Menger sponge

The Menger sponge (also Menger-Sierpinski sponge or, wrongly, Sierpinski sponge),three-dimensional extension ofCantor setSierpinski carpet, isfractalHausdorff dimension (ln 20) / (ln 3) (approx. 2,726833). It was first described by Austrian mathematician Karl Menger1927;construction ofMenger sponge can be visualized as follows:

1. Begin withcube.
2. Cut upcube into 27 smaller cubes, each withside lengthone thirdthat oforiginal one.
3. Removesmall cubes incentereach face oflarge cube, as well asinnermost small cube.
4. Repeatprocesseach ofremaining 20 small cubes.

After an infinite numberiterations,Menger sponge will remain. Formally,Menger sponge can be defined as follows:

where M₀ isunit cube and

Each face ofMenger sponge isSierpinski carpet; furthermore, any intersection ofMenger sponge withdiagonal or medium ofinitial cube M₀ isCantor set. The Menger sponge isclosed set; since italso bounded,theoremHeine-Borel yields thatis compact. Furthermore,Menger spongeuncountablehas Lebesgue measure 0.

As Peitgen, JürgensSaupe showed1992,Menger spongealsosuper-objectall compact one-dimensional object; that is,topological equivalentany compact one-dimensional object can be found inMenger sponge.

See also

• Sierpinski triangle
• Sierpinski tetrahedron

External links

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