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:

### Construction

- 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 = 20
^{n } - Side Length of a Cube,
*s*= (1/3)^{n} - Total Volume,
*V*=*Ns*^{3}=(20/27)^{n} - Surface Area,
*A*= 2(20/9)^{n}+ 4(8/9)^{n}

M x

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