Space-filling surfaces

[msltoe]

What do we know about space-filling surfaces (analagous to space-filling curves)?
A Google/Yahoo image search picks up very little.

I imagine it might yield some nice fractals *and* because the object would technically be a surface,
most complex functions of the surface position would be conformal. Presuming we
didn't subdivide to infinity, moving away from the surface slightly could lead to near-conformal features.

[msltoe]

Here's the simplest example I could think of...

[msltoe]

Ok. So I haven't reached space filling, but it gets interesting when you apply rotations.
BTW, my inspiration is herehttp://en.wikipedia.org/wiki/Space-filling_tree
And here's the code which you can figure out:

Code:

s1=sqrt(2.0)/2.0;

 while ((norm<8)&&(iter<imax)) {
		
   x=fabs(x);
   y=fabs(y);
   z=fabs(z);
   x2=x;y2=y;z2=z;
   
   if (x>2) {norm=1000;}
   if (y>2) {norm=1000;}
   if (z>2) {norm=1000;}
   
   if ((x2<y2)&&(x2<=z2)) {d=1;}
   if ((y2<z2)&&(y2<=x2)) {d=2;}
   if ((z2<x2)&&(z2<=y2)) {d=3;}
   
   if (d==1) {
     if (x2<thresh){iter=imax+1;maxnorm=0.3;} else
     { if (y>0) { y=(y-1); }
        if (z>0) { z=z-1; }
        y2 = s1*y-s1*z;
        z2 = s1*z+s1*y;
        y=y2;z=z2;
     }
   }
   if (d==2) {
     if (y2<thresh){iter=imax+1;maxnorm=0.6;} else
     { if (x>0) { x=(x-1); }
        if (z>0) { z=z-1; }
        x2 = s1*x-s1*z;
        z2 = s1*z+s1*x;
        x=x2;z=z2;
     }	
   }
   if (d==3) {
     if (z2<thresh){iter=imax+1;maxnorm=0.9;} else
     { if (x>0) { x=(x-1); }
        if (y>0) { y=y-1; }
        x2 = s1*x-s1*y;
        y2 = s1*y+s1*x;
        x=x2;y=y2;
     }	
   }
   (*nfunc)++;
   iter++;
   r = x*x+y*y+z*z;
   x = x*2.5;y=y*2.5;z=z*2.5;
  } 
  if (iter==imax){iter=1;}
  
  if (norm>10) {iter=1;}

[Tglad]

It doesn't seem to have received a huge amount of attention, most 3d analogues of space filling curves are just 3d space filling curves (rather than surfaces).
Depending on what you consider a curve/surface there are also many other variations, like space filling dust, space filling tree, space filling sponge, and in 3d only: space filling foam, space filling shell.

[fractower]

An interesting property of the Mandelbrot boundary is that the length of the boundary between any two points is infinite. Is it possible to construct a surface such that the shortest path on the surface between any two points is always infinite?

An counter example is an extrusion of the Mandelbrot boundary. This produces a 3d space filling surface, but distances between points in the direction of extrusion are finite. The Taffy regions of M-Bulbs seem to have the same property.

[cKleinhuis]

The mandelbulb is a bit netter for that but because of the whipped cream not for all directions /locations  the mandelbox might be a better candidate for such, and in fact the real threed mandelbrot should jave exactly such property

[msltoe]

What about the surface of something simple like the Menger sponge?

[laser blaster]

A true space-filling surface (or even a space-filling tree-ish surface) would look very boring, as it would just appear as pretty much a solid cube at high iterations. Cross sections through it might be interesting, though.

But creating a continuous space-filling surface (that's not just an extrusion of a 2D space-filling curve) is much harder than you'd think! I'd be very interested to see what it would look like. I don't know if it's even possible.

[taurus]

What about the surface of something simple like the Menger sponge?

the menger sponge has no surface and no 3-D domain. Hard to imagine, but a menger sponge is a space filling curve, with a topological dimension of 1. A simple wikipedia investigation shows that.

[blob]

It's got no volume but has an infinite surface as one would expect.

[msltoe]

The Wikipedia article says both that the Menger sponge is a curve and that it has infinite surface area. Talk about confusing.
In any case, I'm interested in inspiring new, pretty fractal formulas that we haven't thought of yet.

[Dinkydau]

The surface of the whole thing increases for each iteration. After infinitely many iterations, you could say the surface is infinitely large, but I find it a bit controversial. How can the total surface be infinite when every piece of surface there is is for 100% filled with holes? I think it's like dividing by 0. The closer you get to zero, the larger the division gets, but x/0 is not infinite.

[cKleinhuis]

as far as i understand it it converges towards a point cloud

[Tglad]

The menger sponge could be considered a fractal surface, but it isn't a space filling surface. Nor is it really a surface, I define it as a 3d void-sponge (https://sites.google.com/site/simplextable/what-is-the-simplex-table).
The menger carpet (which is 2d) could be considered a fractal curve, but it isn't a space filling curve, since it doesn't fill space. I define is as a 2d void-sponge.

Is it possible to construct a surface such that the shortest path on the surface between any two points is always infinite

Yes.. for instance my 'volcanic surface' fractal: http://www.fractalforums.com/new-theories-and-research/new-fractal-needs-a-name/msg54859/#msg54859

[taurus]

The menger carpet (which is 2d) could be considered a fractal curve, but it isn't a space filling curve, since it doesn't fill space. I define is as a 2d void-sponge.
Yes.. for instance my 'volcanic surface' fractal: http://www.fractalforums.com/new-theories-and-research/new-fractal-needs-a-name/msg54859/#msg54859

When I understand it right the mengert sponge (3d) isn't space filling either.
Still got difficulties to imagine an infinite area in 3d space, enclosing zero volume and having a lesbesgue covering dimension of one.
At least the first and the last property seem to contradict each other. But it's a fractal and especially those two properties do not exist in reality, they only exist in math idealisation...