Menger Sponge

In the 2006 Taiwanese thriller movie Silk, a scientist creates a Menger Sponge, a special kind of hole-filled cube, to capture the spirit of a child. The Sponge not only functions as an anti-gravity device but seems to open a door into a new world. As fanciful as this film concept is, the Menger Sponge considered by mathematicians today is certainly beautiful to behold, when rendered using computer graphics, and a concept that ought to be more widely known. Certainly, it provides a wonderful gateway to fractals, mathematics, and reasoning beyond the limits of our own intuition.    

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 twenty-seven identical smaller cubes. Next, we remove the cube in the center and the six cubes that share faces with it. This leaves behind twenty cubes. We continue to repeat the process forever with smaller and smaller cubes. The number of cubes increases as 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 Sierpiński carpet. Fractal antennae based on the Sierpiński carpet are sometimes used as efficient receivers of electromagnetic signals. Both the carpets and entire cube have fascinating geometrical properties. For example, the sponge has infinite surface area while enclosing zero volume. Imagine the skeletal remains of an ancient dinosaur that has turned into the finest of dust through the gentle acid of time. What remains seems to occupy our world in a ghostlike fashion but no longer “fills” it.     

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 over 65,000 business cards that weighs about 150 pounds (70 kg).     

The Menger Sponge is an important concept for the general public to become familiar with partly because it reaffirms the idea that the line between mathematics and art can be a fuzzy one; the two are fraternal philosophies formalized by ancient Greeks like Pythagoras and Ictinus and dwelled on by such greats as Fra Luca Bartolomeo de Pacioli (1447–1517), the Italian mathematician and Franciscan friar, who published the first printed illustration of a Leonardo da Vinci’s rhombicuboctahedron, in De divina proportione. The rhombicuboctahedron, like the Menger Sponge, is a beauty to behold when rendered graphically—an Archimedean solid with eight triangular faces and eighteen square faces, with twenty-four identical vertices, and with one triangle and three squares meeting at each.     

Fractals, such as the Menger Sponge, often exhibit self-similarity, which suggests that various exact or inexact copies of an object can be found in the original object at smaller size scales. The detail continues for many magnifications—like an endless nesting of Russian dolls within dolls. Some of these shapes exist only in abstract geometric space, but others can be used as models for complex natural objects such as coastlines and blood vessel branching. The dazzling computer-generated images can be intoxicating, perhaps motivating students’ interest in math as much as any other mathematical discovery in the last century.     

The Menger Sponge reminds students, educators, and mathematicians of the need for computer graphics. As Peter Schroeder once wrote, “Some people can read a musical score and in their minds hear the music.... Others can see, in their mind’s eye, great beauty and structure in certain mathematical functions.... Lesser folk, like me, need to hear music played and see numbers rendered to appreciate their structures.”