What are "3D Burning Ship fractals"?

Question

Fractal images from the Mandelbrot set are well known. Plots of the Burning Ship fractal are a bit lesser known but this answer discusses these and others. They can be viewed by iterating the following on the complex plane and recording the value of the index $n$ for the last iteration that each original point remained below some "escape condition".

I've shown some simple representation of the Burning Ship "interesting area" below.

Recently I found some YouTube videos and blogs showing renderings of the "3D Burning Ship Fractal". Many of these are done using software packages for people who just want to make fractal images. There is some discussion and some examples of this in this blog. Also for background a video including the "great explainer" Arthur C. Clarke can be found here.

Question: What are "3D Burning Ship fractals"? Are they mostly the result of graphics techniques to make interesting looking things, or is this something of substantial academic interest. Since the complex pane is 2D, I have a hunch this is more "art" than a mathematical research field, but I don't know.

Mandelbrot: $ \ \ z_{n+1} = z_n^2 + c$

Burning Ship: $ \ \ z_{n+1} = \left( \lvert Re(z_n) \rvert + i\lvert Im(z_n) \rvert \right)^2 + c$

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above: Burning Ship, $-3, -2i$ to $1, 2i$

above: Burning Ship, $-1.85, -0.1i$ to $-1.65, 0.1i$ The "Radio Towers" above this and other nearby ships are generally the source of the most popular fractal image generation.

A Fractal Zoom Video of the Burning Ship: https://www.youtube.com/watch?v=CD9yNFmb2FE

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A video of a "3D Burning Ship Fractal": https://www.youtube.com/watch?v=yaPTk-DqT1g

above x2: Burning Ship "3D Fractals" from here.

Answer 1

All the formulas of the code fragment of M Benesi combined amount to the transition from $p:(x,y,z)$ to $n:(x_+,y_+,z_+)$ for parameter vector $c:(x_c,y_c,z_c)$ \begin{align} x_+&=x^2-y^2-z^2&&+x_c\\ y_+&=-2\sqrt{x^2+y^2}|z|&&+y_c\\ z_+&=-2\frac{|x^2+y^2-z^2|\,|x|\,|y|}{x^2+y^2}&&+z_c \end{align} which does not really a cross-section that would correspond to the "burning ship" formula.

While superficially setting $y=0$ or $z=0$ does reduce the first terms to $(x^2-z^2, -2|xz|,0)$ resp. $(x^2-y^2,0,-2|xy|)$, the order of the terms is wrong, the results do not lie in the same cross-section, and even alternating the cross-sections does not work as that does require $y_c=z_c=0$.

So it is a combination of mathematical operations that results in visually interesting images, but there is no deeper mathematical structure behind that.

Answer 2

The formula (definitely more art- than science-focused in my opinion) seems to have been developed in 2010 by M Benesi on FractalForums. Here's a translation of the second code block in the first post in that thread into Fragmentarium-flavour GLSL:

#define providesInside
#include "Brute-Raytracer.frag"

#include "Complex.frag"

#group BurningShip3D
uniform bool Julia; checkbox[true]
uniform vec3 JuliaC; slider[(-2,-2,-2),(-1.79,0.02,0.05),(2,2,2)]

const float ER2 = 64.0;
const int Iterations = 20;

bool inside(vec3 position)
{
  vec3 c = Julia ? JuliaC : position;
  vec3 p = position;
  for (int i = 0; dot(p, p) < ER2 && i < Iterations; ++i)
  {
    vec2 victor = vec2(p.x, length(p.yz));
    vec2 bravo = vec2(length(p.xy), p.z);
    vec2 cramden = p.xy;
    float r1 = 1.0 / dot(cramden, cramden);
    victor = cSqr(victor);
    bravo = cSqr(bravo);
    cramden = cSqr(cramden);
    float nx = victor.x;
    float ny = -abs(bravo.y);
    float nz = -abs(bravo.x * cramden.y) * r1;
    p = vec3(nx, ny, nz) + c;
  }
  return dot(p, p) < ER2;
}

There may be a distance estimate formula possible which would make ray-tracing images much more efficient, though I don't know it.

Answer 3

Are they mostly the result of graphics techniques to make interesting looking things, or is this something of substantial academic interest.

I used trial and error to discover that formula, while looking for interesting visual results.

Since the complex pane is 2D, I have a hunch this is more "art" than a mathematical research field, but I don't know.

It (the pursuit of fractal art formulas) led me to some engineering ideas that would be useful. Maybe not this formula in particular, but the pursuit is perhaps valuable, even if mathematically generated artwork wasn't considered to be a research field in the past. I do believe that a friend announced that they were in a postgrad program centered on mathematical art generation, but I'd like to ask them more in case I misunderstood what they said.

That said, did I misunderstand what you said, in the sense that you mean theoretical instead of applied by this being a mathematical research field?

The actual 3D BS formula (take |abs| of each component, add in c) was slightly different.