V9

[jehovajah]

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

[jehovajah]

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!

[jehovajah]

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.

[jehovajah]

"+"	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.

[jehovajah]

"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.

[jehovajah]

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

[jehovajah]

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.

[jehovajah]

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

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

Notice the topology of these finite fields.

<a href="http://www.youtube.com/v/bGY3oUY-Ikw&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/bGY3oUY-Ikw&rel=1&fs=1&hd=1</a>
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

<a href="http://www.youtube.com/v/n3dGwu07kKI&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/n3dGwu07kKI&rel=1&fs=1&hd=1</a>

[jehovajah]

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

<a href="http://www.youtube.com/v/2pXI0b4P2oA&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/2pXI0b4P2oA&rel=1&fs=1&hd=1</a>

[jehovajah]

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.

[jehovajah]

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.

[jehovajah]

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

[jehovajah]

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.

[jehovajah]

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

[jehovajah]

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.


<a href="http://www.youtube.com/v/-6F74TH1i_g&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/-6F74TH1i_g&rel=1&fs=1&hd=1</a>

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!

<a href="http://www.youtube.com/v/F7DjnV6kw90&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/F7DjnV6kw90&rel=1&fs=1&hd=1</a>