This query demonstrates visually the nature and application of approximation. Assuming Scot didn't delete it he has the presentation to go with it: Approximation Theory and Motivation Presentation.ppt.

{{{
begin%Boxes%

Choice(m,n) :- m=0, n=0.
Choice(m,n) :- m=0, n=1.
Choice(m,n) :- m=0, n=2.
Choice(m,n) :- m=1, n=0.
Choice(m,n) :- m=1, n=2.
Choice(m,n) :- m=2, n=0.
Choice(m,n) :- m=2, n=1.
Choice(m,n) :- m=2, n=2.

D(x,y,z) :- x-y=0, z=0.
D(x,y,z) :- D(x1,y,z1), x-x1=1, z-z1=1.


M(x,y,z) :- x=0, y=0, z=0.
M(x,y,z) :- M(x1,y,z1), D(z,z1,y), x-x1=1.
M(x,y,z) :- M(x,y1,z1), D(z,z1,x), y-y1=1.

/* Creates the first set of base lines length=1,3,9...  */

BaseLine(x1,x2):- x1=0, x2=1.
BaseLine(a1,d3):- BaseLine(a1,d1), l=3, M(l,d1,d3).

Box(x,y,s) :- BaseLine(x,b), y-x=0, b-s=0.

Box(x,y,s) :- Box(a,b,l), k=3, M(k,s,l), Choice(m,n),
               M(s,m,z1), D(x,a,z1),
               M(s,n,z2), D(y,b,z2).

XBox(a1,c1,s) :- Box(a,c,l), k=3, M(k,s,l), D(a1,a,s), D(c1,c,s).

fractal1(id,x,y) :- id=1, Box(a,b,l), x-a>=0, D(x1,l,a), x-x1<=0,
                    y-b>=0, D(y1,l,b), y-y1<=0.

fractal2(id,x,y) :- id=2, XBox(a,b,l), x-a>=0, D(x1,l,a), x-x1<=0,
                    y-b>=0, D(y1,l,b), y-y1<=0.

end%Boxes%
}}}