Menger sponge

The Menger sponge (also Menger-Sierpinski sponge or, wrongly, Sierpinski sponge), a three-dimensional extension of the Cantor set and Sierpinski carpet, is a fractal with Hausdorff dimension (ln 20) / (ln 3) (approx. 2,726833). It was first described by Austrian mathematician Karl Menger in 1927; the construction of a Menger sponge can be visualized as follows:

1. Begin with a cube.
2. Cut up the cube into 27 smaller cubes, each with a side length of one third of that of the original one.
3. Remove the small cubes in the center of each face of the large cube, as well as the innermost small cube.
4. Repeat the process for each of the remaining 20 small cubes.

After an infinite number of iterations, a Menger sponge will remain. Formally, a Menger sponge can be defined as follows:

where M₀ is the unit cube and

Each face of the Menger sponge is a Sierpinski carpet; furthermore, any intersection of the Menger sponge with a diagonal or medium of the initial cube M₀ is a Cantor set. The Menger sponge is a closed set; since it is also bounded, the theorem of Heine-Borel yields that is is compact. Furthermore, the Menger sponge is uncountable and has Lebesgue measure 0.

As Peitgen, Jürgens and Saupe showed in 1992, the Menger sponge is also a super-object for all compact one-dimensional object; that is, a topological equivalent of any compact one-dimensional object can be found in the Menger sponge.

See also

• Sierpinski triangle
• Sierpinski tetrahedron

External links

• The Business Card Menger Sponge Project
• An interactive Menger sponge