Cultureblog: Cantor Dust on the Sierpinski Carpet

The Sierpinski Carpet is a plane fractal representing the ternary Cantor set extrapolated into 2-dimensions. Besides being a rather attractive rug pattern, like the other 2-D generalisation, Cantor dust, the 3-D extrapolation Menger sponge and the ternary set itself, it has zero measure.

In the Cantor set, starting with a unit length (ie. any length, with one end of the line being point 0 and the other being point 1) you divide the length into three equal segments and remove the middle third, and continue to infinite iterations.

Since the sum of the removed parts tends to infinity the set has zero measure, although clearly some parts of the segments are never removed, so it also contains an infinite number of points.

In the Carpet the central square piece is removed from a square sliced into thirds horizontally and vertically; the area tends to zero and the total perimeter of the holes tend to infinity. For some reason, this is pretty damn awesome.

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