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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.

The code, for what it's worth: we define a function

makeholethat 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 iteratemakeholefour times, remembering each step, giving us four lists of corners; the line starting withShowmakes 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 usingMathematicaversions 2.2 and 3.0 for the Apple Macintosh.

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