Newtonian Triples

[jehovajah]

The following is the xy view looking down from the z height onto the sculpture. Don't get dizzy now  . This particular code is ready to go to explore coefficient relations. At this moment the coefficient scalars are set to exp(0) or 1.

[jehovajah]

I have exposited my insights in my blog, but this memorial to Newton, De Moivre and Sir Roger Cotes, has turned out to be a satisfying culmination of my original impulse to join in finding the Mandelbulb.

While there is no doubt in my mind of the fundamental significance of the Mandelbulb in once and for all time shifting the paradigm of a mathematics, a physics and a mechanics of the behaviour of our mental, subjective processing of space, it is only one of an uncountable magnitude of dynamically varying reference frame products.

The iconic Mandelbrot set of relations, whose visualisation generated an iconic visual form is the gold Standard for an aesthetic evaluation paradigm. This evaluative standard actually enables several mental processes to be standardised and compared, rather like a Rorsach test http://en.wikipedia.org/wiki/Rorschach_test
From this form I may derive the fundamental process concepts : reference frame, fractal quantification, fundamental dynamic magnitude calculus, based upon a fundamental vector magnitude analysis that relies upon Grassmanian Analysis into Ausdehnung Groesse. Within this analytical scheme particular fundamentals are worthy of note: the Newton, De Moivre, Cotes–Euler analytical and theoretical principles, and the Hamilton Quaternion Algebra; the Turing extension of these thereto disconnected streams of thought into the Universal coding machine paradigms that ultimately reintroduced art and aesthetic apprehension as rigorous elements of a mathematical comprehension of space. Upon the back of these shifting sands Benoit Mandelbrot found the underlying Rock solid coral reef of the Mandelbrot set.
http://en.wikipedia.org/wiki/Alan_Turing
http://en.wikipedia.org/wiki/Benoit_Mandelbrot

From pursuing the aesthetics of this set we, here at fractalforums.com, were able to participate in a remarkable and historical construction process that lead to the revelation of the Mandelbulb. This has spawned several important fractal generators, but none more significant than the Fractint movement that underpins all the work done to generate a z-plot generator, a Cartesian reference frame surface generator.
http://en.wikipedia.org/wiki/Fractint
Certain standards are now possible, but not necessary. The fluidity of the situation should continue as a mark of the actuality, well described by Herakleitos/Plato: All things change, nothing remains still!
http://en.wikipedia.org/wiki/Herakleitos

Let no one fail to recognise what has played out here amongst the members of this forum, nor to be proud of being associated with such a groundswell of collaboration from all levels and nations in the human community.

Steve Jobs has been commemorated by Apple this week. May we commemorate the passing of Benoit Mandelbrot in the days to come.
<a href="http://www.youtube.com/v/LemPnZn54Kw&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/LemPnZn54Kw&rel=1&fs=1&hd=1</a>

[Alef]

Result of quasz looks a bitt distorted. Linear looks of Chaos Pro probably means that render is mathematicaly correct. When I made errors in whatever formula it allways distorted the result.

IMHO, He was the greatest:
http://en.wikipedia.org/wiki/Pythagoras

[Alef]

A had idea about iterating hybridising two brot fractals by mean of means.

Arithmetic mean were strange, but harmonic mean worked well and unlike geometric mean formula of harmonic mean have only quaternion friendly functions.
Result seems to be mean like Cthuhu;) First were initialisated by z=0, second by z=pixel, alsou smoother had fever iterations.

//single iteration
//your brot
zzx=sqr(zx) - 2*zy*zz + Cx;
zzy= - sqr(zz) + 2*zx*zy + Cy;
zzz= sqr(zy) + 2*zx*zz + Cz;

//my brot
      zz=abs(zz);
      zzx2= sqr(zx) - sqr(zy) + 2*zz*zx  + Cx;
      zzy2=  2*zx * zy + 2* zz * zy   + Cy;
      zzz2= sqr(zz) - sqr(zy)  + abs(Cz);


z1= quaternion ( zzx , zzy, zzz, 0);   
z2= quaternion ( zzx2 , zzy2, zzz2, 0);   

   
/// harmonic mean of both
      z3=2/( 1/(z1)+ 1/(z2) );
      

      zx=real(z3);
      zy=imag(z3);
      zz=part_j(z3);         
      z=z3;   

[jehovajah]

Thanks, Asdam.

The differences between Chaos Pro and Quasz come down to differences in Laplacian or Lagrangian constraints. Viewing a fractal generator as a dynamic Lagrangian or Laplacian is useful only if you carry it to its logical extent. Therefore the criteria I use is not "mathematical" correctness, but system congruity. Anyhow, I am not a big fan of " mathematics" per se, since from  the early 19th century it seems to have lost its way.

Your renders and sculptures are every bit as valuable as mine or anybody else's for that matter, and they certainly helped me visualise the formulae.

On the matter of your hybrids, they are very lovely forms, but they lose there authenticity through the quaternion division operation, at least for the Newtonian triples. This is because Newtonian triples are not Quaternions!  I am hijacking the machinery of the quaternion generators to represent non quaternion math visually, but it only works as long as I do not engage the underlying quaternion math.

By specifying which element is conjuncted with which and how it is conjuncted, I impress a foreign outcome on a visualisation platform. Consequently I have to accept some distortion in the images.

To attempt to code a roots of unity modulo 6 generator is something I have no great inclination to do, especially as it will not give me any results that are radically different from the ones I am achieving by baudlerizing the Quaternion reference frame.

<a href="http://www.youtube.com/v/ONZX9GeeygY&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/ONZX9GeeygY&rel=1&fs=1&hd=1</a>
<a href="http://www.youtube.com/v/ONZX9GeeygY&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/ONZX9GeeygY&rel=1&fs=1&hd=1</a>
The quaternion reference frame is the most sophisticated frame we currently use, although Grassmann's Ausdehnungs Groesse are more sophisticated but not implemented in code yet.
 
http://www.ieoie.nl/ produced the animation. To me it reveals the roots of unity.

John Browne is slowly constructing such a code for Mathematica, but how he is progressing I do not currently know.
https://sites.google.com/site/grassmannalgebra/

[jehovajah]

I can confirm Asdam's sculpture of the cubic Newtonian triple, but not the quadratic. The extruded form is in fact a short extrusion, and the flat surface is not due to slicing, but the cubics in x, y and z.
I have checked the code for the quadratic several times, but cannot find any mistakes.

[jehovajah]

http://www.skytopia.com/project/fractal/2mandelbulb.html#epilogue

On this anniversary of Benoit's death, time to take stock of the impact of the catalytic effect of his dogged insistence on the fractality of geometry.

There is a lot more to this notion than just the iconic Mandelbrot Set. This is the time when mathematics was transformed from a dark brown rigid crysalis into a beautiful and flexible butterfly. To do so it had to break through the crust of fuddy duddy mathematicians, the establishment board of Mathematicians, and change into the board of computer scientists!. Mathematicians have never really recovered from that sweeping computational change, nor should they! For God sake do not let the mathematicians of old ever regain control!

I can remember when the Four colour problem was solved! The amount of muttering among traditional mathematicians was huge! I also remember when calculators were allowed into schools, the rumblings that went on then!

As a mathematical student all my life I know how crippling the board of Mathematicians have been. I know how the Hardys have marvelled at the free spirited Raminujans, how the Gauss's have trampled on the Grassmann's and the Lewis Carols have mocked the Hamilton's.

While I was a student in Manchester, the computational department was the poor cousin of the number theorists, who were consigned to endless computings of orders of inaccuracy of computer results to vindicate their colleagues belief in the superiority of the old school! It was trumpeted how old school could give exact solutions while computers could only give approximate! That all changed with the 4 colour solution!

Computational alternatives could be churned through with relentless machine efficiency. Combinatorial results beyond any one mans ability could be arrived at in a number of weeks, then days, then hours and now seconds!

Our arrogance has blinded us as mathematicians from the brute facts: someone centuries ago made computation into a cash cow, called it mathematics and has been defending it ever since! The need to compute, compare proportion is an artistic need, not confined to so called Mathematicians. The development of computer science has enabled that basic need to be met in ways that any individual artist can apprehend.

Today, computer graphics and films and apps owe a tremendous heritage tax to Benoit's pioneering work, Hamilton's promotion of the Mathesis of the imaginary and the development of abstract algebra, and Grassmann's thorough analysis of the theory of Platonic forms in terms of a process algebra that underpins all computation. These 3 in particular are a gateway to philosophers and thinkers and Astrologers of the past since star charts began as cave paintings!

Today I downloaded an app that gives me access to the star charts that began it all! Not only can I see stars now or in the future, but also in the past back to the times when cave paintings were the only records! This is the true legacy of our philosophers and artisans: we have discoverd those elements of constancy even in dynamism that make sense of our experience of our universe, and the fractal, iterative patterns and sequences that encode them.

[Alef]

Looking to video linking to your video.
<a href="http://www.youtube.com/v/21tZI3Hc12E&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/21tZI3Hc12E&rel=1&fs=1&hd=1</a>
This is quind of funny. No wisdom, but with a time b-> v

The Merkava (Hebrew:  מרכבה Chariot) is a main battle tank used by the Israel Defense Forces.
http://en.wikipedia.org/wiki/Merkava


Down with the offtop. I think, my renders should be exact correspondence becouse square and cube version have the same special features. If there were some error, they would not have same features, or would have some sort of distortion like with my first renders of cube version.

[jehovajah]

The Merkaba is not off topic. The topic is about the "mystical", mysterious "roots of unity"

I would have named the topic Hamiltons missing Triples, but really it does not do justice to his attribution to De Moivre and Cotes. Ieoie is also a phonic transliteration of YHWH, which i am sure Rob Hermans knows very well.

This topic is to pursue the many wonderful reference frame structures which can be made from the roots of unity, using a Grassmann analytical method. Hamilton himself recognised that Grassmann was the master in this analytical field. Un fortunately, due to what i can lightly describe as "Gaussian Smoothing" Grassmann was never properly acknowledged publicly until Riemann"s death.
It takes some considerable effort to root out the instilled confusion over the use of the notion of the √-1. In the end it is a mental state of beliefs and manipulation rituals that one comes down to.

I am still exploring this under the heading of The Children of Shunya.

I am of course interested in your renders, and have no qualms about their correctness. I am using them as an aesthetic guide and prompt, and they may inspire me to find mistakes i have repeatedly overlooked. But i am pleased also with my results because they intuitively confirm what i imagined. This may be why i cannot see any mistakes, if any.
At the moment i put it down to the Lagrangian constraints on the 2 differing systems. Where they agree is in the π/4 orientation of the mandelbrot outline, which for me is a remarkable insight into the mechanics of this form.

I have found an A.C.Pickover formulation in Quasz that gives the most stunning Mandelbrot outline when viewed from the side, but the 3d view reveals the disposition of the particles in this spherical distribution in the YZ plane. The degrees of freedom in real space is what we have to expect in the 3d Mandelbrot, but also it gives hope that some arrangement and processing will reveal a holy Grail!

[Alef]

Interesting enought that one of greatest scientist of all times Newton were just a part time scientists, and that on another part he were an alchemist. Just none knows becouse at his time this practices had to be hidden, and at the modern time that would spoil materialistic image of Newton.



I was thinking, maybe 4D spacetime is like like iterated 3D mandelbrot. 3 spatial dimensions corresponds to 3D screen. The 4th dimension of time is unusal, as it moves just in one direction of increased entropy and less energy. This corresponds iterations, iteration number moves in one direction of increased fractal compexity. 0th iteration corresponds to time before creation. 1st dimension is like Big Bang, an act of creation. With next iterations one can notice swirls of galaxies with each new iteration growing more complex but thinner and thinner.

[jehovajah]

I think I have the division coefficients for the Newtonian triples, indicating that they are a field. When I have worked through my current blogposts on polynomials I shall write them out here.

They are based on the cubic, which makes better sense of the extruded mandelbrot solution for the cubic. It makes the square result also interesting and inclines me to the view that the images I have produced on Quasz are along the right lines, if the division coefficients are correct.

[jehovajah]

Interesting enought that one of greatest scientist of all times Newton were just a part time scientists, and that on another part he were an alchemist. Just none knows becouse at his time this practices had to be hidden, and at the modern time that would spoil materialistic image of Newton.

I think that Newton looked into many things. In the Cromwellian climate he lived under Alchemy was not as disreputable as it apparently was in the Catholic dominions. Certainly Robert Boyle had to be very careful hoe he presented his works on electricity and magnetism!

However, apart from the serious charge of occultism, I think Newton was free to investigate Alchemy as a science, within the remit of the Royal Society. One of the remarkable things about Principiae was how studiously he avoided going into chemical detail with regard to matter, and how cleverly he avoided being drawn on the causes of Gravity, for fear of being charged with occultism!

Cotes in fact distinguishes Newton's general approach as being free of occult expositions. However, within the Royal Society, there was a benign view on magnetism , and you find Newton uses it as a metaphor for Gravity. This is not his thinking only, but echoes Boyles dissertations on the same subject.

However when directly asked what the cause of gravity was, Newton refused to be drawn!
I on the other hand have no such qualms!

[Alef]

If Newtonian Triples do constitute a field and there are division, raising in cube and square powers, this mean, that there could be implemented other formulas.
Talis z=z^2/(z+1)+c
 
Or spider:
p = pixel; z = c; c = p;
z=sqr(z)+c+c;
c=0.5*c+z;

Maybe it will create something nice looking with lots of swirls in all directions.

They actualy were afraid not so about religion, but if someone would manage to turn lead into gold, it would ruin economics of monarchs

[jehovajah]

For me Newton, De Moivre and Cotes are the most significant triple in modern mathematical science.

N(x,y,z) = xñ + yč + z$

Where n is the unity in honour of Newton
Č is the Cotes root of unity and $ is the De Moivre root of unity
Č$ = -ñ, č*-ñ= -č and $*-ñ =-$, č²=$ , $²=-č

N²(x,y,z)= x²ñ - z²č +y²$ + 2*( -yzñ + xyč+ xz$)

It is clear that these are not quaternions, but they demonstrate that quaternions are based on the roots of unity modulo 8, these  being based on roots of unity modulo 6.
 As my computer is down at the moment I wonder if anyone would be so kind as to code this up and render a mandelbulb for me. Just replace ñ,ç,$ by 1,i,j using the given transform.
The formula for higher powers I will discuss later.
For the theoretical background and a bit of historical blather see my blog

Modulo 6 for the roots of unity mean the circle is divided into 6 sectors. Thus i can enscribe in one sector an equilateral triangle and that enables me to read off the root of unity.

Actually i have a desktop image that gives me them anyway.
č=(1/2,√3/2)=1/2+i√3/2
conj(č)=(1/2,-√3/2)=1/2-i√3/2=-$

The division in analogy with the dual complex notation is going to be based on the factorisation of x³+y³+z³
The corresponding factorisation is going to be more involved, more polynomial.
let r=(y³+z³)^1/3

x³+y³+z³=(x+r)(x–čr)(x–conj(č)r)

Compare this pattern with the pattern for x²+y²

x²+y²=(x+iy)(x+conj(i)y)

Thus we can expect the same pattern when we factorise r

r=(y³+z³)^1/3={(y+z)(y–čz)(y–conj(č)z)}^1/3

finally, in terms of the axes labels or vectors or even Strecken
r={(y+ñz)(y–čz)(y+$z)}^1/3


x³+y³+z³=(x+ñr)(x–čr)(x+$r)
I have chosen not to label the leading term in each bracket with ñ to provide a clearer mnemonic, but all coefficients are attached to labels!

Applying these factors appropriately should estabish the denominator  of a division as a real value, i just need to check what state it leaves the numerator in.

[jehovajah]

A different tack.

The reciprocal of N(x,y,z) = 1/(xñ + yč + z$) = e₀ñ + f₀č + g₀$

Thus 1=(e₀ñ + f₀č + g₀$)*(xñ + yč + z$)

That is 1ñ + 0č + 0$ ={xe₀ -z f₀ -y g₀}ñ +{ye₀ + xf₀ - zg₀}č + {ze₀+ yf₀ + xg₀}$
Solving the simultaneous equations for the coefficients
e₀=z(x²+zy)/((z²+yx)²+(-y²+zx)(x²+yz))
f₀=z(z²+xy)/((z²+yx)²+(-y²+zx)(x²+yz))

From which g₀ has 3 versions

g₀=(ze₀+yf₀)/-x =(ye₀+xf₀)/z=(xe₀-zf₀-1)/y

Of course this method could be used to solve for N(0,1,0) and N(0,0,1) and so raises fundamental questions about the notion of "number" and group theoretical structures called "number fields", questions which are usually just swept under the carpet! That is not to say they are not explored, but rater they are not allowed to revise the notion of number, or even to falsify it.
I prefer the notion of magnitude and for quantity :bound magnitude. And for process algebra:Conjugacy and Adjugacy in a sequential dynamic.
The process * here is often confused with multiplication. However that should be said in another, more revealing way: the process of multiplication is confused with the combinatorial process  *. Until the process called multiplication is properly distinguished as factorisation bonds this is not at all obvious due to pedagogical miseducation. Factorisation bonds in fact relate better to Euclid's commensurability algorithm (GCD) in which the true combinatorial nature of multipleform is clearly revealed. See my blog for further details.