  Sunday March 26, 2023
 home | about | contact | Donations   Fractals and the Mandelbulb Kenwright

Fractals make you see everything differently. When you first read and learn about fractals, it's like Pandora's box. A hidden magic of unlimited possibilities. Not only are fractals incredibly powerful, but they are also beautiful and fun. In particular, graphical fractals possess infinite detail combined with unpredictability, that is really amazing. Once you get past all of the mathematics and visualization challenges, you'll never look at the world the same again ;)

A particular fractal with interesting visual properties is the Mandelbulb.

## Famous 2D Mandelbrot Equation

2D Mandelbrot equation:

\$z{n+1} = z{n}^2+c\$.

Squaring complex numbers has a simple geometric interpretation: if the complex number is represented in polar coordinates, squaring the number corresponds to squaring the length, and doubling the angle (to the real axis).

Taking the Mandelbrot equation to higher dimensions leads us to what is now known as the Mandelbulb fractal.

```# 2D Cross-section Of (3D) Mandelbrot Fractal import math import random from PIL import Image imgx = 256 # 512 imgy = 256 # 512 image = Image.new("RGB", (imgx, imgy)) pixels = image.load() n = 8 # drawing area (xa < xb & ya < yb) xa = -1.5 xb = 1.5 ya = -1.5 yb = 1.5 maxIt = 256 # 256 # max number of iterations allowed pi2 = math.pi * 2.0 # random rotation angles to convert 2d plane to 3d plane xy = 0.2 * pi2; # random.random() * pi2 xz = 0.2 * pi2; # random.random() * pi2 yz = 0.2 * pi2; # random.random() * pi2 sxy = math.sin(xy) ; cxy = math.cos(xy) sxz = math.sin(xz) ; cxz = math.cos(xz) syz = math.sin(yz) ; cyz = math.cos(yz) origx = (xa + xb) / 2.0 ; origy = (ya + yb) / 2.0 for ky in range(imgy): b = ky * (yb - ya) / (imgy - 1) + ya for kx in range(imgx): a = kx * (xb - xa) / (imgx - 1) + xa x = a ; y = b ; z = 0.5 # 3d rotation around center of the plane x = x - origx ; y = y - origy x0=x*cxy-y*sxy;y=x*sxy+y*cxy;x=x0 # xy-plane rotation x0=x*cxz-z*sxz;z=x*sxz+z*cxz;x=x0 # xz-plane rotation y0=y*cyz-z*syz;z=y*syz+z*cyz;y=y0 # yz-plane rotation x = x + origx ; y = y + origy cx = x ; cy = y ; cz = z for i in range(maxIt): r = math.sqrt(x * x + y * y + z * z) t = math.atan2(math.hypot(x, y), z) p = math.atan2(y, x) rn = r ** n x = rn * math.sin(t * n) * math.cos(p * n) + cx y = rn * math.sin(t * n) * math.sin(p * n) + cy z = rn * math.cos(t * n) + cz if x * x + y * y + z * z > 150.0: break if i > 1 and i <= 50: ss = 128 + int( 128.0 * (i/50.0) ) pixels[kx, ky] = (ss, 0, 0) if i > 50 and i <= 100: ss = 128 + int( 128.0 * ((i-100)/50.0) ) pixels[kx, ky] = (0, ss, 0) if i > 100 and i <= 150: ss = 128 + int( 128.0 * ((i-100)/100.0) ) pixels[kx, ky] = (0, 0, ss) if i == 2: pixels[kx, ky] = (255, 255, 255 ) image.save("Mandelbulb.png", "PNG") ```

Output for the above Python program. Show the cross sectional view of a Mandelbulb. The various depths are shown in different colours.

# MandelBulb Ray Tracer (Stripped)

A minimum working Mandelbulb ray tracing implementation. All you need is a C++ compiler. Of course, you could port it to Python, as it's relatively straightforward, but it would be slow! It's slow in C++, it takes a few seconds to create the image in C++, so it might take a few minutes in Python.

The implementation should hopefully let you see how it all fits together. Also a fun demo for you to tinker with and experiment with as you learn about geometric fractals.

The following implementation creates a ppm image file. You should be able to open this image file in most image editors (e.g., XNView is a free image viewer/formatter if you don't have Photoshop). ``` /* Minimal working (Mandelbulb xbdev.net) PPM image format is a simple and uncomplicated way to generate an image without requiring any external libraries. Easy to open and convert to another format (e.g., open and save as jpg using paint/xnview). */ #define _CRT_SECURE_NO_WARNINGS // constants to tweak and control the fractal output #define BAILOUT 100 #define EPS 0.001 #define mmaxIterations 10 #define maxDetailIter 100 // e.g., try and change this value to 5 instead // and see what the generated output looks like #define myPower 12 #include <math.h> #include <iostream> #include <stdlib.h> #include <stdio.h> using namespace std; #define M_PI 3.14159265358979323846264338327950288 inline double clamp(double x) { return x < 0 ? 0 : x>1 ? 1 : x; } inline double min(double x, double y){ if (x<y) return x; return y; } inline double max(double x, double y){ if (x>y) return x; return y; } inline int toInt(double x) { return int(pow(clamp(x), 1 / 2.2) * 255 + .5); } inline double clamp(const double ff, double lo, double hi) { if (ff > hi) return hi; if (ff < lo) return lo; return ff; } struct vec3 { double x, y, z; // position, also color (r,g,b) vec3(double x_ = 0, double y_ = 0, double z_ = 0) { x = x_; y = y_; z = z_; } vec3 operator+(const vec3 &b) const { return vec3(x + b.x, y + b.y, z + b.z); } vec3 operator-(const vec3 &b) const { return vec3(x - b.x, y - b.y, z - b.z); } vec3 operator*(double b) const { return vec3(x*b, y*b, z*b); } vec3 mult(const vec3 &b) const { return vec3(x*b.x, y*b.y, z*b.z); } vec3& norm() { return *this = *this * (1 / sqrt(x*x + y*y + z*z)); } double dot( const vec3 &b) const { return x*b.x + y*b.y + z*b.z; } vec3 cross (vec3&b) { return vec3(y*b.z - z*b.y, z*b.x - x*b.z, x*b.y - y*b.x); } }; struct vec4 { double x, y, z, w; vec4(const vec3& v, double _w){ x=v.x, y=v.y, z=v.z, w=_w; } vec4(double x_ = 0, double y_ = 0, double z_ = 0, double w_ = 0) { x = x_; y = y_; z = z_; w = w_; } vec4 operator+(const vec4 &b) const { return vec4(x + b.x, y + b.y, z + b.z, w + b.w); } vec4 operator-(const vec4 &b) const { return vec4(x - b.x, y - b.y, z - b.z, w - b.w); } vec4 operator*(double b) const { return vec4(x*b, y*b, z*b, w*b); } vec4 mult(const vec4 &b) const { return vec4(x*b.x, y*b.y, z*b.z, w*b.w); } vec4& norm() { return *this = *this * (1 / sqrt(x*x + y*y + z*z + w*w)); } double dot(const vec4 &b) const { return x*b.x + y*b.y + z*b.z, w*b.w; } }; double dot(const vec3&a, const vec3 &b) { return a.dot(b); } double length(const vec3& v) { return sqrt(v.dot(v)); } double length(const vec4& v) { return sqrt(v.dot(v)); } vec3 normalize(const vec3& v) { double len = length(v); return vec3( v.x/len, v.y/len, v.z/len ); } /* These two functions, MainBulb(..) and IntersectMBulb(..) do all of the work. Responsible for the fractal shape/details */ vec4 MainBulb(vec4 v) { double r = length( vec3(v.x, v.y, v.z) ); if (r>BAILOUT) return v; double theta = acos(clamp(v.z / r, -1.0, 1.0))*myPower; double phi = atan2(v.y, v.x)*myPower; v.w = pow(r, myPower - 1.0)*myPower*v.w + 1.0; double zr = pow(r, myPower); vec3 vv = vec3(sin(theta)*cos(phi), sin(phi)*sin(theta), cos(theta))*zr; return vec4(vv.x, vv.y, vv.z, v.w); } double IntersectMBulb(vec3& rO, vec3& rD, vec4& trap) { double dist; for (int i = 0; i<maxDetailIter; i++) { double r = 0.0; vec4 v = vec4(rO.x, rO.y, rO.z, 1.0); vec3 va (v.x, v.y, v.z); trap = vec4(abs(v.x), abs(v.y), abs(v.z), dot(va, va)); for (int n = 0; n<mmaxIterations; ++n) { va = vec3(v.x, v.y, v.z); r = length(va); if (r>BAILOUT) break; v = MainBulb(v) + vec4(rO.x, rO.y, rO.z, 0.0); va = vec3(v.x, v.y, v.z); vec4 v4 (abs(va.x), abs(va.y), abs(va.z), dot(va, va)); trap.x = min(trap.x, v4.x); trap.y = min(trap.y, v4.y); trap.z = min(trap.z, v4.z); trap.w = min(trap.w, v4.w); } dist = 0.5*log(r)*r / v.w; rO = rO + rD * dist; if (dist < EPS) break; } return dist; } /* Hacky code to work out an approximate normal for our geometry (without lighting it looks horrible) */ vec3 NormEstimateMB(vec3 p) { double ddd = 0.000001; vec4 g0 = vec4(p, .0); vec4 gx = vec4(p + vec3(ddd, 0, 0), .0); vec4 gy = vec4(p + vec3(0, ddd, 0), .0); vec4 gz = vec4(p + vec3(0, 0, ddd), .0); vec4 v0 = g0; vec4 vx = gx; vec4 vy = gy; vec4 vz = gz; double ln; for (int i = 0; i < 100; i++) { g0 = MainBulb(g0) + v0; gx = MainBulb(gx) + vx; gy = MainBulb(gy) + vy; gz = MainBulb(gz) + vz; ln = length( vec3(g0.x, g0.y, g0.z) ); if (ln>BAILOUT) break; } //float ln = length( vec3(g0.x, g0.y, g0.z) ); double gradX = length( vec3(gx.x, gx.y, gx.z) ) - ln; double gradY = length( vec3(gy.x, gy.y, gy.z) ) - ln; double gradZ = length( vec3(gz.x, gz.y, gz.z) ) - ln; //N = normalize(vec3(length(gx-g0), length(gy-g0), length(gz-g0))); return normalize(vec3(gradX, gradY, gradZ)); } /* Good old Phong lighting function */ vec3 Phong(vec3 light, vec3 eye, vec3 pt, vec3 N) { vec3 diffuse = vec3(0.40, 0.95, 0.25); // base color of shading const int specularExponent = 10; // shininess of shading const double specularity = 0.45; // amplitude of specular highlight vec3 L = normalize(light - pt); // find the vector to the light vec3 E = normalize(eye - pt); // find the vector to the eye double NdotL = dot(N, L); // find the cosine of the angle between light and normal vec3 R = L - N * 2 * NdotL; // find the reflected vector diffuse = diffuse + N*0.1; // add some of the normal for 'effect' return diffuse * max(NdotL, 0.0) + specularity*pow(max(dot(E, R), 0), specularExponent); } /* Program entry point - loop over all the image pixels and work out the color (i.e., based on a ray we shoot at the Mandelbulb fractal/intersection). */ int main(int argc, char *argv[]) { int w = 1024, h = 1014; // # default resolution if ( argc == 2 ) w = h = atoi(argv); // # resolution vec3 *c = new vec3[w*h]; // output colors #pragma omp parallel for schedule(dynamic, 1) private(r) // OpenMP for (int y=0; y < h; y++ ) { // Loop over image rows fprintf(stderr, "\rRendering (%dx%d res) %5.2f%%", w, h, 100.*y / (h - 1)); for (unsigned short x=0; x < w; x++ ) // Loop cols { // index for this pixel into color array c const int i = (h - y - 1)*w + x; // we 'orientate' the ray so it's not just facing // down the z-axis. Instead of implementing all the rotation // code, we use some simple vector trigonometry to work out // the look at point, direction and origin of the ray vec3 off(0,-2,-2); vec3 dir = vec3(0,0,0) - off; dir = normalize(dir); off = dir*-2.5; vec3 left = vec3(0,1,0).cross( dir ); left = normalize(left); vec3 up = left.cross( dir ); up = normalize(up); // pixels for the screen -0.5 to 0.5 double dx = -0.5 + ((x / double(w - 1)) * 1.0); double dy = -0.5 + ((y / double(h - 1)) * 1.0); vec3 rO = off; vec3 rD = dir + left*dx + up*dy; rD = rD.norm(); #if 0 // debug - if we want the camera to look down the z axis dx = -0.5 + ((x / double(w - 1) ) * 1.0 ); dy = -0.5 + ((y / double(h - 1)) * 1.0 ); rD = vec3(dx, dy, 0.9).norm(); rO = vec3(0,0,-2.3); #endif vec4 cols(0,0,0,0); // distance from the ray to the geometry double dist = IntersectMBulb(rO, rD, cols); c[i] = vec3(); // default black if ( dist < 0.001 ) { // if we get here, then we have 'hit' the geometry, so we can work // out some lighting information. // If we just set the color to some 'constant' we won't see any of // that sexy detail/geometry that is fractals possess vec3 light (2,2,-5); // light position vec3 N = NormEstimateMB(rO); // Compute the Phong illumination at the point of intersection. vec3 color = Phong(light, rD, rO, N); c[i] = color; } } } // details on the ppm image format // ref: https://en.wikipedia.org/wiki/Netpbm#PPM_example FILE *f = fopen("image.ppm", "w"); // Write image to PPM file. fprintf(f, "P3\n%d %d\n%d\n", w, h, 255); for (int i = 0; i < w*h; i++) fprintf(f, "%d %d %d ", toInt(c[i].x), toInt(c[i].y), toInt(c[i].z)); } ```

 [Wikipedia Mandelbulb](https://en.wikipedia.org/wiki/Mandelbulb) - Explains the history and the mathematics (helps you understand where and why of the equations in the code)
 [WebGL MandelBulb Ray Marching 3D rendering](https://github.com/BrutPitt/wglMandelBulber) [Nice web-based demo on GitHub]
 Dual-Quaternion Julia Fractals (Kenwright) - Exploring fractals (alternative mathematical models and methods for representing and defining geometric surfaces)
 [Python Mandelbulb Slice Receipe](http://code.activestate.com/recipes/578198-mandelbulb-fractal/) Sweet little python program to show a minimal working example of a mandelbulb 'slice'
 [Daniel White's site](https://www.skytopia.com/project/fractal/mandelbulb.html) Mandelbulb and its history, very well documented details, and continue to discuss various distance estimators for the concept
 [WebGPU WGSL Demo - Ray Tracing](https://webgpulab.xbdev.net/index.php?page=editor&id=mandelbulbbasic2&)