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Concept
Menger sponge
Summary
In mathematics, the Menger sponge (also known as the Menger cube, Menger universal curve, Sierpinski cube, or Sierpinski sponge) 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.
The construction of a Menger sponge can be described as follows:
Begin with a cube.
Divide every face of the cube into nine squares, like Rubik's Cube. This sub-divides the cube into 27 smaller cubes.
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).
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.
The th stage of the Menger sponge, , is made up of smaller cubes, each with a side length of (1/3)n. The total volume of is thus . The total surface area of is given by the expression . Therefore, the construction's volume approaches zero while its surface area increases without bound. Yet any chosen surface in the construction will be thoroughly punctured as the construction continues so that the limit is neither a solid nor a surface; it has a topological dimension of 1 and is accordingly identified as a curve.
Each face of the construction becomes a Sierpinski carpet, and the intersection of the sponge with any diagonal of the cube or any midline of the faces is a Cantor set. The cross-section of the sponge through its centroid and perpendicular to a space diagonal is a regular hexagon punctured with hexagrams arranged in six-fold symmetry. The number of these hexagrams, in descending size, is given by , with .
The sponge's Hausdorff dimension is log 20/log 3 ≅ 2.
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