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