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Fractal Math, Chaos Theory & Research => (new) Theories & Research => Topic started by: msltoe on August 22, 2013, 12:27:42 AM




Title: Space-filling surfaces
Post by: msltoe on August 22, 2013, 12:27:42 AM
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.


Title: Re: Space-filling surfaces
Post by: msltoe on August 22, 2013, 03:01:28 AM
Here's the simplest example I could think of...




Title: Re: Space-filling surfaces
Post by: msltoe on August 22, 2013, 03:17:11 AM
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 (http://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;}


Title: Re: Space-filling surfaces
Post by: Tglad on August 23, 2013, 05:44:28 AM
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.


Title: Re: Space-filling surfaces
Post by: fractower on August 23, 2013, 05:18:33 PM
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.


Title: Re: Space-filling surfaces
Post by: cKleinhuis on August 23, 2013, 06:27:29 PM
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


Title: Re: Space-filling surfaces
Post by: msltoe on August 24, 2013, 02:28:27 AM
What about the surface of something simple like the Menger sponge?


Title: Re: Space-filling surfaces
Post by: laser blaster on August 25, 2013, 04:03:17 AM
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.


Title: Re: Space-filling surfaces
Post by: taurus on August 25, 2013, 01:04:30 PM
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.


Title: Re: Space-filling surfaces
Post by: blob on August 25, 2013, 04:44:58 PM
It's got no volume but has an infinite surface as one would expect. :dink:


Title: Re: Space-filling surfaces
Post by: msltoe on August 25, 2013, 08:21:03 PM
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.


Title: Re: Space-filling surfaces
Post by: Dinkydau on August 25, 2013, 10:41:57 PM
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.


Title: Re: Space-filling surfaces
Post by: cKleinhuis on August 25, 2013, 10:48:16 PM
as far as i understand it it converges towards a point cloud


Title: Re: Space-filling surfaces
Post by: Tglad on August 26, 2013, 07:38:09 AM
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.

Quote
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


Title: Re: Space-filling surfaces
Post by: taurus on August 26, 2013, 10:06:57 AM
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...


Title: Re: Space-filling surfaces
Post by: fractower on August 27, 2013, 10:10:53 PM
Quote
Quote
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

This is an interesting fractal, but the operation appears to preserve edges. If this is a general property then points on an edge have a finite connection.

Since the edges added at each generation are exponentially shrinking, it seems that the path between any two arbitrary points will also have a finite connection.



Title: Re: Space-filling surfaces
Post by: Tglad on August 28, 2013, 09:51:34 AM
Ah true, certain paths are finite in this case. Well, other fractals have the property that any two points have an infinite surface path between them... like cumulus clouds, broccoli, fractal mountains (e.g. diamond square algo), etc.