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menger fractal


Intro

Producing simple analytic shapes with for a raymarching engine is very simple, as seen in the in this reference article in this very webstite. Besides simple combination of shapes to construct more complex compound shapes, there is also the possibility to do this composition algorithmically. Probably the most basic way of algorithmic composition, is the recursive introduction of regular smaller details. This naturally produces classic Cantor fractals. A good example of that is the "untraceable" 1 kilobyte demo by TBC.


Probably the most trivial example is that of the Menger Spone. Let's take this as an example and so some quick raymarching experiment. You can see the final result, running realtime, here at the Shadertoy page. I assume you already have some code working, and that you most have your basic raymarching intersection routine look looking somthing like this:

vec3 intersect( in vec3 ro, in vec3 rd )
{
    for(float t=0.0; t<10.0; )
    {
        vec3 h = map(ro + rd*t);
        if( h.x<0.001 )
            return vec3(t,h.yz);
        t += h;
    }
    return vec3(-1.0);
}
We will now define the map() funcion. We start by computing the distance to the unit cube: 
vec3 map( in vec3 p )
{
    float d = sdBox(p,vec3(1.0));
    return vec3(d,0.0,0.0);
}
We now construct a cross made of 3 infinite boxes, by doing their union: 
float sdCross( in vec3 p )
{
  float da = sdBox(p.xyz,vec3(inf,1.0,1.0));
  float db = sdBox(p.yzx,vec3(1.0,inf,1.0));
  float dc = sdBox(p.zxy,vec3(1.0,1.0,inf));
  return min(da,min(db,dc));
}
This can be optimized by using 2D boxes instead: 
float sdCross( in vec3 p )
{
  float da = sdBox(p.xy,vec2(1.0));
  float db = sdBox(p.yz,vec2(1.0));
  float dc = sdBox(p.zx,vec2(1.0));
  return min(da,min(db,dc));
}
Now we scale the cross by one third, and perform the substraction of the cross to the box 
vec4 map( in vec3 p )
{
   float d = sdBox(p,vec3(1.0));
   float c = sdCross(p*3.0)/3.0;
   d = max( d, -c );
   return vec4(d,1.0,1.0,1.0);
}
Finally, we can do this substraction many times, iterativelly: 
vec3 map( in vec3 p )
{
   float d = sdBox(p,vec3(1.0));

   float s = 1.0;
   for( int m=0; m<3; m++ )
   {
      vec3 a = mod( p*s, 2.0 )-1.0;
      s *= 3.0;
      vec3 r = 1.0 - 3.0*abs(a);

      float c = sdCross(r)/s;
      d = max(d,-c);
   }

   return vec3(d,0.0,0.0);
}
For the sdCross() function, we can furhter optimize by noticing that we only care about the negative values of the 2d box functions. Therefore, 
float sdCross( in vec3 p )
{
  float da = maxcomp(abs(p.xy));
  float db = maxcomp(abs(p.yz));
  float dc = maxcomp(abs(p.zx));
  return min(da,min(db,dc))-1.0;
}
sdBox()





sdCross()





sdBox() - sdCross()





sdBox() - iterate(sdCross())

or even better, 
vec3 map( in vec3 p )
{
   float d = sdBox(p,vec3(1.0));

   float s = 1.0;
   for( int m=0; m<3; m++ )
   {
      vec3 a = mod( p*s, 2.0 )-1.0;
      s *= 3.0;
      vec3 r = abs(1.0 - 3.0*abs(a));

      float da = max(r.x,r.y);
      float db = max(r.y,r.z);
      float dc = max(r.z,r.x);
      float c = (min(da,min(db,dc))-1.0)/s;

      d = max(d,c);
   }

   return vec3(d,1.0,1.0);
}

Last step is, to compute the material id based on the iteration count, and some fake occlusion too (see images on the right of the article to see how it looks like when all these elements are put together): 
vec3 map( in vec3 p )
{
   float d = sdBox(p,vec3(1.0));
   vec3 res = vec3( d, 1.0, 0.0, 0.0 );

   float s = 1.0;
   for( int m=0; m<3; m++ )
   {
      vec3 a = mod( p*s, 2.0 )-1.0;
      s *= 3.0;
      vec3 r = abs(1.0 - 3.0*abs(a));

      float da = max(r.x,r.y);
      float db = max(r.y,r.z);
      float dc = max(r.z,r.x);
      float c = (min(da,min(db,dc))-1.0)/s;

      if( c>d )
      {
          d = c;
          res = vec3( d, 0.2*da*db*dc, (1.0+float(m))/4.0, 0.0 );
       }
   }

   return res;
}






Some neat tricks can be applied dueing the iterative substraction of cubes, suh as translating and rotating the point p a little bit in each iteration. That produces less symmetrical patterns, as can be seen in the images to the right of this article or here below (which was rendered with a simple pathtracer):