Mitch Richling: Collatz Fractal

Author:	Mitch Richling
Updated:	2024-10-31

1. Introduction
2. Algorithms
3. Gallery
4. References

1. Introduction

The Collatz conjecture is a famous unsolved math problem. In studying the Collatz conjecture, mathematicians have defined the Collatz map and it's smooth continuation:

$f (z) = \frac{1}{2} z \cos^{2} (\frac{π}{2} z) + \frac{3 x + 1}{2} z \sin^{2} (\frac{π}{2} z)$

Of course, whenever fractal-heads see a smooth complex function, they want to iterate the thing and make pretty pictures. :)

2. Algorithms

We render the Collatz Fractal 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. Coding up this idea leads to the following C++ program:

#include "ramCanvas.hpp"

int main(void) {
  std::chrono::time_point<std::chrono::system_clock> startTime = std::chrono::system_clock::now();
  const int    NUMITR = 164;
  const int    IMXSIZ = 7680/2;
  const int    IMYSIZ = 7680/2;

  //mjr::ramCanvas3c8b theRamCanvas(IMXSIZ, IMYSIZ, -3.75, 3.75, -3.75, 3.75);            // 56  centered 0+0i
  //mjr::ramCanvas3c8b theRamCanvas(IMXSIZ, IMYSIZ, -0.99-0.15, -0.99+0.15, -0.15, 0.15); // 28  centered near 1+i
  mjr::ramCanvas3c8b theRamCanvas(IMXSIZ, IMYSIZ, -0.99-0.03, -0.99+0.03, -0.03, 0.03);   // 28  same, zoomed in

  for(int y=0;y<theRamCanvas.getNumPixY();y++) {
    for(int x=0;x<theRamCanvas.getNumPixX();x++) {
      std::complex<double> z = theRamCanvas.int2real(x, y);
      int count = 0; 
      while((std::norm(z)<1e10) && (count<=NUMITR)) {
        z = 0.25 * (2.0 + 7.0*z - (2.0 + 5.0 * z) * std::cos(z * std::numbers::pi));        
        count++;
      }
      if(count < NUMITR)
        theRamCanvas.drawPoint(x, y, mjr::ramCanvas3c8b::colorType::csCCsumRGB::c(static_cast<mjr::ramCanvas3c8b::csIntType>(count*28)));
    }
  }
  theRamCanvas.writeTIFFfile("collatz.tiff");
  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 this picture:

3. Gallery

4. References

Check out the fractals section of my reading list.

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

Mitch Richling: Collatz Fractal

Table of Contents

1. Introduction

2. Algorithms

3. Gallery

4. References