Newtonian Triples

[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$ = -n, č*-n= -č and $*-n =-$, 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

[kram1032]

So what you're saying is:

(x+iy+jz)² = x²-iz²+jy²+2(-yz+xyi+xzj)

so...

Code:

x =  x²-2yz + a
y = -z²+2xy + b
z =  y²+2xz + c

As proposed MSet?

[jehovajah]

Almost.

The labels 1,i,j have several meanings in such a formula. In my case they do not represent the underlying algebra, the coefficients however do. It may be that you have to code it with 1,i,j to get your app to render the kind of image I want to inspect. However, if you do not use i,j in your coding you shoul be ok to use the coefficient rules as you have stated.

I expect the generated image to be distorted into the Cartesian reference frame, but at this stage I do not want to apply correction values to portray exactly how it sits in space. Anyway I have not worked those out yet!

See what you can do with what you have thanks. I will be interested in your results.

[jehovajah]

So what you're saying is:

(x,y,z)² = x²-iz²+jy²+2(-yz+xyi+xzj)

so...

Code:

Newx =  x²-2yz + Cx
Newy = 2xy-z² + Cy
Newz = 2xz+ y²+Cz

As proposed MSet?

Just to clarify it for any code warriors not used to this early stage of implementing a new formula.

[jehovajah]

The quaternion 8 group is not isomorphic to the roots of unity modulo 8 because of the non commutativity. This means it is homeomorphic to a related group, possibly the direct sum of the cyclic groups 2 and 4. While the logarithmic addition is commutative, the underlying product is not, making it truely a remarkable insight to combine the group elements precisely that way, and also explaining why it took ten years for Hamilton to realise this combination. There was no group theory then to help him out, and this is why he is, along with Grassmann, generally regarded as the fathers of modern Algebra.

[jehovajah]

My recent blog post explains how Hamilton's quaternions came to be non commutative. I also posit that a commutative version of the quaternions underpins it. Group theoretical structures are referenced, but I would refer the reader to Norman's excellent overview of group theory on his YouTube channel.

The Quaternions are a viable functioning reference frame, so why fix what works, right? Well would you not like them to be commutative, providing they do not mess up the fractals? That is all i am saying! Don't mess with the fractals, baby!

[Alef]

So what you're saying is:

(x+iy+jz)² = x²-iz²+jy²+2(-yz+xyi+xzj)

so...

Code:

x =  x²-2yz + a
y = -z²+2xy + b
z =  y²+2xz + c

As proposed MSet?

Here I rendered m-brot and j-set of formula. Side extruded 3D fractal. All are rendered using Log Trichrome colouring, colour lines are z value gradients during all iterations.

IMHO it needs someting more, maybe some hidden coeffitients or something.
 You could test different formula versions, just download Chaos Pro and place your formula in formula file ( forFormula.cfm in forFormula.zip at the bottom), which actualy are text file. Just change zzx= zx*zx + .... All another needed code alredy are there. It have the most easy to use compiler of all, unless you can use assambler language and edit machine codes, then you can test with m3d;) Or GPU programming with no math libraries;) Of corse it don't means best renders ever.

[Alef]

Here is 2D, m-brot inside and cutout on z=-0.0001. All the same colout method. IMHO, it needs much more.

[jehovajah]

Thanks Asdam! Some great colouring.

I will be able to work with what you have done, but I am very pleased to see the Mandelbrot like cross sections.
I suspect a few things may need adjusting to counter the fractal generator bias toward ijk but Good Job!

[jehovajah]

Ok, I am excited!
My original request was for a 3 d render to produce a mandelbulb. I think Asdams work is based on the notion of extruded fractals, but I am not familiar with Chaospro.

The render I am looking for may have to be placed in a quaternion frame  or a bespoke 3d renderer. What I expected was a twistedlathe image with lots of swirls. This can be controlled by getting the correct parameters even if it flattens the image.
Asdam,Good Job! Thank you so much!

[Alef]

Ok, I am excited!
My original request was for a 3 d render to produce a mandelbulb. I think Asdams work is based on the notion of extruded fractals,mbut I am not familiar with Chaospro.

The render I am looking for may have to be placed in a quaternion frame  or a bespoke 3d renderer. What I expected was a twistedlathe image with lots of swirls. This can be controlled by getting the correct parameters even if it flattens the image.
Asdam,Good Job! Thank you so much!

There are lots of swirls and twists, but only in single plane and in XY cutout;)
If you want image with twisted swirls you'll need triplex math or maybe trigonometry. Or more complex polynomials;)

Best so far what I got using quadratic polynomials and modulus function is this:

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

zzx = zx*zx - zy*zy + 2*zz*zx  + Cx;
zzy = 2* zx * zy + 2* zz * zy   + Cy;
zzz= zz*zz - zy*zy  + abs(Cz);

z= quaternion(zzx, zzy, zzz, 0);

[jehovajah]

There are lots of swirls and twists, but only in single plane and in XY cutout;)
If you want image with twisted swirls you'll need triplex math or maybe trigonometry. Or more complex polynomials;)

Thanks again Asdam. Interesting form, but not what I expected to see from my formula. However your comments pinpointed the notation issue. I am used to Quasz so xyz are real and imaginary coefficients of 1,i,j!

Try
zx^2-2*zy*zz + cx=zzx
2*zx*zy-zz^2 +cy=zzy
Zy^2+2*zx*zz +cz =zzz
In the quaternion bracket.
Please allow zz to have its full variability.

The swirls in one plane refer to the extrusion render done previously, or have you other mandelbulbs not published?
Thanks for your support 

[M Benesi]

  Hey there.   Ended up with something very similar to Asdam's version.  Basically a very thick 2d Mandy.

[jehovajah]

  Hey there.   Ended up with something very similar to Asdam's version.  Basically a very thick 2d Mandy.

Hiya Matt! Howya doing?

Yes this happens sometimes with me to! I remember when I first attempted to make a 3d version of Fracmonks formula. The important thing is the relationship between the coefficients and the frame used to display those coefficients. And then you do have to fiddle a bit .
Asdam's quaternion offering is very exciting, because this is what I might get with Quasz when I get my computer back.
It is such a long time since we did this we forget that simple mistakes in programming our fractal generators cause unexpected results, and differences in notation too!
Like your new work by the way, and I think it is important.

Thanks again guys

Just an update, while i am analysing your results: I think I might try z=z^3 +c this being modulo 6 and all! Catchya later. Lol!

[jehovajah]

Restrict y and z to<<1 to pick out Mandy's.

Working on cubic now.

RIP computer! repair man told me it died.