Sierpinski carpet perimeter

Plane fractal built from squares

"Sierpinski snowflake" redirects here. For other uses, see Sierpiński curve.

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 such generalization is the 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.

Sierpinski carpet perimeter

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