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 Author Topic: V9  (Read 7885 times) Description: The general group structure on a circle with a specific example 0 Members and 1 Guest are viewing this topic.
jehovajah
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May a trochoid in the void bring you peace

 « on: May 08, 2013, 04:41:21 AM »

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
 « Last Edit: February 11, 2015, 08:21:36 AM by jehovajah » Logged

May a trochoid of 去逸 iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
jehovajah
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May a trochoid in the void bring you peace

 « Reply #1 on: May 08, 2013, 10:02:14 PM »

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!
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jehovajah
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May a trochoid in the void bring you peace

 « Reply #2 on: May 12, 2013, 11:14:14 AM »

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.
 « Last Edit: June 10, 2013, 08:33:23 AM by jehovajah » Logged

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jehovajah
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May a trochoid in the void bring you peace

 « Reply #3 on: May 16, 2013, 04:51:02 PM »

 "+" 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.
 « Last Edit: August 09, 2013, 03:49:43 AM by jehovajah » Logged

May a trochoid of 去逸 iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
jehovajah
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May a trochoid in the void bring you peace

 « Reply #4 on: May 17, 2013, 09:56:21 AM »

 "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.
 « Last Edit: June 21, 2016, 06:19:53 PM by jehovajah » Logged

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jehovajah
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May a trochoid in the void bring you peace

 « Reply #5 on: May 17, 2013, 10:32:57 AM »

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
 « Last Edit: June 21, 2016, 09:31:00 PM by jehovajah » Logged

May a trochoid of 去逸 iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
jehovajah
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May a trochoid in the void bring you peace

 « Reply #6 on: May 23, 2013, 09:47:39 AM »

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.
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jehovajah
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May a trochoid in the void bring you peace

 « Reply #7 on: June 09, 2013, 01:01:11 AM »

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

Notice the topology of these finite fields.

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

 « Last Edit: June 09, 2013, 05:54:58 AM by jehovajah » Logged

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jehovajah
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May a trochoid in the void bring you peace

 « Reply #8 on: June 29, 2013, 07:58:36 AM »

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

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jehovajah
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May a trochoid in the void bring you peace

 « Reply #9 on: July 08, 2013, 11:47:06 AM »

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.
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jehovajah
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May a trochoid in the void bring you peace

 « Reply #10 on: August 12, 2013, 11:58:05 AM »

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

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.
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jehovajah
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May a trochoid in the void bring you peace

 « Reply #11 on: August 17, 2013, 12:13:30 PM »

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
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jehovajah
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May a trochoid in the void bring you peace

 « Reply #12 on: October 20, 2013, 11:42:28 AM »

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.
 « Last Edit: May 25, 2014, 04:23:42 AM by jehovajah » Logged

May a trochoid of 去逸 iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
jehovajah
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May a trochoid in the void bring you peace

 « Reply #13 on: January 24, 2014, 10:22:12 AM »

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.

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
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jehovajah
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May a trochoid in the void bring you peace

 « Reply #14 on: February 26, 2014, 03:32:55 PM »

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.