The Sierpiński carpet is a plane fractal first described by Wacław Sierpiński in 1916. The carpet is a generalization of the Cantor set to two dimensions; another is Cantor dust. The technique of subdividing a shape into smaller copies of itself, removing one or more copies, and continuing recursively can be extended to other shapes. For instance, subdividing an equilateral triangle into four equilateral triangles, removing the middle triangle, and recursing leads to the Sierpiński triangle. In three dimensions, a similar construction based on cubes is known as the Menger sponge. Construction The construction of the Sierpiński carpet begins with a square. The square is cut into 9 congruent subsquares in a 3-by-3 grid, and the central subsquare is removed. The same procedure is then applied recursively to the remaining 8 subsquares, ad infinitum. It can be realised as the set of points in the unit square whose coordinates written in base three do not both have a digit '1' in the same position, using the infinitesimal number representation of . The process of recursively removing squares is an example of a finite subdivision rule. Properties The area of the carpet is zero (in standard Lebesgue measure). Proof: Denote as the area of iteration . Then . So , which tends to 0 as goes to infinity. The interior of the carpet is empty. Proof: Suppose by contradiction that there is a point in the interior of the carpet. Then there is a square centered at which is entirely contained in the carpet. This square contains a smaller square whose coordinates are multiples of for some . But, if this square has not been previously removed, it must have been holed in iteration , so it cannot be contained in the carpet – a contradiction. The Hausdorff dimension of the carpet is . Sierpiński demonstrated that his carpet is a universal plane curve. That is: the Sierpinski carpet is a compact subset of the plane with Lebesgue covering dimension 1, and every subset of the plane with these properties is homeomorphic to some subset of the Sierpiński carpet. This "universality" of the Sierpiński carpet is not a true universal property in the sense of category theory: it does not uniquely characterize this space up to homeomorphism. For example, the disjoint union of a Sierpiński carpet and a circle is also a universal plane curve. However, in 1958 Gordon Whyburn uniquely