Menger sponge

Menger Sponge animation

A Menger Sponge is a three-dimensional fractal, which can be made by taking a cube and cutting out a square section through the centre in each of the three directions; then each of the resulting smaller cubes is cut out in the same way, and so on until you’ve removed infinitely many pieces. So each Menger Sponge is made from twenty identical-but-smaller Menger Sponges. This results in an object which has zero volume but infinite surface area!

iterative construction of a Menger sponge

The construction of a Menger sponge can be described as follows:

  1. Begin with a cube (first image).
  2. Divide every face of the cube into 9 squares, like a Rubik’s Cube. This 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 center of the larger cube, leaving 20 smaller cubes (second image). This is a level-1 Menger sponge (resembling a Void Cube).
  4. Repeat steps 2 and 3 for each of the remaining smaller cubes, and continue to iterate ad infinitum.

The second iteration gives a level-2 sponge (third image), the third iteration gives a level-3 sponge (fourth image), and so on. The Menger sponge itself is the limit of this process after an infinite number of iterations.

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