Lab 6: Sierpinski's Carpet
Sierpinski was also in the carpet business! The most famous carpet for which he is known today is called, appropriately enough, Sierpinski's Carpet.
He began with a large square cloth (stage 0), pretended that it was subdivided into 9 equal squares and cut out the center square (stage 1).
Next, he considered each of the eight remaining squares, imagined they were subdivided into 9 equal squares and cut out each center square again (stage 2).
He continued on in this fashion, through as many stages as he could, always imagining that he was removing the center square of the remaining solid squares, which were getting progressively smaller.
Begin with a 27x27 square "carpet" on a sheet of graph paper, and construct your own version of Sierpinski's Carpet up to stage 3.
Waclaw Sierpinski (1882-1969) was a famous Polish mathematician who introduced his "carpet" to the world in 1916. Actually Sierpinski's Carpet is not a physical carpet, it is not even something that we can draw with paper and pencil, because it continues for infinitely many stages. The actual carpet is the object that would be left if you were able to continue removing the "center squares" forever.
Sierpinski, along with many other mathematicians of his time, was trying to understand the finer points of the notion of dimension, and the true purpose of his construction was to create a universal object, that is, an object that could contain any closed and bounded one dimensional object that could exist in the plane.
