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Cantor dust

Cantor dust, named after the mathematician Georg Cantor, is the diadic (two-dimensional) version of the Cantor set.

In the limit, starting from a square, this produces a set with an infinite number of square sections each having zero area--the sum of all areas also decreases to zero in the limit.

The triadic (three-dimensional) form of this is called the Menger sponge. An alternate diadic generalization of the Cantor set produces the Sierpinski carpet.