Menger sponge

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A Sierpinski carpet is the set of all points $(x,y)$ such that $x$ or $y$ is inthe Cantor set. An equivalent and perhaps simpler definition is:

Let $S_0$ be a unit square. Let $S_{n+1}$ be $S_n$, with each square dividedinto ninths, by being divided into thirds horizontally and vertically, and the central resulting square removed, and the other resulting squares treated separately in further divisions. The limit as $n\to \mathrm{\infty }$ of $S_n$ is a Sierpinski carpet. An approximation is shown in figure 1.

Figure 1: An approximation of the Sierpinski carpet

The Menger sponge is a fractal embedded in 3-dimensional space. It can be seen as a 3-d generalization of the Sierpinski carpet, which is itself a 2-dimensional generalization of the Cantor set. The Menger sponge is almost always represented as being constructed from Cantor sets using the “middle third” rule.

The Menger sponge consists of all points $(x,y,z)$ such that $(x,y)$, $(y,z)$, and $(x,z)$ are all in Sierpinski carpets. Each “face” is a Sierpinski carpet.

Similarily to the Sierpinski carpet the Menger sponge can be constructed in the following way:

Start with a unit cube and split it into 27 smaller cubes of equal size. Remove the central cube and the ones joining a face with it. Then start over with the remaining smaller cubes.

Figure 2: An iteration of the Menger sponge – created in Blender 2.35.(The Blender filemenger.blend for this picture.)