Mitch Richling: Multibrot Fractal

Author:	Mitch Richling
Updated:	2024-10-31

1. Introduction
2. When $k$ is not an integer
3. When $k$ is an integer
4. References

1. Introduction

The Multibrot Fractal is a family of Mandelbrot'alikes in that the generating functions are a minor modification of the one used for the Mandelbrot Set:

$\begin{array}{rclr} (1) & f (z) & = & z^{2} + c & Mandelbrot \\ (2) & f (z) & = & z^{k} + c & Multibrot \end{array}$

Notice the only difference is the power, $k$ . Also note that $k$ need not be an integer.

We render Multibrot Fractals using the $L$ function just like we did with the Mandelbrot Set. The general idea is to map an array of pixels making up the raster image to a rectangular region of the complex plane, and color each pixel of the image based upon a color scale for the computed value of the $L$ function.

2. When $k$ is not an integer

When $k$ not an integer, we interesting results. I think the most instructive way to look at them is to make a movie for values of $k$ from about $0.9$ up to $4$ . About 200 frames captures the evolution quite nicely. The following bit of C++ will draw the frames, which may then be rendered into a movie.

#include "ramCanvas.hpp"

int main(void) {
  std::chrono::time_point<std::chrono::system_clock> startTime = std::chrono::system_clock::now();
  const double MINPOW = 0.9;
  const double MAXPOW = 4.0;
  const int    NUMFRM = 192;
  const int    NUMITR = 255;
  const int    IMXSIZ = 7680/8;
  const int    IMYSIZ = 4320/8;

  mjr::ramCanvas3c8b theRamCanvas(IMXSIZ, IMYSIZ, -2.75, 2.75, -1.55, 1.55);

  for(int f=0;f<NUMFRM;f++) {
    double p = MAXPOW*static_cast<double>(f)/static_cast<double>(NUMFRM) + MINPOW;
    std::chrono::time_point<std::chrono::system_clock> frameStartTime = std::chrono::system_clock::now();
    theRamCanvas.clrCanvasToBlack();
    for(int y=0;y<theRamCanvas.getNumPixY();y++) {
      for(int x=0;x<theRamCanvas.getNumPixX();x++) {
        std::complex<double> c(theRamCanvas.int2realX(x), theRamCanvas.int2realY(y));
        std::complex<double> z(0.0, 0.0);
        int count = 0; 
        while((std::norm(z)<4) && (count<=NUMITR)) {
          z=std::pow(z, p) + c;
          count++;
        }
        if(count < NUMITR)
          theRamCanvas.drawPoint(x, y, mjr::ramCanvas3c8b::colorType::csCColdeFireRamp::c(static_cast<mjr::ramCanvas3c8b::csIntType>(count*20)));
      }
    }
    theRamCanvas.writeTIFFfile("multibrotMovie_" + mjr::math::str::fmt_int(f, 3, '0') + ".tiff");
    std::chrono::duration<double> frameRunTime = std::chrono::system_clock::now() - frameStartTime;
    std::cout << "Frame " << f << " Runtime " << frameRunTime.count() << " sec" << std::endl;
  }
  std::chrono::duration<double> runTime = std::chrono::system_clock::now() - startTime;
  std::cout << "Total Runtime " << runTime.count() << " sec" << std::endl;
}

The above program will generate 192 frames of a movie which we can assemble into a GIF:

3. When $k$ is an integer

When $k$ is an integer, we see nice symmetric results. The following bit of C++ will draw the cases for $k$ from $1$ up to $9$ .

#include "ramCanvas.hpp"

int main(void) {
  std::chrono::time_point<std::chrono::system_clock> startTime = std::chrono::system_clock::now();
  const int MINPOW = 2;
  const int MAXPOW = 9;
  const int NUMITR = 1024;
  const int IMXSIZ = 7680/2;
  const int IMYSIZ = 4320/2;

  mjr::ramCanvas3c8b theRamCanvas(IMXSIZ, IMYSIZ, -2.75, 2.75, -1.55, 1.55);

  for(int p=MINPOW;p<=MAXPOW;p++) {
    std::chrono::time_point<std::chrono::system_clock> frameStartTime = std::chrono::system_clock::now();
    theRamCanvas.clrCanvasToBlack();
    for(int y=0;y<theRamCanvas.getNumPixY();y++) {
      for(int x=0;x<theRamCanvas.getNumPixX();x++) {
        std::complex<double> c(theRamCanvas.int2realX(x), theRamCanvas.int2realY(y));
        std::complex<double> z(0.0, 0.0);
        int count = 0; 
        while((std::norm(z)<4) && (count<=NUMITR)) {
          z=std::pow(z, p) + c;
          count++;
        }
        if(count < NUMITR)
          theRamCanvas.drawPoint(x, y, mjr::ramCanvas3c8b::colorType::csCColdeFireRamp::c(static_cast<mjr::ramCanvas3c8b::csIntType>(count*20*log(p))));
      }
    }
    //theRamCanvas.autoHistStrech();
    theRamCanvas.writeTIFFfile("multibrotSnaps_" + mjr::math::str::fmt_int(p, 3, '0') + ".tiff");
    std::chrono::duration<double> frameRunTime = std::chrono::system_clock::now() - frameStartTime;
    std::cout << "Frame " << p << " Runtime " << frameRunTime.count() << " sec" << std::endl;
  }
  std::chrono::duration<double> runTime = std::chrono::system_clock::now() - startTime;
  std::cout << "Total Runtime " << runTime.count() << " sec" << std::endl;
}