In mathematics, the Menger sponge (also known as the Menger cube, Menger universal curve, Sierpiński cube, or Sierpiński sponge) is a fractal curve. It is a three-dimensional generalization of the two-dimensional Sierpinski carpet. It was first described by Karl Menger in 1926, in his studies of the concept of topological dimension. It has similar properties as the Cantor set and the Cantor dust, because the construction requires in both cases the removal of the inner third.
Construction
The construction of a Menger sponge can be described as follows:
Begin with a cube. Divide every face of the cube into nine squares in a similar manner to a 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 larger 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.
Properties
The n {\displaystyle n} th stage of the Menger sponge, M n {\displaystyle M_{n}} , is made up of 20 n {\displaystyle 20^{n}} smaller cubes, each with a side length of (1/3)n. The total volume of M n {\displaystyle M_{n}} is thus ( 20 27 ) n {\textstyle \left({\frac {20}{27}}\right)^{n}} . The total surface area of M n {\displaystyle M_{n}} is given by the expression 2 ( 20 / 9 ) n + 4 ( 8 / 9 ) n {\displaystyle 2(20/9)^{n}+4(8/9)^{n}} . 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 the following recurrence relation: a n = 9 a n − 1 − 12 a n − 2 {\displaystyle a_{n}=9a_{n-1}-12a_{n-2}} , with a 0 = 1 , a 1 = 6 {\displaystyle a_{0}=1,\ a_{1}=6} . The sponge's Hausdorff dimension is log 20/log 3 ≅ 2.727. 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. Similarly, 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. In 2024, Broden, Nazareth, and Voth proved that all knots can also be found within a Menger sponge. The Menger sponge is a closed set; since it is also bounded, the Heine–Borel theorem implies that it is compact. It has Lebesgue measure 0. Because it contains continuous paths, it is an uncountable set. Experiments also showed that cubes with a Menger sponge-like structure could dissipate shocks five times better for the same material than cubes without any pores.
Formal definition Formally, a Menger sponge can be defined as follows (using set intersection):
M := ⋂ n ∈ N M n {\displaystyle M:=\bigcap _{n\in \mathbb {N} }M_{n}}
where M 0 {\displaystyle M_{0}} is the unit cube and
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