Wednesday, April 30, 2008

Menger sponge

As the Sierpinski carpet is a generalization of the Cantor set from one dimension into two dimension, the Menger sponge is a generalization of the Sierpinski carpet into three dimensions. Sometimes this three-dimensional fractal called Menger-Sierpinski sponge or Sierpinski sponge. It was first described by Austrian mathematician Karl Menger in 1926.

Like the Sierpinski carpet begins from square, Menger sponge begins from cube. Every face of the cube is divided into 9 smaller squares. This operation divide the cube into 27 smaller cubes. Then center cubes from all faces and the inner center cube are removed, leaving 20. This is a level 1 Menger Sponge. The next levels forms by repeating these steps to all 20 cubes rest. Below you can see first four levels.

Below you can see the Menger sponge with cut off corner, which was designed by Seb Przd.

There's also similar three-dimensional fractal based on tetrahedron, which is a generalization of the Sierpinski triangle into three dimensions. Below, you can see two versions of the Sierpinski pyramid and the Menger sponge in a single image.

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