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Mitch Richling: Phoenix Fractal

Author: Mitch Richling
Updated: 2024-09-29

phoenix_00_2.png phoenix_04_2.png phoenix_01_2.png

Table of Contents

1. Introduction

The Phoenix Fractal is given by the following iteration:

\[ z_n = z_{n-1} + c + p\cdot z_{n-1} \]

Where \( c \) and \( p \) are both complex parameters we may set as we wish.

This formula is iterated for each point in the complex plane (z) with the initial conditions:

\[ \begin{align*} z_{-1} & = \Im(z) + \Re(z)\cdot i \\ z_{-2} & = 0 \end{align*} \]

An exterior distance estimator is given by:

\[ d_{n} = 2\cdot d_{n-1}\cdot z_{n-1} + p\cdot d_{n-2} \]

With the initial conditions:

\[ \begin{align*} d_{-1} & = 1 \\ d_{-2} & = 0 \end{align*} \]

2. Static Image: Escape Time

These fractals are generated in the same manner as Mandelbrot Julia Sets.

2.1. Code

#include "ramCanvas.hpp"

typedef mjr::ramCanvas3c8b::colorType ct;

std::vector<std::array<double, 9>> params {
  /*       cr,       ci,       pr,       pi,    k, x-min, x-max, y-min, y-max */
  {  0.566700,  0.00000, -0.50000,  0.00000, 10.0, -1.35,  1.35, -1.35,  1.35}, //  0 
  {  0.544992,  0.00000, -0.47000,  0.00000, 10.0, -1.35,  1.35, -1.35,  1.35}, //  1
  {  0.269000,  0.00000,  0.00000,  0.01000,  5.0, -1.10,  1.10, -1.00,  1.00}, //  2
  { -0.400000,  0.10000,  0.29550,  0.00000, 10.0, -1.10,  1.10, -1.50,  1.50}, //  3
  {  0.400000,  0.00000, -0.25000,  0.00000, 10.0, -1.30,  1.20, -1.00,  1.00}, //  4
  {  0.100000,  0.60000, -0.35000,  0.00000, 10.0, -1.30,  1.30, -1.30,  1.30}, //  5
  {  0.400000,  0.40000,  0.20500,  0.00000, 30.0, -1.30,  1.30, -1.30,  1.30}, //  6
  {  0.400000,  0.50000,  0.20500,  0.00000, 30.0, -1.30,  1.30, -1.30,  1.30}, //  7
};

int main(void) {
  std::chrono::time_point<std::chrono::system_clock> startTime = std::chrono::system_clock::now();
  const int    WIDTH  = 1920*4;
  const int    HEIGHT = 1920*4;
  const int    NUMITR = 500;
  const double MAXZ   = 4.0;

# pragma omp parallel for schedule(static,1)
  for(decltype(params.size()) j=0; j<params.size(); ++j) {
    const std::complex<double> c(params[j][0], params[j][1]);
    const std::complex<double> p(params[j][2], params[j][3]);
    mjr::ramCanvas3c8b theRamCanvas(WIDTH, HEIGHT, params[j][5], params[j][6], params[j][7], params[j][8]);
    for(int y=0;y<theRamCanvas.getNumPixY();y++) {
      for(int x=0;x<theRamCanvas.getNumPixX();x++) {
        std::complex<double> z1(theRamCanvas.int2realY(y), theRamCanvas.int2realX(x));
        std::complex<double> z2(0.0, 0.0);
        int count = 0;
        while((std::norm(z1)<MAXZ) && (count<=NUMITR)) {
          std::complex<double> z = z1*z1+c+p*z2;
          z2 = z1;
          z1 = z;
          count++;
        }
        if(count < NUMITR)
          theRamCanvas.drawPoint(x, y, ct::csCCfractal0RYBCW::c(static_cast<ct::csIntType>(count*params[j][4])));
      }
    }
    theRamCanvas.writeTIFFfile("phoenix_" + mjr::fmtInt(j, 2, '0') + ".tiff");
    std::cout << "ITER(" << j <<  "): " << "DONE" << std::endl;
  }

  std::chrono::duration<double> runTime = std::chrono::system_clock::now() - startTime;
  std::cout << "Total Runtime " << runTime.count() << " sec" << std::endl;
}

Source Link: phoenix.cpp

The above program will generate several images includeing this classic:

phoenix_00_10.png

3. Static Image: Distance Estimator (Outside)

3.1. Code

#include "ramCanvas.hpp"

typedef mjr::ramCanvas3c8b::colorType ct;

std::vector<std::array<double, 9>> params {
  /*       cr,       ci,       pr,       pi,     k, x-min, x-max, y-min, y-max */
  {  0.566700,  0.00000, -0.50000,  0.00000, 100.0, -1.35,  1.35, -1.35,  1.35}, //  0 
  {  0.544992,  0.00000, -0.47000,  0.00000, 100.0, -1.35,  1.35, -1.35,  1.35}, //  1
  {  0.269000,  0.00000,  0.00000,  0.01000, 100.0, -1.10,  1.10, -1.00,  1.00}, //  2
  { -0.400000,  0.10000,  0.29550,  0.00000, 100.0, -1.10,  1.10, -1.50,  1.50}, //  3
  {  0.400000,  0.00000, -0.25000,  0.00000, 100.0, -1.30,  1.20, -1.00,  1.00}, //  4
};

int main(void) {
  std::chrono::time_point<std::chrono::system_clock> startTime = std::chrono::system_clock::now();
  const int    WIDTH  = 1920*4;
  const int    HEIGHT = 1920*4;
  const int    NUMITR = 500;
  const double MAXZ   = 4.0;

# pragma omp parallel for schedule(static,1)
  for(decltype(params.size()) j=0; j<params.size(); ++j) {
    const std::complex<double> c(params[j][0], params[j][1]);
    const std::complex<double> p(params[j][2], params[j][3]);
    mjr::ramCanvas3c8b theRamCanvas(WIDTH, HEIGHT, params[j][5], params[j][6], params[j][7], params[j][8]);
    for(int y=0;y<theRamCanvas.getNumPixY();y++) {
      for(int x=0;x<theRamCanvas.getNumPixX();x++) {
        std::complex<double> d1(1.0, 0.0);
        std::complex<double> d2(0.0, 0.0);

        std::complex<double> z1(theRamCanvas.int2realY(y), theRamCanvas.int2realX(x));
        std::complex<double> z2(0.0, 0.0);
        int count = 0;
        while((std::norm(z1)<MAXZ) && (count<=NUMITR)) {
          std::complex<double> z = z1*z1+c+p*z2;
          std::complex<double> d = 2.0*d1*z1+p*d2;
          z2 = z1;
          z1 = z;
          d2 = d1;
          d1 = d;
          count++;
        }
        if(count < NUMITR)
          theRamCanvas.drawPoint(x, y, ct::csCCfractal0RYBCW::c(static_cast<ct::csIntType>(dst*params[j][4])));
      }
    }
    theRamCanvas.writeTIFFfile("phoenixD_" + mjr::fmtInt(j, 2, '0') + ".tiff");
    std::cout << "ITER(" << j <<  "): " << "DONE " << mxd << std::endl;
  }

  std::chrono::duration<double> runTime = std::chrono::system_clock::now() - startTime;
  std::cout << "Total Runtime " << runTime.count() << " sec" << std::endl;
}

Source Link: phoenixD.cpp

4. Static Image: Distance Estimator (Inside)

4.1. Code

#include "ramCanvas.hpp"

typedef mjr::ramCanvas3c8b::colorType ct;

std::vector<std::array<double, 9>> params {
  /*       cr,       ci,       pr,       pi,      k, x-min, x-max, y-min, y-max */
  {  0.566700,  0.00000, -0.50000,  0.00000,  300.0, -1.35,  1.35, -1.35,  1.35}, //  0 
  {  0.544992,  0.00000, -0.47000,  0.00000, 1000.0, -1.35,  1.35, -1.35,  1.35}, //  1
  { -0.400000,  0.10000,  0.29550,  0.00000,  300.0, -1.10,  1.10, -1.50,  1.50}, //  2
};

int main(void) {
  std::chrono::time_point<std::chrono::system_clock> startTime = std::chrono::system_clock::now();
  const int    WIDTH  = 1920*4*4;
  const int    HEIGHT = 1920*4*4;
  const int    NUMITR = 500;
  const double MAXZ   = 6.0;

# pragma omp parallel for schedule(static,1)
  for(decltype(params.size()) j=0; j<params.size(); ++j) {
    const std::complex<double> c(params[j][0], params[j][1]);
    const std::complex<double> p(params[j][2], params[j][3]);
    mjr::ramCanvas3c8b theRamCanvas(WIDTH, HEIGHT, params[j][5], params[j][6], params[j][7], params[j][8]);
    for(int y=0;y<theRamCanvas.getNumPixY();y++) {
      for(int x=0;x<theRamCanvas.getNumPixX();x++) {
        std::complex<double> d1(1.0, 0.0);
        std::complex<double> d2(0.0, 0.0);

        std::complex<double> z1(theRamCanvas.int2realY(y), theRamCanvas.int2realX(x));
        std::complex<double> z2(0.0, 0.0);
        int count = 0;
        while((std::norm(z1)<MAXZ) && (count<=NUMITR)) {
          std::complex<double> z = z1*z1+c+p*z2;
          std::complex<double> d = 2.0*d1*z1+p*d2;
          z2 = z1;
          z1 = z;
          d2 = d1;
          d1 = d;
          count++;
        }
        if(std::norm(z1)<MAXZ)
          theRamCanvas.drawPoint(x, y, ct::csCCfractal0RYBCW::c(static_cast<ct::csIntType>(std::log(std::abs(d1)+1)*params[j][4])));
      }
    }
    theRamCanvas.scaleDownMean(4);
    theRamCanvas.writeTIFFfile("phoenixI_" + mjr::fmtInt(j, 2, '0') + ".tiff");
    std::cout << "ITER(" << j <<  "): " << "DONE " << std::endl;
  }

  std::chrono::duration<double> runTime = std::chrono::system_clock::now() - startTime;
  std::cout << "Total Runtime " << runTime.count() << " sec" << std::endl;
}

Source Link: phoenixI.cpp

5. Making Movies

Becasue the phoenix has a couple free parameters, we can easily make movies by sweeping these parameters through a sequence of values.

5.1. Sweeping p

The following code will sweep \(p\) across a tiny range: 

#include "ramCanvas.hpp"

typedef mjr::ramCanvas3c8b::colorType ct;

int main(void) {
  std::chrono::time_point<std::chrono::system_clock> startTime = std::chrono::system_clock::now();
  const int    WIDTH  = 1920/2;
  const int    HEIGHT = 1920/2;
  const int    NUMITR = 500;
  const double MAXZ   = 4.0;

  const int    NUMFRM = 24*2;
  const double ANGMIN = 0.0;
  const double ANGMAX = std::numbers::pi*2;
  const double RADIUS = 0.0001;

# pragma omp parallel for schedule(static,1)
  for(int frame=0; frame<NUMFRM; frame++) {
#   pragma omp critical
    std::cout << "Frame: " << frame << std::endl;
    double angle = frame*(ANGMAX-ANGMIN)/NUMFRM+ANGMIN;

    const std::complex<double> c(0.566700, 0.00000);
    const std::complex<double> p(-0.50000+RADIUS*std::cos(angle),  0.00000+0*RADIUS*std::sin(angle));
    mjr::ramCanvas3c8b theRamCanvas(WIDTH, HEIGHT, -0.75, -0.5, 0.13, 0.32);

    for(int y=0;y<theRamCanvas.getNumPixY();y++) {
      for(int x=0;x<theRamCanvas.getNumPixX();x++) {
        std::complex<double> z1(theRamCanvas.int2realY(y), theRamCanvas.int2realX(x));
        std::complex<double> z2(0.0, 0.0);
        int count = 0;
        while((std::norm(z1)<MAXZ) && (count<=NUMITR)) {
          std::complex<double> z = z1*z1+c+p*z2;
          z2 = z1;
          z1 = z;
          count++;
        }
        if(count < NUMITR)
          theRamCanvas.drawPoint(x, y, ct::csCCfractal0RYBCW::c(static_cast<ct::csIntType>(count*10)));
      }
    }
    theRamCanvas.writeTIFFfile("phoenixM_" + mjr::fmtInt(frame, 2, '0') + ".tiff");
#   pragma omp critical
    std::cout << "FRAME(" << frame <<  "): " << "DONE" << std::endl;
  }

  std::chrono::duration<double> runTime = std::chrono::system_clock::now() - startTime;
  std::cout << "Total Runtime " << runTime.count() << " sec" << std::endl;
}

Source Link: phoenixM.cpp

The result is the following movie:

phoenixM.gif

5.2. Sweeping both c & p

The following code will sweep \(p\) and \c\) around tiny circles in the complex plane. 

#include "ramCanvas.hpp"

typedef mjr::ramCanvas3c8b::colorType ct;

int main(void) {
  std::chrono::time_point<std::chrono::system_clock> startTime = std::chrono::system_clock::now();
  const int    WIDTH  = 1920/2;
  const int    HEIGHT = 1920/2;
  const int    NUMITR = 40;
  const double MAXZ   = 4.0;

  const int    NUMFRM = 24*2;
  const double ANGMIN = 0.0;
  const double ANGMAX = std::numbers::pi*2;
  const double RADIUS = 0.06;

# pragma omp parallel for schedule(static,1)
  for(int frame=0; frame<NUMFRM; frame++) {
#   pragma omp critical
    std::cout << "Frame: " << frame << std::endl;
    double angle = frame*(ANGMAX-ANGMIN)/NUMFRM+ANGMIN;

    const std::complex<double> c(-0.400000+RADIUS*std::cos(angle),  0.10000+RADIUS*std::sin(angle));
    const std::complex<double> p(0.29550+RADIUS*std::cos(angle),  0.00000+RADIUS*std::sin(angle));
    mjr::ramCanvas3c8b theRamCanvas(WIDTH, HEIGHT, -1.10,  1.10, -1.50,  1.50);

    for(int y=0;y<theRamCanvas.getNumPixY();y++) {
      for(int x=0;x<theRamCanvas.getNumPixX();x++) {
        std::complex<double> z1(theRamCanvas.int2realY(y), theRamCanvas.int2realX(x));
        std::complex<double> z2(0.0, 0.0);
        int count = 0;
        while((std::norm(z1)<MAXZ) && (count<=NUMITR)) {
          std::complex<double> z = z1*z1+c+p*z2;
          z2 = z1;
          z1 = z;
          count++;
        }
        if ((count < NUMITR) && (count > 5))
          theRamCanvas.drawPoint(x, y, ct::csCCfractal0RYBCW::c(static_cast<ct::csIntType>(std::log(count)*110)));
      }
    }
    theRamCanvas.writeTIFFfile("phoenixM2_" + mjr::fmtInt(frame, 2, '0') + ".tiff");
#   pragma omp critical
    std::cout << "FRAME(" << frame <<  "): " << "DONE" << std::endl;
  }

  std::chrono::duration<double> runTime = std::chrono::system_clock::now() - startTime;
  std::cout << "Total Runtime " << runTime.count() << " sec" << std::endl;
}

Source Link: phoenixM.cpp

The result is the following movie:

phoenixM2.gif

6. References

  • S. Ushiki, "Phoenix" in IEEE Transactions on Circuits and Systems, vol. 35, no. 7, pp. 788-789, July 1988, doi: 10.1109/31.1825.

Check out the fractals section of my reading list.

All the code used to generate everything on this page may be found on github.