Maths Bite: Menger Sponge

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”.

Sierpinski Carpet

The Menger sponge has an infinite surface area and zero volume.

Inside a Menger Sponge:


  1. Begin with a cube.
  2. Divide every face of the cube into 9 squares (like a Rubik’s Cube), which will sub-divide the cube into 27 smaller cubes.
  3. 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.
  4. 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 = 20
  • 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

M x


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