Print Page - ix: possibly the holy grail fractal (in fff lore)

Title: ix: possibly the holy grail fractal (in fff lore)
Post by: mfg on March 28, 2016, 05:51:36 PM

The generalization of the quadratic iteration in complex algebra to three or higher dimensions has been an elusive longstanding quest.

The number class sequence: natural → integer → rational → real → complex can be extended to higher dimensions in different ways.
The foremost group property that is lost in higher dimensional division algebras is commutativity. Such is the case in (four dimensional) quaternion algebra and more generally Clifford algebras.
In higher dimensions, the associative property is no longer satisfied. Such is the case in the octonion normed algebra.
In dimensions other than 1,2 or 4, the structure is no longer a division algebra, that is, there are elements without multiplicative inverse.
Some of these algebras give interesting confined sets under the quadratic iteration. However, they fall short from the richness obtained in the (two dimensional) complex plane.

However, a property that has been mostly left untouched is distributivity of the product over addition.
Imaginary scator algebra is a non-distributive extension of complex numbers to higher dimensions.
The product is not bilinear but it is well defined in a subset of the 1+n dimensional space.
It is equipped with an order parameter that in addition to the sum of the squared components it also involves terms with the inverse squared of the scalar component.
To put it in plain terms, the bailout condition in fractal generation has to be fundamentally modified.
A three dimensional rendering of the bound set under the quadratic iteration reveals a highly complex structure of its boundary.

(http://c2i0E-1+2(-0.6;.03,-.15)(-.8;-1.65,1.65)1920x1080.jpg)

The imaginary scators bound set in parameter space is codenamed 'ix' (pronounced 'eesh' in english)

1/ cont.

Title: Re: ix: possibly the holy grail fractal (in fff lore)
Post by: mfg on March 28, 2016, 06:17:51 PM

The ix set has an extremely intricate boundary. Views from different perspectives and/or magnifications yield very diverse images.
In order to have some guidance as to where the rendering is, the following notation is suggested.

Notation used to label visualizations:

$\underset{\textrm{confined quadratic iteration}}{\underbrace{c2i}}\qquad\overset{\textrm{parameter space}}{\overbrace{0}}\qquad\underset{\textrm{scator algebra and dimension}}{\underbrace{\mathbb{S}_{-}^{1+2}}}\qquad\overset{\textrm{fractal location}}{\overbrace{\left(s;x,y\right)}}\qquad\underset{\textrm{viewpoint}}{\underbrace{\left(p_{0};p_{1},p_{2}\right)}}$

(http://c2i0E-1+2(-0.67;.007,-.25)(-1.91;-.002,.3)1920x1080.jpg)

ix: c2i0S-1+2(-0.67;.007,-.25)(-1.91;-.002,.3)

detailed explanation in http://luz.izt.uam.mx/index.html/?q=node/47 (http://luz.izt.uam.mx/index.html/?q=node/47)

2/ cont.

Title: Re: ix: possibly the holy grail fractal (in fff lore)
Post by: mfg on March 28, 2016, 06:37:40 PM

The square of a scator $\overset{o}{\varphi}=s+x\check{\mathbf{e}}_{x}+y\check{\mathbf{e}}_{y}$ , is
$<br />\begin{equation} \overset{o}{\varphi}^{2}=s_{\diamond}+x_{\diamond}\check{\mathbf{e}}_{x}+y_{\diamond}\check{\mathbf{e}}_{y}=s^{2}\left(1-\frac{x^{2}}{s^{2}}\right)\left(1-\frac{y^{2}}{s^{2}}\right)+2sx\left(1-\frac{y^{2}}{s^{2}}\right)\check{\mathbf{e}}_{x}+2sy\left(1-\frac{x^{2}}{s^{2}}\right)\check{\mathbf{e}}_{y}. \end{equation}$

(http://c2i0E-1+2(-0.5;0,.3)(-0.5;-1.6,.38)1920x1080-00slice,y=0,z=0.jpg)
image ix: c2i0S-1+2(-0.5;0,.3)(-0.5;-1.6,.38)

The quadratic mapping for the scalar component in imaginary scator algebra is,

$\begin{equation} s_{m+1}=s_{m}^{2}\left(1-\frac{x_{m}^{2}}{s_{m}^{2}}\right)\left(1-\frac{y_{m}^{2}}{s_{m}^{2}}\right)+s, \end{equation}$

and for the director components, the recurrence relationship is

$\begin{equation} x_{m+1}=2s_{m}x_{m}\left(1-\frac{y_{m}^{2}}{s_{m}^{2}}\right)+x, \end{equation}$

$\begin{equation} y_{m+1}=2s_{m}y_{m}\left(1-\frac{x_{m}^{2}}{s_{m}^{2}}\right)+y. \end{equation}$

The square magnitude of an imaginary scator $\overset{o}{\varphi}$ is

$\begin{equation}\overset{o}{\varphi}\overset{o}{\varphi}^{*}=s^{2}+x^{2}+y^{2}+\frac{x^{2}y^{2}}{s^{2}}. \end{equation}$

With this preamble of imaginary scator algebra, it is possible to perform the quadratic iteration. It is then possible to visualize the bound sets in two or three dimensions as well as the escape velocities of unbound points.

The url http://luz.izt.uam.mx/index.html/?q=node/89 (http://luz.izt.uam.mx/index.html/?q=node/89) has a link to an introductory article about fractals with imaginary scators.

3/ cont.

Title: Relationship of the imaginary scator algebra product with triplex multiplication
Post by: mfg on March 28, 2016, 06:56:09 PM

Recall that the square operation with triplex numbers is given by [White and Nylander (2009) www.fractalforums.com/theory/triplex-algebra/ (http://www.fractalforums.com/theory/triplex-algebra/), Bonzini (2010) www.fractal.org/mbulb-paolo-bonzini.pdf (http://www.fractal.org/mbulb-paolo-bonzini.pdf)]

$\left\{ s,x,y\right\} ^{2}=\left\{ \left(s^{2}-x^{2}\right)\left(1-\frac{y^{2}}{s^{2}+x^{2}}\right),2sx\left(1-\frac{y^{2}}{s^{2}+x^{2}}\right),\mp2y\sqrt{s^{2}+x^{2}}\right\} .$

To facilitate comparisons, the variables in [White and Nylander (2009)www.fractalforums.com/theory/triplex-algebra/ (http://www.fractalforums.com/theory/triplex-algebra/)] have been rewritten with our notation ( $x_{1}=x_{2}\rightarrow s$ , $y_{1}=y_{2}\rightarrow x$ and $z_{1}=z_{2}\rightarrow y$ ). The triplex product definition was inspired on a sequence of rotations together with an Euclidean quadratic form $r=\sqrt{s^{2}+x^{2}+y^{2}}$ . For $x$ much smaller than $s$ , the triplex product in the $x$ -paraxial approximation is

$\left\{ s,x,y\right\} ^{2}\approx\left\{ \left(s^{2}-x^{2}\right)\left(1-\frac{y^{2}}{s^{2}}\right),2sx\left(1-\frac{y^{2}}{s^{2}}\right),\mp2ys\left(1+\frac{x^{2}}{2s^{2}}\right)\right\}.$

Recall that the scator square function is given by

$\overset{o}{\varphi}^{2}=\left(s^{2}-x^{2}\right)\left(1-\frac{y^{2}}{s^{2}}\right)+2sx\left(1-\frac{y^{2}}{s^{2}}\right)\check{\mathbf{e}}_{x}+2sy\left(1-\frac{x^{2}}{s^{2}}\right)\check{\mathbf{e}}_{y}.$

So, as can be seen from direct comparison of the above two expressions, the triplex proposal was not too far off from the imaginary scator product in 1+2 dimensions. It is possible that for this reason, rather intricate surfaces with fractal like properties were obtained with the triplex product.

Details in http://luz.izt.uam.mx/index.html/?q=node/97 (http://luz.izt.uam.mx/index.html/?q=node/97)

4/ cont.

Title: Re: ix: possibly the holy grail fractal (in fff lore)
Post by: mfg on March 28, 2016, 07:15:41 PM

As the readers may have guessed, the renderings shown in this thread were performed with the fantastic programme Mandelbulber v.2.07 modified accordingly to cope with scator algebra. The file modifications are described in the following link
luz.izt.uam.mx/index.html/?q=node/95&language=en (http://luz.izt.uam.mx/index.html/?q=node/95&language=en)

I tried to attach the modified files in this post. Unfortunately, .cpp .hpp .ui and .ts files are not allowed to be uploaded. An example of initial settings .fract file is attached.

Time to play!

Enjoy!

5/ last

Title: Re: ix: possibly the holy grail fractal (in fff lore)
Post by: cKleinhuis on March 28, 2016, 09:53:50 PM

Hey manuel, heavy stuff, had no time to check it out yet, uploading zip files as attachments should work, beside of some visible discontinuities the results look very promising

Title: Re: ix: possibly the holy grail fractal (in fff lore)
Post by: zebastian on March 29, 2016, 12:58:53 PM

hi mfg,

your website is currently broken due to mysql server down or unreachable:
luz.izt.uam.mx/index.html/?q=node/95&language=en

your formula sounds interesting. O0
i can put your code into mainline of buddhis repository or i can give you instructions how to fork the repository and make a pull request. this way you can make updates in the future.

20 formulas sounds a little bit much for scator formula.
in general we try to keep
one conceptual idea --> one formula
if there are applicable dimensions of adjustments we try to parameterize this into a variable adjustable in the ui.
but if they are much different in calculation, they should of course be two distinguishable formulas.

Title: Re: ix: possibly the holy grail fractal (in fff lore)
Post by: mclarekin on March 29, 2016, 02:22:39 PM

@ zebastian

Quote

20 formulas sounds a little bit much for scator formula.

In the infinity of possibilities there is possibly a possibility of more then 20 scator formulas. ;D

I think Manuel is wisely reserving some numbers, to avoid the problem he had between V2.06 and V2.07 where we added formulas using the numbers he had already used. A good idea to have some reserved

space in the list to experiment in, whether or not you eventually add all the formulas to Mandelbulber.

I just want, asap, to get behind the controls and see what the formula(s) do, :) :) :)

Title: Re: ix: possibly the holy grail fractal (in fff lore)
Post by: mfg on March 29, 2016, 03:58:09 PM

Mandelbulber modified files in compressed .zip attachment to compile and visualize 'ix'.

ps. Thank you for your help and kind replies.

Title: Re: ix: possibly the holy grail fractal (in fff lore)
Post by: mclarekin on April 08, 2016, 03:23:24 AM

This is basic formula with no addition of constants. Distance Estimation (deltaDE log) is not good.

Title: Re: ix: possibly the holy grail fractal (in fff lore)
Post by: mclarekin on April 08, 2016, 03:25:18 AM

And here we have a box fold applied.

Title: Re: ix: possibly the holy grail fractal (in fff lore)
Post by: mclarekin on April 08, 2016, 04:45:33 AM

and here is this formula with Cpixel added in three symmetrical ways

Welcome to Fractal Forums

Fractal Math, Chaos Theory & Research => (new) Theories & Research => Topic started by: mfg on March 28, 2016, 05:51:36 PM