OpenGL and Direct3D are for sissies...

A level-4 Menger sponge.

...real men do their 3-d graphics in PostScript! :-)

What the Cantor dust is in one dimension and the Sierpinski carpet in two, the Menger sponge (also known as the Sierpinski cube) is in three dimensions. Start with a cube and divide it into 3*3*3 smaller cubes; then remove those that do not touch any of the edges of the original cube. This leaves 20 out of the 27 smaller cubes. Now the same step can be performed on them, resulting in an object composed of 400 even smaller cubes, and so on. The volume of the object tends towards zero, but its surface increases beyond all bounds. The picture shown here is the object obtained after we perform this step four times.

Anyway, I thought it would be fun to draw this in PostScript. It doesn't have any particular support for 3-d graphics, so we have to project things by ourselves. The surface of the sponge consists of a number of small squares; we just have to be careful to draw them in the correct order, so that the ones which appear closer to the viewer will be drawn last (and thus won't be obscured by other things). The correct order depends on the angle from which we are looking at the object; the one used by the showCube procedure below is suitable for the particular angles (lat and lon) used, but if these were changed considerably, the procedure might need to be rearranged a bit.

%!PS-Adobe-2.0 EPSF-2.0
%%BoundingBox: 109 199 495 600

/n 4 def   % approximation level of the fractal
/xyColor { 1 0.5 0.5 setrgbcolor } def
/xzColor { 0.5 1 0.5 setrgbcolor } def
/yzColor { 0.5 0.5 1 setrgbcolor } def
/distanceFromCamera 2.5 def
/lon 50 def /lat 36 def  % to rotate the object
/scale 600 def % larger values = larger object

/pow { % a b pow -> a^b
    2 dict begin /b exch def /a exch def
    1 { b 0 le { exit } if
        a mul
        /b b 1 sub def
    } loop
    end
} def

/isPresent { % x y z isPresent --> true/false
    20 dict begin 
    /z exch def /y exch def /x exch def

    x 0 lt y 0 lt or z 0 lt or
    x n3 ge or y n3 ge or z n3 ge or
    { false }
    {
        {   n 0 le { true exit } if
            x 3 mod 1 eq { 1 } { 0 } ifelse
            y 3 mod 1 eq { 1 } { 0 } ifelse
            z 3 mod 1 eq { 1 } { 0 } ifelse
            add add 2 ge { false exit } if
            /x x 3 idiv def
            /y y 3 idiv def
            /z z 3 idiv def
            /n n 1 sub def
        } loop
    } ifelse
    end
} def

/map3dTo2d { % x y z map3dTo2d --> xMapped yMapped
    30 dict begin
    /zi exch   n3 2 div  sub n3 div def
    /yi exch   n3 2 div  sub n3 div def
    /xi exch   n3 2 div  sub n3 div def

    % Rotate by 'lon' degrees around the z-axis.
    xi lon cos mul   yi lon sin mul   sub  % new xi
    xi lon sin mul   yi lon cos mul   add  % new yi
    /yi exch def /xi exch def
    % Rotate by 'lat' degrees around the y-axis.
    xi lat cos mul   zi lat sin mul   sub  % new xi
    xi lat sin mul   zi lat cos mul   add  % new zi
    /zi exch def /xi exch def
    % Project.
    /xi xi distanceFromCamera add def
    yi xi div scale mul
    zi xi div scale mul

    end
} def

/xyfacet {
    30 dict begin 
    /zi exch def /yi exch def /xi exch def
    newpath
        xi yi zi              map3dTo2d moveto
        xi yi 1 add zi        map3dTo2d lineto
        xi 1 add yi 1 add zi  map3dTo2d lineto
        xi 1 add yi zi        map3dTo2d lineto
        xyColor fill
    end
} def

/xzfacet {
    30 dict begin 
    /zi exch def /yi exch def /xi exch def
    newpath
        xi yi zi              map3dTo2d moveto
        xi yi zi 1 add        map3dTo2d lineto
        xi 1 add yi zi 1 add  map3dTo2d lineto
        xi 1 add yi zi        map3dTo2d lineto
        xzColor fill
    end
} def

/yzfacet {
    30 dict begin 
    /zi exch def /yi exch def /xi exch def
    newpath
        xi yi zi              map3dTo2d moveto
        xi yi zi 1 add        map3dTo2d lineto
        xi yi 1 add zi 1 add  map3dTo2d lineto
        xi yi 1 add zi        map3dTo2d lineto
        yzColor fill
    end
} def

/showCube
{
    30 dict begin

    0 1  n3 1 sub {
        % We will now show level r 
        % (z-coordinate from r to r+1).
        /r exch def 

        % First show the bottom facets.
        0 1  n3 1 sub  { /yi exch def 
        0 1  n3 1 sub  { /xi exch def
            xi yi r 1 sub isPresent 
            xi yi r isPresent 
            not and { xi yi r xyfacet} if
        } for } for

        % Then show the facets which are not
        % perpendicular to the z-axis.
        0 1  n3  { /yi exch def
            n3 -1 0  { /xi exch def
                xi yi 1 sub r isPresent 
                xi yi r isPresent 
                not and { xi yi r xzfacet } if
            } for
            n3 -1 0  { /xi exch def
                xi yi r isPresent 
                xi 1 sub yi r isPresent 
                not and { xi yi r yzfacet } if
            } for
        } for

        % For the last (topmost) layer,
        % also show the top facets.
        r  n3 1 sub  eq {
            0 1  n3 1 sub  { /yi exch def
            0 1  n3 1 sub  { /xi exch def
                xi yi r isPresent 
                { xi yi r 1 add xyfacet } if
            } for } for
        } if
    } for

    end
} def

gsave
0.2 setlinewidth
/n3 3 n pow def
% Center it on the A4 sheet of paper.
0.5 595 mul 0.5 842 mul translate
showCube
grestore

showpage

You can copy the above code into a file and send it to your favourite PostScript printer, or open it in a viewer such as Ghostview.

For a level-4 sponge, such as we see in the picture above, this draws 168192 facets. When I tried converting this PostScript file to PDF, the resulting PDF file was several megabytes long, because PDF is not really a programming language with loops and so forth, and the PDF file simply contained all the objects drawn by the above program. However, of these 168192 facets, most aren't really visible in the end because they are obscured by other facets. I wrote a C++ program to determine which ones are completely obscured by others and may be removed; only approx. 22000 facets were left, and the image when rendered at 1200 DPI was completely unchanged; the resulting PDF file was only about 400 KB long.

P.S. Since the title mentions sissies, this comic should not be missed.