Menger Sponge - The Math Book

Menger Sponge by Jeannine Mosely, at the Institute for Figuring. Photo: Ravi Apte

Karl Menger (1902 - 1985) - The Menger sponge is a fractal object with an infinite number of cavities - a nightmarish object for any dentist to contemplate. The object was first described by Austrian mathematician Karl Menger in 1926. To construct the sponge, we begin with a "mother cube" and subdivide it into 27 identical smaller cubes. Next, we remove the cube in the center and the six cubes that share faces with it. This leaves behind 20 cubes. We continue to repeat the process forever. The number of cubes increases by 20n, where n is the number of iterations performed on the mother cube. The second iteration gives us 400 cubes, and by the time we get to the sixth iteration, we have 64,000,000 cubes.

Each face of the Menger sponge is called a Sierpinski carpet. Fractal antennae based on the Sierpinski carpet are sometimes used as efficient receivers of electromagnetic signals. Both the carpets and the entire cube have fascinating geometrical properties. For example, the sponge has an infinite surface area while enclosing zero volume.

According to the Institute for Figuring, with each iteration, the Sierpinski carpet face "dissolves into a foam whose final structure has no area whatever yet possesses a perimeter that is infinitely long. Like the skeleton of a beast whose flesh has vanished, the concluding form is without substance - it occupies a planar surface, but no longer fills it." This porous remnant hovers between a line and a plane. Whereas a line is one-dimensional and a plane two-dimensional, the Sierpinski carpet has a "fractional" dimension of 1.89. The Menger sponge has a fractional dimension (technically referred to as the Hausdorff Dimension) between a plane and a solid, approximately 2.73, and it has been used to visualize certain models of a foam-like space-time. Dr. Jeannine Mosely has constructed a Menger sponge model from more than 65,000 business cards that weights about 150 pounds (70 kilograms).