb0

Commentary on the crystalnomics foreword.

There is a lot of in these verses of plain song or plain verse. Justus has had 12 or more years to craft nearly every verse as his response to the lack of interest or understanding of his major work the Space Doctrine!

There are many literary and historical allusions and words have a double or deeper meditative meaning.

"Vene,vidi,vici" for example is alluded to in how the strict labelling for combinatorics has led to a major mis-comprehension of how combinatorics actually enfolds lines and lineal figures and forms in space.. Justus came, he saw the mis-relation and the labelling took the combinatorics captive! " gekommen,gehört, genommen".

There is also an allusion to the revolutionary times or the Moment in which he lived. The reference to the moment of Inequality draws on the spirit of the times, the French revolution and the Prussian Spring or Rennaisance. The Humboldt reforms if the Prussian Education system were in full Swing and Bismark was busy consolidating his power base and industrialisation programme to rebuild Prussian pride and dominance on the geopolitical stage. So an opportunity presented itself for alternative views to be looked at to rebuild the self actuating capability of Prussian children and scientists.


As a loyalist Justus did his part and waited patiently to be heard, but his work was overlooked! And so it remained for nearly half a century. If it were not for the loyalty of his sons Hermann and Robert it would have been buried in the historical turmoil.

We shall see how doggedly he pursued his conviction that combinatorics of the line segment enfolded in the plane was the key to setting geometry on the correct basis!

As a child I played with all manner of bricks, rods and sticks and mud. How I combined those materials was intuitively understood to be important to an apprehension of geometry. However nobody study the rules of combinations directly related to such materials. It is that aspect that Justus picked up on. Their is a centrally acting property to the combinatorics of spatial objects that is analogous to arithmetic! Indeed arithmetic derives its being and rules from this a priori property of combinations of space-like things. The trick is how one teases that out of our subjective processing! It was not easy!

Again at this stage I would remind the reader that the line segment should necessarily even principally include the circular arc segment. The predilection is and was to exclude the circle from these initial concepts, but in fact an examination of the Stoikeia does not justify this. Historically we may suspect that Apollonius ofbPerga criticised geometers for doing just this and redacted the texts accordingly.

The unending line and the unending circle are a fundamental analogy, by which all counting and measurement is ultimately deriveable.

The term "comprehending enfolding" is a conflation of notions associated with " umfasst" from"umfassen" . The importance of this notion is derived here: it is the enfolding of planes in space as oriented notions and lines in planes as oriented notions( and if you like points enfolded in lines, but it is orientation and direction which is important for geometry). However the revolutionary intuition thatvJustus intuited was the analogous and prior abstract notions of combinatorics of line segments with their orientations and directions enfolded into the space of a plane which itself is enfolded into an orientation in space.

This freedom is what Hermannmwas indoctrinated in and why he was able to derive thevAusdehnungslehre from his fathers ideas.

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.

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.

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!

"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.

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!

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?

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.
http://youtu.be/M_p5NksDCCg
http://www.youtube.com/watch?v=M_p5NksDCCg

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
http://youtu.be/lQ2m-GSaX8E
http://www.youtube.com/watch?v=lQ2m-GSaX8E

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.

The videos by Frederico Ardili are useful, but particularly lectures 1 (http://youtu.be/pCJNjW8kMIg) ,2 (http://youtu.be/Ss9ukTUJlCo) and 19 (http://youtu.be/YpbbJC51yQk).

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.

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!

Randy Powells V9 lines everyway knitting Entity

(http://nocache-nocookies.digitalgott.com/gallery/17/410_11_05_15_12_28_11.jpeg)


magnetic intensities

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=
http://www.youtube.com/watch?v=kxuU8jYkA1k

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
http://www.youtube.com/watch?v=rG6aIVGquOg

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

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

http://www.youtube.com/watch?v=MInlPBLOaro

This is what fractal combinatorics can achieve!

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.

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.

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.

Combinatorics overview.

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

http://www.youtube.com/watch?v=7kcO8EYY7xs

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.

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 .

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

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!

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


The Contents Guide of the first Booklet

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                                                                                                                                                 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

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

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The Contents Guide of the first Booklet

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                                            _xx_
  Fore-running  Construction of the completely perfect appointed Shapes      
  Quick  Review on the general Combination Doctrine

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.

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The Contents Guide of the first Booklet

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                                                 –XXII—

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.

Toward the physical Crystalnomics and  the geometrical doctrine of Combination 1829
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The Contents Guide of the first Booklet

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                                                 –XXIII—
                                                                                                                                                  Page
General Magnitude Developing

Magnitude developing for the shapes of the regularly ruled system
                                 Hypotheses
Closing Remarks
                                           __XIV__

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!

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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.

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                                              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*

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.

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 .

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.

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.

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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 !
----------------------------------------------
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

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 .

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 .

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 .

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 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 .

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

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.

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

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

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

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

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

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

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

Andbthisb1-1 correspondence is crucial to the effectiveness of the whole symbolic system as adjusts goes on to find out.
Norman repeatedly explores these issue from many angles, but he is unwilling to give upon mathematics! Hermann and I recognise the inevitability of the demise of Mathematics, especially with the rise of computational systems that can model so many extensive and intensive magnitudes.

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

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

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

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

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

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

Despite his brilliant devotion to Newton and Aristotle, Hamilton was not able to approach the generality of Insight that Hermann, free from Academia was able to synthesise from his lights .

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

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

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

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

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

The application to electromagnetic wave patterns naturally describes this kind of dynamic growth. Sinusoidal wave forms are a mathematical/ physical myth, that misdescribed a natural growth and decay process in a fluid dynamic .

Justus Grassmann resource link

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

Enjoy!

This continues Normans careful construction of an Algebra. We see how the motivation to mimic the combinatorial structur of the Arithmetic of the Arithmoi is divined or distilled via the judicious use of symbols and or labels!
This is called Algebra from the Arabic Al gibr! This means the twisting . In the vernacular it refers to the " mind fluff!" involved in this distillation process. Most of us can relate to that! But actually it refers to the spiralling thought patterning required to distill the essence of any calculation!
 
Calculation is derived from the notion of calculus using a small stone to represent or symbolise an entity and ultimately a quantity of entities. By extension it also represents a quality of an entity by an intensive magnitude experience.

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

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

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

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

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

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

http://m.youtube.com/watch?v=oWJIQdo1vpQ

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

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

http://m.youtube.com/watch?v=SGc4Wn3D6jE

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

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

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

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

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

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

Vielen Danken !!

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


Translating

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

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

This case specifically is the one by considering which the three lines, as concording to the three dimensions of space, are going perpendicularly onto one another

Hallo Jehovajah,

I wish you a good new year.

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

May be you like it.

Hermann

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

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

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

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

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

Another possible use is in generating secure codes?

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

Happy 2016 to you and your family and great success in our work and hobbies xxx

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

Typo warning; it could improve my typo rate ... Or not!!! Lolxx

Hallo Jehovajah,

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

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

The Ada-Software did only some basics works, mainly converting Ada-Matrixes into HTLM-Tables giving me a tool to convert Ada-Datastructures into HTML-Pages.
http://www.wackerart.de/mathematik/geometric_algebra/doc/src_html.ads.html (http://www.wackerart.de/mathematik/geometric_algebra/doc/src_html.ads.html)
http://www.wackerart.de/mathematik/geometric_algebra/doc/src_html.adb.html (http://www.wackerart.de/mathematik/geometric_algebra/doc/src_html.adb.html)

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

A nother issue lies in the use of real numbers in mathematics and in software development.
Writing 1/3 is far more elegant then writing 0.33333333333... !
So a long time ago I desided, that it was far more elegant to use rational numbers in software development then real numbers.
I then discoverd that Donal Knuth in his famous book the Art of Computer Programming had written about this Issue.

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

For this reasons I used the following package for implementing rational numbers.
http://www.wackerart.de/Ada/ada-data-structure.html#rational (http://www.wackerart.de/Ada/ada-data-structure.html#rational)
But using only standard integer numbers as generic parameters for instance of the generic packages was not very satisfying when I tried to approximate Pi or e.
So bigger Integers are required for further progress!

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


Hermann

http://m.youtube.com/watch?v=_1cQLPWKE5k
The history of Indin Mathematics.

Thank you for your contribution to the thread Hermann.

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

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

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

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

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

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

Many lies have been told about Indian Mathematics

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

These tables of relationships, matrices or arrays code a practice of measurement and counting and combining forms. This combinatorial base is the foundation on which topology is based.

http://m.youtube.com/watch?v=N5gjX_zuPLU
The shame is that Justus and Hermann and Robert were publishers whose works achieved limited circulation! But Peano acknowledges that he read a translation of Ausdehnungslehre1844 prior or in tandem with his own work!

Thus all these " axioms" are thoroughly discussed in the Einleitung of 1844

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

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

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

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

To get a parallelogram from an orientation process clearly involves more than just relative rotation about a point. Thus the complexity of composition should never be underestimated, and it should be no surprise that commutativity is not a fundamental compositional expectation! After all, we do not expect to build a brick wall using commutativity! We expect different outcomes if we compose quanta differently!

http://m.youtube.com/watch?v=peYrSP8cke8
The interaction of mineral structure with the force surfaces we perceive in Nature .
YHWH cause you to feel grateful .
Kapharim 2016 temple calendar .

http://youtu.be/KJEJUGz1ReE
http://m.youtube.com/watch?v=KJEJUGz1ReE
Tesla was using different cyclus or modular arithmetics to explore frequency ratios for his AC research . However Grassmann used the cyclus to design a 3 dimensional product for line segments capturing rotational motions .
Hamilton designed a quaternion product using i which has a cyclus of 4 .Grassmann got 4 by designing a product that had a rational product and a non commutative product which nevertheless could be congruent to the sign switched commutated  form

This is a google translation of the photographed page of Justus page 6

I have yet to read the German myself and produce my meditative translation, but this is a better starting point for continuing this thread and for others to contribute their own translations.

Please contribute if you can xxx

My recent meditative working on the text

http://m.youtube.com/watch?v=qhbuKbxJsk8
Quite a lot of detail in this video, but what underlies it all is rotational motion discretised with straight lines emphasised.

These are the line figures Justus begins to analyse in his work I am translating here

Kunde refers to those who are subscribers to or patrons of a certain interest group or activity . Thus Justus is writing to those communities that love Mathematical and Natural items or topics of interest and are willing to pay to maintain that subject interest and its development.
It seems not many were that interested in Justus ideas, except his publisher and his sons

The Grassmanns were a great publishing family. They lived near to one of the great publishing houses in all of Europe, and Robert even started and maintained his own publishing business.

As primary educators they were expected to prepare and publish their own teaching material for their students, and so not only were they very aware of the need to I Struct a printer very carefully, but also of the necessity of providing accessible material for a diverse range of abilities.

Here Justus prepares an article or a tract to attract subscribers. He hoped to publish in one or more of the well known magazines with reliable subscription bases but was unsuccessful in getting his material accepted for print!

Nevertheless he printed the material off himself for use with his students and any interested parties.
His sons were educated in the schools he had responsibility for, and in a manner akin to the Pestslozzi style.
He also collaborated briefly with Jakov Steiner, the European greatest synthetic geometer since Newton! To synthetic geometers the magnitudes they worked with we're dynamic elemental entities, not as we find today algebraic symbols dealing more with quantity of extension rather than the extensive magnitude itself.

Thus naturally a line segment is a living dynamic entity not captured by a number, but by an endless dynamic line whose intension is as dynamic as its extension . Continuity like curvature is an irreducible un analysable quality or property of magnitude. The creative point is a dynamic conception of position . It's dynamic creations the magnitudes of lines surfaces and space itself are mysteries to which we may assign one creation as a measure of all the others by an inductive method or system . Numbers as precise symbols in their own right do not measure or count. We consciously utilise metrons to count and measure and announce the results as counts of a specified Metron. We call the Metron a standard unit in the SI system, and recognise it as a fundamental di- mension, that is a split in the measuring process of a magnitude , usually split by orientation, but also by sensory perceptibility. Thus mass, length, time are sensorily split by kinaesthetic, visual, and motion/ change perceptions.

It is the combinatorial aspects of synthesis that so intrigued Justus, and this is why precise Numbering is so secondary to the overall method of description or depiction.

Norman J Wildberger has made a stab at making mathematic dependent on numerals, but shows time and again the problems of doing so!

The old wisdom is to use metrons of which the Arithmoi are the most noteable creation of the mind. These metrons are not numbers but rather extensible magnitudes arranged into patterns called mosaics. It was what the Pythagorean school was most noted for, and something Atistotle failed to understand, preferring a rhetorical basis to interacting with the cosmos.

The particular mosaic Justus is intrigued by is one constructed principally from lines!

These mosaics are dealt with in depth by Eudoxus in books 5 and 6 of the Stoikeia . The modern mathematician is unaware of the difference between a line segment and a length! The methods of Eudoxus are taught by using length and then horrors using numerals! These methode do not rely on length or numerals but on counting!

Thus Logoi are different relationships of counting comparable things of which the most commonly known is the Analogos method! In book 5 Eudoxus defines several more methods of counting comparables ( homogenous by their dynamic generation sic "the creative point") and in each case the synthesis of the comparables is set out positionaly. Thus they may be alongside each other, underneath each other as sides of any dynamic form , relative to some reference line or locus or point or any combination of those.( for example the legs of a point on the circle perimeter that stand on the diameter )

Many interesting properties of form were found by these synthetic comparisons, the most well known being Pythagoras theorem .
We see why algebra is definitely symbolic arithmetic, because the use of letters or marks to denote extensible forms is a natural tool in the synthetic geometers tool box. However, the system Justus and others pioneered was robust enough to replace symbols by other symbols as long as it was done in a conistent and coherent manner.
DesCartes method, though influential was not consistently worked out. Wallis later inspired a consisten set of orthogonal lines called axes, and researchers like Abel, Steiner and the Grassmanns popularised this consistent and coherent system by demonstrating its wide utility in classifying forms.

The risebofbthe Cartesian coordinate system to denote position only took a long time , and much was lost by adopting it as  superior fom of geometry ! It made the vector( Träger) a mystery instead of a natural extensive magnitude.

Everything that Hermann needed to construct ( synthesise) his LINEAL algebra was and is worked out in this booklet by Justus Grassmann. What Hermann did and to a less well known extent Robert , was show the applicability of this method to physics in particular beyond the classification of crystal forms, and to a general geometry of any number of dimensions, defined by the system or method.

Another experimental post.
I have restarted working on the translation of Justus work with failing eyesight . However modern accessibility applications mean I can tackle it with hope of a readable outcome .

I have previously studied two methods of interacting with the text. Dragon speak allows me to dictate my meditative thoughts on a passage or a word, while Google translate supplies me with a reasonable guide translation of the original photographed old Prussian German .
I will post both as I work through some  250+ pages of detailed thinking and teaching by Justus


It is clear that proof reading of the OCR is a necessary part of this process !

Today's translation

also in dieselbe Lage, in welcher sich der Astronom in Beziehung auf den Himmel befindet, oder in Rücksicht auf das Planetensystem auf den heliocen- trischen Standpunkt, Die Krystallographen sind ge wohnt, von Aussen zu beschreiben, welches ich für sen weniger vortheilhaft halte. Ihre Ansicht verhält sich zu der unsrigen, wie die eines künstlichen Himmels- globus zur Ansicht des. Himmels unmittelbar oder in jener am Rücksicht, wie das Ptolomäische Weltsystem zum Copernikanischen. Da wir es hier allein mit der Gestalt zu thun haben so kann es gleichgültig sein ob sie die Grösse eines Sandkorns hat, oder an dem unermesslichen Gewölbe des Him mels beschrieben ist. Das unmessbar Grosse in der Astronomie kann mit dem unmessbar Kleinen in der Krystallographie (etwa Hauy's integrirendem Molecul), füglich, unter einer Form betrachtet werden. Beides gelangt zur Anschauung, indem wir ihm end liche Abmessungen geben, und wir werden finden, dass die Kugel für beide Arten der Entwickelung mit gleichem Vortheile angewandt werden kann. Von den 3 auf einander senkrechten Linien sei nun die eine vertical, OU (Fig. 1.), die beiden an- dern horizontal, und zwar die eine nach rechts und links, RL, die andere nach vorwärts und rückwärts, VH, alle drei durch denselben Punct M gehend. Durchschneiden wir nun jede in gleicher Entfernung vom gemeinschaftlichen Durchschnituspuncte M aus mit einer darauf senkrechten Ebene, so erhalten wir den Würfel. Man stelle sich in die Mitte seines Zimmers, das Gesicht gegen die eine Wand gekehrt, und denke sich durch diesen Standpunct die beschrie- benen Linien gezogeh, so ist klar, dass, abgesehn von etwanigen Unregelmässigkeiten, die Wände sammt

That is, in the position in which the astronomer is related to heaven, or in consideration of the planetary system, on the heliocentric standpoint, The crystallographs are habited, from outside, which I consider to be less advantageous. Their view is analogous to ours, like that of an artificial heavenly globe to the view of the heaven, either directly or in the latter, as the Ptolomian world system to the Copernican. Since we have only to deal with the form here, it can be indifferent whether it has the size of a sand-grain, or is described on the immense vault of heaven. The immeasurable greatness in astronomy can be regarded with the immeasurable little in crystallography (such as Hauy's integrating Molecul), feasible, under a form. Both attains to the point of view, by giving it final dimensions, and we shall find that the sphere can be applied with the same advantage for both kinds of development. Of the three lines which are perpendicular to each other, let the one vertical, the other two horizontal, one to the right and the left, the other to the forward and backward, to the other, to the three Through the same point M. If we now cut each at a uniform distance from the common intersection point M with a plane perpendicular to it, we obtain the cube. Let us stand in the middle of his room, his face turned against the one wall, and if this line of sight see the lines described, it is clear that, apart from some irregularities, the walls are mingled

Today's translation


Decke und Fussboden jene Ebenen, so weit sie sich gegenseitig begrenzen vorstellen können. Man möge dieses immer nahe Versinnlichungsmittel nicht vornehm. verwerfen; es erleichtert die Orientirung ungemein, und kann nicht leicht zu einer falschen Ansicht führen. Jede Linie zerfällt nun vom Durchschnittspunkte aus in zwei Theile, welche ich, in sofern sie zur Construc tion der Ebenen dienen, und dieselben tragen, "radii con- structores", oder tragende Strahlen, auch schlechthin Träger nenne. Die Richtungen, nach welchen wir die Träger des Würfels angenommen haben, heissen mir, die Hauptrichtungen, und es werden hier, wie in jedem andern Systeme, je zwei, die zu derselben Linie gehören, mit demselben Buchstaben bezeichnet, von denen aber der eine den Accent erhält. Auf diese Weise sind in Fig. 1 die Träger der Würfelflächen bezeichnet: MO= b, MU=b', MR=c, ML=c', MV=d, MH=d'. Ueberhaupt sollen diejenigen Träger, welche den Combinationen ursprünglich zum Grunde liegen, und ihre Elemente geben, Elementarträger genannt werden. Man kann sich nun vorstellen, die Ebenen lägen anfangs alle durch den Mittelpunkt M, so dass also je zwei zusammenfielen, und fingen nun an sich auf ihren Trägern gleichmässig fortzubewegen, so dass in jedem beliebigen Moment alle Träger von M aus gleich gross waren, so wird man immer einen fort und fort wachsenden Würfel haben, und diese Construction kann in einem beliebigen Puncte ge- hemmt werden, um einen Würfel von bestimmter Grösse zur Anschauung zu bringen. Die Gestalten hören dadurch gewissermassen auf, starre und unver- änderliche Producte zu sein, und erscheinen, wenig-

Ceiling and floor, those levels, as far as they can mutually limit each other. One should not take this near-natural remedy. discard; It greatly facilitates the orientation, and can not easily lead to a false view. Each line now divides from the average point into two parts, which, in so far as they are used for the construction of the planes, are called radii constructores, or bearing rays, also simply bearers. The directions according to which we have assumed the carriers of the cube are called the principal directions, and here, as in any other system, two each belonging to the same line are designated by the same letter, Accent. In this way, the carriers of the cube surfaces are designated in FIG. 1: MO = b, MU = b ', MR = c, M L = c', MV = d, MH = d '. In general, those carriers which originally form the basis of the combinations, and their elements, are called elementary carriers. It may be imagined that the planes lie at first all through the center M, so that two each have collapsed, and they began to move on their carriers, so that at any given moment all the carriers of M were of the same magnitude We shall always have an ever-increasing cube, and this construction can be inhibited at any point in order to render a cube of a certain size. The structures thus cease to be rigid and unalterable products, and appear to be little-


To be
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Commentary

The details that Justus establishes here are of fundamental importance.
Many may know about Felix Klein , the Erlangen Proeject and the notion of "Dynamic Geometry" or transformational geometry, but not realise that this was a reactive movement to the way geometry was taught as a static perfection!
In fact , geometry as a term is a misnomer if applied to the Stoikeia or to the Pythagoreans

Panta Rhei, and it took a revolution in France to free the academic mind from centuries of encrusted misinformation, and to give modern, young fresh minds the chance to directly engage with the Pythagorean Text.
Of these, the most profound thinkers and educators were the Grassmanns and the Natural Philosophical school, who attempted to start again from first principles.

Thus the construction of form within a spherical overjacket was of ancient origin. And the notion of a vector( Träger) that is a radius of construction an ancient practice! That Hamilton called it a Vector and Justus a Träger attests to the uniformity of this practice amongst geometers.

And so now I see that a vector carries the final line segment of a form when it is precisely constructed using a compass and ruler or straight edge method.

The line of construction is and always has been for the purpose of orientation and drawing direction of a line segment. The magnitude marked off on this construction line is called the length of the line segment.
Ths a construction line carries the magnitude of the line segment, it imparts to the line segment it's orientation, and allows the constructor to decide upon a direction of carry!

Justus says it is a " bad idea" to call it just a vector/ Träger!  However lazy as we are we just do it anyway!!

The other point to be made is that construction arcs are Also to be called vectors/Träger!

Now pay attention to the Dynamism of Justus construction and conception! Thus continuity is encapsulated as a dynamic motion, not a static property of a line segment at all. Continuity is imparted to a line segment by the vector/ Träger too!

The conception of dynamic Strecken leaps out of the page when you read the very first Words of the Ausdehnungslehre 1844

Today's translation

wenigstens ihrer Grösse nach, in einer unendlichen Ent- wickelung begriffen, gleichsam fliessend, und sind nur für einen Moment dieser Entwickelung zur be- stimmten Anschauung gebracht. Durch diese vor stellung tritt es klarer vor dem Geist hin, dass nicht das endliche Product, sondern der Act der Produc tion es ist, welcher in dieser Entwickelung das Haupt- moment ausmacht Die Träger geben uns also nur die Richtungen und Verhältnisse gewisser Bewegungen an. Zwischen diesen soll nun combinirt werden. Diese Combina tion ist eine Combination der Bewegungen, muss also nach der reinen Bewegungslehre bestimmt und beurteilt werden. Ich nenne deshalb diesen Tieil der geometrischen Combinationslehre die pho ro nomische Combinationslehre Eine Conbi nation in diesem Sinne ist also von ganz anderer Art und Beschaffenheit, als die von welcher vorhin die Rede war und bei welcher man auf das Raum- liche sah, was den verbundenen Elementen gemein war, oder doch durch sie bestimmt wurde. Hier ist es die Richtung und relative Grösse einer Bewegung, welche aus zwei oder mehr andern zusammengesetzt ist. Zwei oder mehr Bewegungen lassen sich aber zusammensetzen oder combiniren, wenn man die eine einem Punct auf einer Linie, die andere der Linie selbst beilegt. Kommt noch eine dritte Bewe- gung hinzu, so kann man diese der Fläche, in wel cher die beiden erstern vor sich gehen, zuschreiben. Man könnte die zweifach zusammopgesetzte Bewe- gung eben so als die Bewegung oder den Weg des Durchschnittspunctes zweier sich bewegenden Linien, die dreifache als den Weg des Durchschnittspunctes dreier sich bewegenden Ebenen ansehn. Wir be-

At least in size, in an infinite development, are, as it were, flowing, and have been brought to a definite view only for a moment of this development. By this representation, it is clearer in the mind that the finite product is not the final product, but the act of production, which is the main moment in this development. The carriers, therefore, give us only the directions and relations of certain movements. We shall now combine them. This combination is a combination of the movements and must therefore be determined and judged according to the pure theory of motion. I therefore call this part of the geometrical theory of combination the phoronomic theory of combination. A combination in this sense is, therefore, quite different in nature and composition from that which was mentioned earlier, and in which one looked at the space, Elements, or was determined by them. Here it is the direction and relative magnitude of a movement composed of two or more others. Two or more movements, however, can be composed or combined even if the one movement laying by  a point on one line, the other movement laying by the line itself. If a third motion is added, this can be ascribed to the surface in which the two former occur. The two-fold aggregate motion may be regarded as the motion or the path of the intersection point of two moving lines, which are threefold as the path of the intersection point of three moving planes. We are

To be processed further

Commentary

Note that Justus, in combining movements, does not refer to a resultant line between the two movement lines or vectors or construction lines , instead he describes a process of laying one movement to a point in its line, and the other movement to its line . .

While this is not truly clear it is clarified by the reference to the intersection point. One movement is laid by one line up to the intersection point with the other line against which the other movement is laid. .

Later this is clarified as placing the beginning and end points of Strecken or line segments in a specific pattern to depict the movement. This is the idea of vector sums!  There is no vector resultant mentioned here, nor in Hermanns opening discussion of his first apprehension of summing line segments.

The phoronomic reference is arcane to most, but stereoscope and Opthamology are derived from this once vibrant field of study. The topic is how the eyes focus and track the objects and motions in the field of vision , and how 3 dimensional vision arises. . Thus a lot of these construction line demands and conventions are imported from that doctrine. Into Justus thinking here.

Today's translation

halten die erste dieser beiden Ansichten als die übliche und darum geläufigere bei.. Dass die Lehre von der zusammengesetzten Bewegung ganz der rei- nen Mathematik angehöre, unterliegt, sobald man dabei von den bewegenden Kräften ganzlich absieht, keinem Zweifel, wiewohl die reine Bewegungslehre bis jetzt in dem Lehrbüchern derselben keine Auf- nahme gefunden hat. Für unsern Zweck genügt das, was Fischer im 4ten Capitel seines Lehrbuchs der mechanischen Naturlehre (3te Auflage, Berlin 1826) über die Theorie der gleichformigen Bewe gung, von der hier allein die Rede sein wird, vor- getragen hat. Man kann die wenigen Lehrsätze, auf welche es dabei ankommt, auch aus jedem Lehrbuehe der Mechanik nehmen. Die, von denen hier Anwen- dung gemacht werden soll, sollen spälerhin auch noch zusammengestellt werden. Als Lehnsäze führen wir hier nur an, dass, wenn die einfachen Bewegungen gleichformig sind, die zusammengesetzte bei zweien nach der Diagonale des Parallelo gramms, bei dreien nach der Transversale des Parallele pipedums so nenne ich die gerade Linie von einer Ecke des Parallelepipedums in die gegenüberstehende) geht welches darch Richlung und Verhältniss der einfachen Bewegungen be- stimmt ist. Die Combinationen der Träger lassen sich nun, wie andere Combinationen mit und ohne Wieder holung, und nach verschiedenen Anordnungen auf- stellen, wobei nur zu bemerken ist, dass eine Com- bination zwischen den gleichnamigen Trägern, wie b und b' niemals slattfinden darf, weil diese entge- gengesetzte Bewegungen bezeichnen, welche sich entweder ganz oder theilweise aufheben, und die

Hold that the first of these two views is the usual one, and therefore more common. The fact that the doctrine of the composite movement belongs entirely to the pure mathematics is, no doubt, as far as the moving forces are concerned Now in the textbooks of which he has not found a copy. For our purpose, what Fischer, in the fourth chapter of his textbook of mechanics of nature (3rd edition, Berlin, 1826), presented the theory of uniform motion, of which we shall be concerned here. One can take the few doctrines, which are important, from every doctrine of mechanics. The ones which are to be applied here are also to be compiled. As lemmings, let us assume here that, if the simple movements are uniform, the two are parallel to the diagonal of the parallelogram, and three to the transverse of the parallel pipedum, I call the straight line from a corner of the parallelepipedal to the opposite ) Is the determination and the relation of the simple motions. As with other combinations with and without repetition, the combinations of the carriers can be set up according to different arrangements, with the exception of the fact that a combination between the carriers of the same name, such as b and b ' Because these represent opposing movements, which either cancel wholly or partly;

To be processed

Today's translation


Natur entgegengesetzter Grossen haben. Die Auf- gabe, die Complexionen der Elemente bcd b'c'd' zu entwickeln, gehört also in die Classe derjenigen, bei welchen die Verbindung gewisser Elemente ver- boten ist.

Vorläufige Construction der vollkommen be- stimmten Gestalten.

Wir wollen nun, gleichsam zur Probe, und um den Leser nicht durch Vorbereitungen, deren Zweck er noch nicht übersehen kann, zu ermüden, aus den gegebenen Elementen die Complexionen ohne Wie derholung entwickeln, und die durch sie bestimmten Gestalten construiren,— damit wenigstens an einem Beispiele erhelle, wohin diese Combinationen führen, —und sodann den Faden da, wo wir ihn fallen liessen, wieder aufnehmen.

Die Combinationen aus den gegebenen Elemen- ten sind nun folgende:

Unionen b. b'. c. c'. d. d'.

Binionen bc, bc', bd. bd'
b'c.b'c'.b'd.b'd'.
 cd. cd'. c'd. c'd'.  

Ternionen: bcd. bcd' bc'd. bc'd'. b'cd. b'cd'. b'c'd. b'c'd'.

Eine höhere Classe kann es nicht geben, da die gleichnamigen Elemente nicht verbunden werden dürfen.

Nature of the opposite size. The task of developing the complexions of the elements bcd b'c'd 'thus belongs to the class of those in which the connection of certain elements is forbidden.

Provisional construction of perfectly defined figures. Let us, as it were, to the rehearsal, and not to weary the reader by means of preparations whose object he can not overlook, to develop the complexes without repetition from the given elements, and to construct the figures determined by them Give an example of where these combinations lead, and then resume the thread where we let it fall.

The combinations of the given elements are now as follows:

Uni onen  B '. C. C '. D '.

Bin ions bc, bc', bd. bd'
b'c. b'c'. b'd. b'd'.
 cd. cd'. c'd. c'd'.
Ternionen: bcd. bcd'. bc'd. bc'd'. b'cd. b'cd'. b'c'd. b'c'd'.

A higher class can not exist, since the elements of the same name can not be connected.


To be processed

Commentary
Note Justus points to the diagonal and the "transversal" as associate determinations or resultants of simple movement sums. We take this for granted , as a doctrine from Mechanics. He sets Fischers work as a textbook. This is a direct reference to Newtons Astrological principles! Newton established this association in this book. However this is too advanced for primary school teachers and hard to get, so Fischer is recommended.

The sources of vector/ Träger concepts are mundanely revealed by this reference.

We see, then how closely Hermann followed his Fathers ideas and presentation. But Hermann made one crucial change and observation and that was with regard to 3 line segments in the plane. It revealed to him that the main planar shape for study was the parallelogram and it's 3d equivalents,Parallelepipeds  not the cube!

On d you make that determination, then 2 products / multiplications become fundamental: the spreading apat product, that is the sides of the parallelogram , which really evaluate when the sides are orthogonal ; and the colliding together product , that is a dropped perpendicular onto one side from the tip of the other side, which evaluates according to the trigonometric evaluation of the perpendicular and the receiving side. The outcome is the variation of the " outer sides" product by a trig ratio. This cane to be called the " inner line segment"  product, or just the inner product. Both evaluate the so called area of the figure, but , and it is a big but, the answer is not important! What is important is the ratios of accessible figures and forms. So the area of a parallelogram can be expressed as multiples of a unit parallelogramm( a rhombus) which then makes the count invariant irrespective of the angle between the vectors/ Trägers. The inner product then enable a comparison relative to squares and cubes not the count but the shapes!!

Freed from standardised evaluations, the geometer can manipulate and synthesis the pure forms that are appropriate to the circumstances.

What about angles? They are not necessary. Trigonometry is about ratios in right nglrd triangles, boy angles.

So why are angles made to seem so important?
In a student Maths set you have the set squares and the protractor. The protractor reduces the number of set pieces required to construct a figure. Using a ruler, a compass and a protractor a wide range of figures can be constructed. In addition, using trig tables standard values can be evaluated.

The length of a circular arc is NOT problematic, but it is often inaccessible to Astrologers . The arc length can be calculated by using Analogous ratios and trig tables.. But where it is accessible it is most accurately measure by rolling the arc on a flat surface without slipping. This moves the circle centre along the dimeter parallel to the flat surface the required displacement .

Again this is a curved measurement apparently straightened out, but it is an unmoveable axiom, demand that curves can never be straightened!! So this measure is in fact a ratio of an arc to a straight line on a flat surface. The ratio changes for curved and trochoidal surfaces dramatically.

So Justus establishes a method of synthesis that requires counting not measuring , later he deals with the trigonometric ratios that becom necessary only then! It is the last few chapters of his booklet, and is simply traditional higher mathematics related to his method. Hermann, following his fathers set out plan hoped to produce a second volume to express this part of his fathers life long work, but the poor reception of his 1844 work meant he nearly gave up on ever achieving that goal. However his brother Robert, also promoting his fathers work and insight redacted and reprinted Hermanns work in a format that he judged would go own better with its target audience, and he was right, but it was never going to burn the house down while Gauss and Riemann were alive. Once they died then Hermanns work came into its own, and outshone Roberts several publications on the subject!

http://youtu.be/bMlX28tIqzA
Norman develops the planar geometry using the dot product. And a defined cross product for the plane.
Norman spends a great deal of time filling in the gaps Herrmann and almost everyone else left . So he defines natural numbers using data structure notation , the concept of anti matter rather than opposing directions or mirror reflections or rotations through Pi radians.  He gets as far as integers but has defined rational numbers in an alternative series.

The theory of numbers he promotes is firmly based on the logoi cases in the Stoikeia.

However he misunderstands the role of numbers and the role of magnitudinally extensive forms in the Stoikeia. He correctly identifies the generality of the form over the expressions using labels and so algebraic forms ( symbolic arithmetic )
It reminds me of the fantastic development of real no.s by Hamilton, using the concept of flowing or propagating time.
It was also brilliant but flawed! The issue always is continuity, and that is a motion concept. Like curvature can not be straightened out, so continuity can not be divorced from  motion even jerky motion.

Any ways Norman is coming back to Justus ideas on geometry and kinematics.

http://m.youtube.com/watch?v=x6UofU0QJI4
This series of videos in which Norman explains the spread , relates it, at last to the unit circle.
Of course for any circle an arc on the perimeter is unique and thus any regular gigue that contains that arc as a diagonal will also be unique in area. Thus area can be a measure of arc in a proportional way, related to the square of an arc " length" .

Norman points out that arc length is more naturally related to sector area, but that calculation is more complicated.

We should not let calculative difficulty o scurr what is observation ally simple!

Of course the utility of arc length is questionable. In fact Normans whole premise for rational trigonometry is that arc length is restrictive! This is true because Pi is an incommensurable ratio, and rarely do we need to construct perfect spheres or circles.

However, the deeper mysteries of physical motion are inaccessible to straight lines, even though we can confidently establish metrons with them. In point of fact we can establish excellent metro s only through the use of circuar construction lines! And similarly we can construct better curve approximations to general curves using circular arc elements.

We ca not make what is curved straight so we need to expand our notion to u its of curved extensions as well as straight ones.

We use a fudge factor called Pi to reveal that the algebra is the same even if the metrons are curved.

So the area of a sector is just so e fudge factor multiple of the area of a parallelogram and both are a measure of the contained arc squared.

Squating an arc seems a novel idea, but in fact it is procedurally identical to Squaing any lineal element. There is no unique measure called area. We standardise a process of counting a Metron that " divides" or we place down upon a region in a contiguous manner, very much a ong and dance process!

So whether we use a line or an arc squared as a measure of rotation is a trchnicallity of calculation. The most direct measure is to draw a standard circle and ompare arcs on that circle, but comparing areas volumes etc does provide a more accessible measure in many instances even if it is a none li Ealing method that distorts the numerical results from uniformity.  Evenso it does provide a unique up to sign, measure of rotation , but not of distance travelled by a rotating disc with no slippage. For that you do need the most direct measure which is the arc to diameter ratio..
For a perfectly flat surface the centre of the disc moves along the diameter ib a direct proportion to the arc turned. This is a possible definition of the notion of arc length : the arc: length ration of movement measured at the diameter.

You will note that this length varies with the surface the disc rolls over, in particular that the circle within or upon circles motion produces not only incredible patterns but incredible arc lengths.

http://m.youtube.com/watch?v=TsMWT7U_AIc
In this seminar Norman illustrates the combinatorics of line segments .
Of particular interest is the Cyclus groups, for which the permutation count formulae is degenerative. It is not n! , but nx1x....x1

I think that complexions is the word used to describe each instance of a permutation .
Everything set out for straight lines of course can be set out for circular arcs. The additional requirement of construction radii to determine arc centres introduces more intersecting lines whic then  may become the subjects of study . It is useful therefore to distinguish between construction lines or radii and the lines of interest which they carry( vectors / Träger)

In seeking to understand the notation Justus introduced and the term geometric product I have concretely identified it with standard area formulae.
But Justus inspiration was from the geometry of his time which redacted heavily the Greek geometry, rincipally that  of Aristotle as passed down by Islamic scholars. Later Greek sources of the Stoikeia were found but they had little impact except on a few like Newton and Barrows fir example.

The Stoikeia takes the line segment as its fundamental magnitude , but not as it's fundamental measure. The fundamental measure is the arallelogram and the rectangle.
The parallelogram is defined by two contiguous line segments. They form a corner that contains the Rectilineal forms.

This is the product notated by ab where a and b are notation for general line segments.

The triangle is formed by using a third line segment that is a diagonal of the parallelogram , or a diameter of a Rectilineal form. This is usually a factor of a half.

The next product is that formed by drawing a right triangle in a figure that isbrectilineal. This is called dropping a perpendicular from a given point,vertex onto a Rectilineal line. . This product is a right triangle and is denoted by a|b in Hermann Grassmans notation.  While this is associated with the altitude or height of a general triangle, in fact it can be a line dropped any where on another segment , by extending the segment from which it is dropped. . The various products formed in this way are similar or in a proportion so the factor is not restricted to a 1/2. The factor for this product is determined fro trigonometric tables.

There are other products like for examp,e trapezoids  and regular polygons. These products have distinctive firms but are made up of some combination of the basic 2, some combination of parallelograms and right triangles.

The Middler product is thus some combination of right triangles and parallelograms, the factors of each are determined by the form .

In 3d does the general form become a parallelepiped?  Yes, but the form is still dealt with by appropriate right triangles nd parallelograms in the appropriate planes.

The thing to remember from the Greek Stoikeia is that all these products are dynamic. And so a rotational dynamic is inherent in these geometrical products.

That rotational dynamic s form
Malised in the spheres which in general construct the segments and forms by intersection of their surfaces.

The importance if parallelograms as the general form for products was missed by Justus. He after all was trying to logically found arithmetic, and that is based on the rectangular parallelogram, and particularly the square.
However this is a misconception carried over by Islamic cultures from Aristotelian analysis.

The Sumerians had a greater respect for factors, factoring , the rectangle and the parallelogram inscribed in a circle.
Recent work by Norman Wildberger and his colleague is uncovering the richness of their sexagesimal system.

http://youtu.be/J5Ug3Cr8RUE

http://m.youtube.com/watch?v=J5Ug3Cr8RUE
Factors become the bedrock for apprehending scale, proportions( Analogos) squares( duoplassoi) cube ( triplassioi) and other logoi( dynamic ratios) relationships found in book 5 of the Stoikeia. These ratios are all based on the intersection of chords and secants associated with the circle, in the most general propositions of books 1 to 4

Book 2 is the clue that is overlooked its propositions are based on the general parallelogram with a diameter, the general half parallelogram. And the special rectangular parallelogram, and it's half.

Tables have since Sumer been associated with quick and handy calculation. The process using the tables was therefore key to being competent. In a real sense algebra( symbolic arithmetic) is really an expression of the general process for using a set of tables!

Justus understood that his ring theoretical approach had to be based on this dynamic geometry of the Greeks, but he was not clear how multiplication old be logically founded. There was no proposition for multiplication, only the rectangular form.

Un fact Hermann his son realised that the proposition for multiplication is based on the dynamic parallelogram. Not the static rectangular form. Like all propositions every type of multiplication had to be constructed to demonstrate its validity.

Several products can be constructed geometrically. Books 1 and 2 deal with those as foundational concepts. Pythagoras theorem is a product, intersecting chords is a product, gnomon, parallelograms and rectangles are products.

Where multiplication enters the discussion is in the notion of factoring. Factoring is breaking into parts, and it is this notion of breaking into regular or irregular parts that is emphasised over and over. Those parts that are equal are called artios or perfect, those that are not equal are called perisos or approximate( near to some perfect part) .

The process of multiplication is founded on duplication. The number of times a form is duplicated( isaskis)  is punted and this product is called a polyplassios or multiple form .

The discourse in book 6 is about dynamic ratios . Skesis is a held stage in that dynamic process( kinesis) . The ratios are thus studied in proportions, that is in some relationship of 2 or more ratios. . The first relationship is called Analogos, and involvesc2 ratios . Book 6 goes on to define squared, cubed, inverse and other relationships rarely mentioned.
All of these have a foundation in a geometrical form inscribed in or exterior to a citcle.

Measuring the circular arc was done by using the chord bow.. Remarkably the radius of a citcle is a chord that cuts the circle precisely 6 times by definition! From a proportion perspective one might think  twice the radius( diameter) should cut it 3 times, but in fact it cuts the cicle in equal semicircles. What cuts it into 3 is the diagonal of a parallelogram called a rhombus .
As you study chords more you fing these rhomboids appealing regularly in the dissection of chords. So it is most diagonal length that is important, but what rhomboid form is it that bisects, trisecting quadrasects and quintasects a chord bow.
Proportions in a circle are based on area not length .

http://youtu.be/56gzV0od6DU
Benoit Mandelbrot RIP

http://youtu.be/1-ssKUf4-mk
The logoi that are Analogos ø.
Here they are presented as segment , but in fact they should be presented as rectangular parallelograms.

http://m.youtube.com/watch?v=1-ssKUf4-mk

http://youtu.be/CVS4_9EXgbs
http://m.youtube.com/watch?v=CVS4_9EXgbs

I am going to be copying the translation of Justus Grassmans pamphlet  or exercise book directed to the subscribers to "Mathematics and Nature", a proposed magazine title he wished to found, over into the new forum .
I will try to complete the table before this forum version goes into archive mode .
Hopefully this vides gives you a flavour of why V9 ( and it's scale le versions and parts) are a significant group or ring for apprehending aether dynamics or rotationl dynamics, when mapped out on the surfaces of torii . I might add that a torus is a simple trochoidal surface in the family of trochoidal surfaces. The complexity we have to model goes beyond the toroid as a surface in most cases, but a torus is a great pragmatic tool, like the sphere.