In mathematics, the Menger sponge is a fractal curve. It is a three-dimensional generalization of the Cantor set and Sierpinski carpet. It was first described by Karl Menger in 1926, in his studies of the concept of topological dimension.
The Menger sponge simultaneously exhibits an infinite surface area and zero volume.
The construction of a Menger sponge can be described as follows:
- Begin with a cube (first image).
- Divide every face of the cube into 9 squares, like a Rubik's Cube. This 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 center of the larger cube, leaving 20 smaller cubes (second image). This is a level-1 Menger sponge (resembling a Void Cube).
- Repeat steps 2 and 3 for each of the remaining smaller cubes, and continue to iterate ad infinitum.
The second iteration will give you 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.
The number of cubes is 20n, with n being the number of iterations performed on the first cube.
The Lebesgue covering dimension of the Menger sponge is one, the same as any curve. Menger showed, in the 1926 construction, that the sponge is a universal curve, in that every curve is homeomorphic to a subset of the Menger sponge, where a curve means any compact metric space of Lebesgue covering dimension one; this includes trees and graphs with an arbitrary countable number of edges, vertices and closed loops, connected in arbitrary ways. In a similar way, the Sierpinski carpet is a universal curve for all curves that can be drawn on the two-dimensional plane. The Menger sponge constructed in three dimensions extends this idea to graphs that are not planar, and might be embedded in any number of dimensions.
The Menger sponge has infinite surface area but zero volume. In spite of this, there exists a homeomorphism of the cube having finite distortion that "squeezes the sponge" in the sense that the holes in the sponge go to a Cantor set of zero measure (Iwaniec & Martin 2001, §6.5.6).
The sponge has a Hausdorff dimension of (log 20) / (log 3) (approximately 2.726833).
Formally, a Menger sponge can be defined as follows:
where M0 is the unit cube and
In 2014, twenty level-three Menger sponges were constructed, which combined would form a distributed level-four Menger sponge.
- Apollonian gasket
- Cantor cube
- Koch snowflake
- List of fractals by Hausdorff dimension
- Sierpiński tetrahedron
- Sierpiński triangle
- Menger, Karl (1928), Dimensionstheorie, B.G Teubner Publishers
- Menger, Karl (1926), "Allgemeine Räume und Cartesische Räume. I.", Communications to the Amsterdam Academy of Sciences. English translation reprinted in Edgar, Gerald A., ed. (2004), Classics on fractals, Studies in Nonlinearity, Westview Press. Advanced Book Program, Boulder, CO, ISBN 978-0-8133-4153-8, MR 2049443
- "Menger sponge", Wolfram Alpha, retrieved 2013-12-12
- "MegaMenger". Retrieved 15 February 2015.
- Iwaniec, Tadeusz; Martin, Gaven (2001), Geometric function theory and non-linear analysis, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, ISBN 978-0-19-850929-5, MR 1859913.
- Zhou, Li (2007), "Problem 11208: Chromatic numbers of the Menger sponges", American Mathematical Monthly 114 (9): 842
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