First steps toward the Sierpinski carpet

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We start with a square, dividing it into a 3-by-3 grid, and remove the center square; we then repeat the process for each remaining square. The limiting shape (that is, the shape remaining "after" iterating the process an infinite number of times) is the Sierpinski carpet.

 [ sorry -- not much fun without pictures ]

The code, for what it's worth: we define a function makehole that takes the lower-left and upper-right corners of a square, and returns a list of eight squares arranged about the center of the original square (that is, it returns the original square with the middle ninth erased); we iterate makehole four times, remembering each step, giving us four lists of corners; the line starting with Show makes the lists of corners into renderable rectangles, which it then shows. It made sense at the time.

makehole[{{x0_, y0_}, {x1_, y1_}}] :=

  With[{xr = Abs[x1 - x0], yr = Abs[y1 - y0]},
    
      { {{x0, y0},               {x0 + xr/3, y0 + yr/3}},
        {{x0, y0 + yr/3},        {x0 + xr/3, y1 - yr/3}},
        {{x0, y1 - yr/3},        {x0 + xr/3, y1}},
        {{x0 + xr/3, y0},        {x1 - xr/3, y0 + yr/3}},
        {{x0 + xr/3, y1 - yr/3}, {x1 - xr/3, y1}},
        {{x1 - xr/3, y0},        {x1, y0 + yr/3}},
        {{x1 - xr/3, y0 + yr/3}, {x1, y1 - yr/3}},
        {{x1 - xr/3, y1 - yr/3}, {x1, y1}}
       }];

Show[
  GraphicsArray[
    Partition[
      Map[ Graphics[#, AspectRatio -> 1]&,
        Apply[
          Rectangle,
            Rest@
              NestList[ Flatten[ Map[ makehole, # ], 1 ]&,
                {{{0, 0}, {1, 1}}},
               4],
         {2}]],
     2]]];
Designed and rendered using Mathematica versions 2.2 and 3.0 for the Apple Macintosh.

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