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Repeat steps two and three for each of the remaining smaller cubes, and continue to iterate ad infinitum. The second iteration gives a level-2 sponge, the third iteration gives a level-3 sponge, and so on. The Menger sponge itself is the limit of this process after an infinite number of iterations.

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Remove the smaller cube in the middle of each face, and remove the smaller cube in the center of the more giant cube, leaving 20 smaller cubes. This is a level-1 Menger sponge (resembling a void cube).Divide every face of the cube into nine squares, like a Rubik's Cube. This sub-divides the cube into 27 smaller cubes. The n {\displaystyle n} th stage of the Menger sponge, M n {\displaystyle M_{n}} , is made up of 20 n {\displaystyle 20 An illustration of the iterative construction of a Menger sponge up to M 3, the third iteration Properties [ edit ] Hexagonal cross-section of a level-4 Menger sponge. (Part of a series of cuts perpendicular to the space diagonal.) Three-dimensional fractal An illustration of M 4, the sponge after four iterations of the construction process In mathematics, the Menger sponge (also known as the Menger cube, Menger universal curve, Sierpinski cube, or Sierpinski sponge) [1] [2] [3] is a fractal curve. It is a three-dimensional generalization of the one-dimensional Cantor set and two-dimensional Sierpinski carpet. It was first described by Karl Menger in 1926, in his studies of the concept of topological dimension. [4] [5] Construction [ edit ]