#mandelbulb #animation created with #WolframLanguage

#fractal #mandelbrot #math

Cover art for an album that hasn’t been released yet.

#mandelbulb #mandelbulb3d #fractalart #fractal #gimp #gmic #opensource #opensourcetools #mastodonartists #mastodonart

A render of Adam Burch’s A-Wing model. Using a fractal render for the background generated in #mandelbulb. Rendered in #Lightwave3D and pixel poked in Photoshop #StarWars #StarWarsArt

https://math.stackexchange.com/questions/3713960/eigenvalues-of-jacobian-of-mandelbulb-triplex-power-formula
I asked a question about #Mandelbulb #Jacobian #Eigenvalues because I got stuck and #SageMath wasn't helping by spewing thousands of lines of formulas that I struggled to persuade it to simplify.

#Magnet #Mandelbrot set #fractal mashed up with #ThreeD #Triplex algebra (a la #Mandelbulb) rendered with #DualNumber #DistanceEstimation in #FragM fork of #Fragmentarium

My highly over-engineered extravagant framework of shaders including each other multiple times with different things defined (to emulate C++ templates with #GLSL function overloading without polymorphism) takes significantly longer to link the #shader than it does to render the #animation.

First attempts with typos gave 100k lines of cascaded errors in the shader info log, which which the Qt GUI list widget was Not Happy At All. Luckily the log went to stdout too, so I could pipe to a file and see the start where I missed a return statement or two.

Thinking about power p #Mandelbulb #DistanceEstimate. There is a long history of just using the 1/2 |z| log |z| / |z'| formula that is justified by complex dynamics theory (Koebe 1/4 theorem etc) for the 2D Mandelbrot set and just plugging it into non-conformal 3D "triplex" arithmetic, without using a Jacobian matrix. It has even been found that using a Jacobian matrix is harmful, due to excess stretching.

Case (x,y) -> (0,0) at iteration n, r = sqrt(x^2+y^2+z^2) = |z| and it depends on r<>1 whether it will go to infinity (exterior) or 0 (interior) like (0,0,r^p^k) at iteration n+k, which is locally like r->r^p in 1 dimension so the scalar dr := p r^{p-1} dr update is justified.

Case sin(n phi) -> 0 at iteration n,, at the very next iteration (x,y)->(0,0) and z-> r^p and then the previous case applies. This step is also like r->r^p in one dimension, so the scalar dr := p r^(p-1) dr update is justified.

1/2

Using #SageMath instead of #Maxima was more productive (the syntax is much nicer, though I think #Sage may use Maxima under the hood).

Managed to derive the symbolic Jacobian of the #Mandelbulb iteration (#Juliabulb variant, where the `+C` is a constant), and substituted in some numerical values to get the condition number at various points.

The even-power bulbs have singularities at the poles (near x = y = 0) with the condition number increasing without bounds, and the odd-power bulbs also have singularities, e.g. power 3 bulb near $(x = 0, y = t cos(pi/6), z = t sin(pi/6))$.

This can probably be deduced from the Jacobian determinant which I think works out as:
$$ (n r^n)^3 cos(n \phi) log(r/cos(\phi)) / 3$$
When it is 0, problems...

The blank screen seemed to come from the $a$ scaling factor in $z \to a z^p$ being miscalculated; I think that led to overflows in the arguments to the Lambert W function implementation used when $a != 1$ and $p != 1$. Toggling an override switch and forcing $a = 1$ seemed to give me a visible #Mandelbulb but it seems not quite canonical - maybe the non-associative triplex algebra is throwning it off as I compute $z^8$ as $((z^2)^2)^2$. Will try the long-winded way to compare.

#raytracing #mandelbulb #fragmentarium #threedee #fractals

I added light sampling to Raymond: for diffuse opaque surfaces it is vastly more efficient (less noise for given rendering time) than random sampling, at the cost of double the number of rays to trace (an additional one from each surface hit directly towards a light).

Not so beneficial for shiny opaque surfaces in one test. And it breaks rendering of transparent objects (refractive caustics disappear).

https://code.mathr.co.uk/raymond/commitdiff/636d2fc5dc19dbbe0c4c657f63d1f2271f7d6877#patch9

#mandelbulb #autostereogram

got centred pattern fill working

print at 180dpi (A4) and view cross-eyed from about 7 inches

texture is based on https://rdex.mathr.co.uk/kiblix/texture/1324 #reactiondiffusion

Ha ha ha I found the Mandelbulb renderer I wrote back in 2011. :)

#Mandelbulb #Annihilation

I wish I could have suspended my prior knowledge of the Mandelbulb for the duration of the film though.
#Annihilation #Mandelbulb

https://en.wikipedia.org/wiki/Mandelbulb