The Menger Sponge is a fractal curve, which was first described by Karl Menger in 1926 during his studies of the concept of topological dimension.
It is a “three-dimensional generalisation of the Cantor set and Sierpinski carpet”.
The Menger sponge has an infinite surface area and zero volume.
Inside a Menger Sponge:
- Begin with a cube.
- Divide every face of the cube into 9 squares (like a Rubik’s Cube), which will sub-divide the cube into 27 smaller cubes.
- Remove the smaller cube in the middle of each face, and remove the smaller cube in the very centre of the larger cube, leaving 20 smaller cubes. This is a level-1 Menger sponge.
- Repeat steps 2 and 3 for each of the remaining smaller cubes, and continue to iterate ad infinitum.
If n is the number of iterations performed on the first cube with unit side length, then:
- Number of Cubes, N = 20n
- Side Length of a Cube, s = (1/3)n
- Total Volume, V = Ns3 =(20/27)n
- Surface Area, A = 2(20/9)n+ 4(8/9)n