Menger sponge

3d curve

last updated: 2005-01-07

where I3 is the unit cube, and S the Sierpinski square.
In fact, the sponge can be seen as the three-dimensional analog of the Sierpinski square. It is a fractal.

The cube can be constructed as follows: take a cube, divide it into 3 x 3 x 3 smaller cubes of equal size. Then remove the cube in the center, and also the six cubes that share sides with it.
Then, repeat the process on each of the remaining twenty cubes. And again, and again, infinitely.

The first picture shows the square after three iterations, the second and third picture show the square after six iterations.
The 2nd picture has been constructed with Open Inventor at the Interdisciplinary Center for Scientific Computing (IWR) in Heidelberg/Germany by C. Dartu and D. Volz in 1997.
The 3rd picture has also been made in Heidelberg.

The curve is the most popular creation of the mathematician Karl Menger, while working on dimension theory.
The curve has as alternative names: the Menger universal curve, or the Sierpinski sponge.

The curve is a three-dimensional variant on the Cantor set and the Sierpinski curve.